An Introduction to the Mathematical Aspects of Quantum Mechanics:

Universidade Federal de Goiás Instituto de Física e Química

An Introduction to the Mathematical Aspects of Quantum Mechanics: Course Notes

Petrus Henrique Ribeiro dos Anjos Paulo Eduardo Gonçalves de Assis

Catalão,Go 2015

Contents 1 Quantum States and Observables 1.1

Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1

5 6

Uncertainty Principle . . . . . . . . . . . . . . . . . . . 11

1.2

Inner Product and Hilbert Spaces . . . . . . . . . . . . . . . . 12

1.3

Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4

Probability and Functions of Observables . . . . . . . . . . . . 18

1.5

Self-adjoint operators . . . . . . . . . . . . . . . . . . . . . . . 20

1.6

Riez Representation . . . . . . . . . . . . . . . . . . . . . . . . 22

1.7

Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.8

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 The Spectrum

29

2.1

Spectrum and Resolvent . . . . . . . . . . . . . . . . . . . . . 30

2.2

Finding the spectrum . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.1

2.3

The spectrum of the Hamiltonian . . . . . . . . . . . . 36

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Quantum Dynamics

43

3.1

Time evolution and Schrödinger Equation . . . . . . . . . . . 43

3.2

Applications to Two-Level Systems . . . . . . . . . . . . . . . 49

3.3

Schrödinger’s wave equation . . . . . . . . . . . . . . . . . . . 52

3.4

Time dependence of Expected Values . . . . . . . . . . . . . . 57 3.4.1

Newton’s 2nd Law and Quantum Mechanics . . . . . . 58

CONTENTS

4 3.5 Quantum Pictures

. . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4 Approximation Methods

63

4.1 Non-perturbative methods . . . . . . . . . . . . . . . . . . . . 63 4.1.1

Variational Methods . . . . . . . . . . . . . . . . . . . 63

4.1.2

Extension to Excited States . . . . . . . . . . . . . . . 66

4.1.3

A 2 × 2 example . . . . . . . . . . . . . . . . . . . . . 67

4.1.4 4.1.5

Method os Successive Powers . . . . . . . . . . . . . . 68 WKB - Semiclassical approximation . . . . . . . . . . . 68

4.2 Time-independent Perturbation Theory . . . . . . . . . . . . . 71 4.2.1

Time-independent perturbation: Non-degenerate . . . . 72

4.2.2

Time-independent perturbation: Degenerate . . . . . . 75

4.3 The Anharmonic Oscillator

. . . . . . . . . . . . . . . . . . . 77

4.4 Time-dependent Perturbation . . . . . . . . . . . . . . . . . . 79 4.4.1

Time-dependent Perturbation Theory . . . . . . . . . . 80

4.5 Furhter Applications and Fermi’s Golden Rule . . . . . . . . . 83 4.6 Dirac’s interaction picture . . . . . . . . . . . . . . . . . . . . 84 4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Referências

87

Chapter 1 Quantum States and Observables In classical physics the mathematical description of a phenomenon is somewhat clear. From the early days of modern science, the movement of a macroscopic body could be completely characterized by the specification of its position at a given instant of time. This process was easily achieved with the use of simple tools such as rulers and clocks. Predictions could be computed based on formulations relying on simple mathematical objects called functions. The development of differential and integral calculus, in fact, owes a lot to the works of scientists like Isaac Newton. With the advance of our understanding of the microscopic world, physicist were forced to abandon the classical formalism to match experiments. Intead, a new theory came to light and both the basic objects to describe it and the mathematical tools to formulate it had to be revised. Part of this mathematical toolbox was already well stablished at the beginning of the last century but some of it had to be developed alongside the results of experiments. Here we present an introduction to the mathematical aspects of quantum mechanics. 5

CHAPTER 1. QUANTUM STATES AND OBSERVABLES

6

1.1

Quantum States

Definition 1.A. The state of a single particle 1-D quantum system is a complex valued continuous function ψ(x, t) such that i. The probability Pψ (x ∈ I) of finding that the position of the particle belong to the interval I ⊂ R at time t is given by Z Pψ (x ∈ I) = |ψ(x, t)|2 dx I

ii. The probability Pψ (p ∈ I) of finding that the momentum of the particle belong to te interval I ⊂ R at time t is given by Z 1 ˜ p , t)|2 dp, |ψ( Pψ (p ∈ I) = ~ I ~

where f˜ is the Fourier transform of ψ, given by Z 1 ˜ f (k) = √ f (x)eikx dx 2π R A general property of the Fourier transform called Parceval’s identity [1], show us that Note that Z

R

2

|ψ(x, t)| dx =

Z

˜ p , t)|2 dp . ψ( ~ ~ R

And to our probability interpretation hold, we that Z |ψ(x, t)|2 dx = 1,

(1.1.1)

(1.1.2)

R

which means that the particle lies somewhere in the real line. As an exercise, we suggest that you show that the Pψ defined on 1.A

actually obeys other probabilities rules. More precisely, you should show that with our definition, for any countable sequence of disjoint intervals I1 , I2 , ..., we have that Pψ (x ∈

[ n

In ) =

X n

Pψ (x ∈ ∪In ).

(1.1.3)

1.1. QUANTUM STATES

7

So the position x (or the momentum p) of the particle can assume any value in the real line, and to each interval we assign a probability Pψ (x ∈ I) that the value of x lies in I. We can show how to compute the mathematical

expectation of x. As a warm up, assume that x is restricted to a bounded interval [a, b]. We can divide [a, b] into smaller subintervals Ik , and consider the following object:

X

xk Ik ,

k

where xk is an arbitrary point of Ik . We desire that this sum converge to a limit as the maximum length goes to zero, and furthermore the convergence is independent of our choices of intervals Ik and point xk . If all this holds, we call the limit x¯ the mathematical expectation of x. If x is not restricted to a bounded interval, we can fix an arbitrary bounded interval [a, b], calculate (as above) a limit for this bounded interval, and finally take the limit where [a, b] goes to the real line. Now, in the latter case, if this limit (a) exist, (b) is independent of our choice of interval [a, b] and (c) is independent of how we grow the interval; we call this limit x¯ the mathematical expectation of x. There are a lot of “if” ’s here, and in order to our physical theory works we need to start to prove some things. Fortunately the following holds: Lemma 1.1.1. Let the system be in a state ψ and f (x) be a continuous function such that

Z

R

|f (x)||ψ(x, t)|2 dx < +∞,

then f¯ the mathematical expectation of f (x) is given by f¯ =

Z

f (x)|ψ(x, t)|2 dx

R

Proof. Let I ⊂ R be a bounded interval and I1 , ...IN be a partition of I into

smaller non-overlaping intervals with maximum length δ. For let for each k pick an arbitrary xk ∈ Ik , then

8


Z XZ X 2 f (xk )Pψ (x ∈ Ik ) 6 |f (x)−f (xk )||ψ(x, t)|2 dx. f (x)|ψ(x, t)| dx − I Ik k

k

The continuity of f (x) implies that we can make |f (x) − f (xk )| 6 ǫ R by taking δ sufficiently small. Therefore Ik |f (x) − f (xk )||ψ(x, t)|2 dx 6 R ǫ Ik |ψ(x, t)|2 dx, then Z Z X 2 f (xk )Pψ (x ∈ Ik ) 6 ǫ |ψ(x, t)|2 dx. f (x)|ψ(x, t)| dx − I I k

To conclude the demonstration we note that Z +l Z 2 f (x)|ψ(x)|2 dx f (x)|ψ(x)| dx = lim l→+∞

R

−l

A direct consequence of the lemma above is the Corolary 1.1.1. Let the system be in a state ψ, the following holds Z x¯ = x|ψ(x, t)|2 dx R

Analogue statements holds concerning the momentum, as follows. Their proof are left as exercises. Lemma 1.1.2. Let the system be in a state ψ and g(p) denote a continuous function such that

Z

˜ p , t)|2 dp < +∞, |g(p)||ψ( ~ ~ R

the following holds

Z

˜ p , t)|2 dp = g¯ g(p)ψ( ~ ~ R

Corolary 1.1.2. Let the system be in a state ψ, the following holds Z ˜ p , t)|2 dp p¯ = pψ( ~ ~ R

1.1. QUANTUM STATES

9

Since we use Fourier transform to describe the probabilities concerning the momentum, it will be helpful to recall some properties of this integral transformation. We use Fourier transform here without too much concern on the conditions that f (x) must satisfy in order that the transforated function exists, for a complete discussion of Fourier transform see Ref. [1]. First we recall an useful property of Fourier transform: g ∂ f (k) = −ik f˜(k) ∂x

(1.1.4)

Z

(1.1.5)

This property follows directly form integration by parts and the observation that if f˜ exists then |f (x)| → 0 as |x| → 0. We also need an inverse Fourier

transform, and it is given by

1 f (x) = √ 2π

f˜(k)e−ikx dk,

R

. To have some insight into equation 1.1.5, it worth consider a very helpful identity:

1 δ(x − x ) = 2π ′

Z

′

e−ik(x−x ) dk,

(1.1.6)

R

where δ(x) denotes the quite ubiquitous entity that physicist call “Dirac Delta Function. Formally, this object can be defined as a “function” such that for any continuous function f and ǫ > 0 Z +ǫ f (x)δ(x)dx = f (0).

(1.1.7)

−ǫ

When dealing with “Dirac Delta Function”, we need to keep in mind that: 1. It is not a function: Even an initial analysis show that δ(x) is not a function, it is actually an entity that the mathematicians call a distribution, and we suggest to see ref. [2] for a rigorous treatment. 2. Dirac is not the first to use it: For exemple, an infinitesimal formula for an infinitely tall, unit impulse delta function explicitly appears in a work of Cauchy in 1827 (see ref [3]). And indeed, several other


10

authors have dealt with objects with similar characteristics (among them Poisson , Kirchoff , Lord Kelvin , again see ref. [3]). And in particular, in the late nineteenth century Heaviside had derived the main properties of δ(x). Despite these considerations , it should be noted that the notation introduced by Dirac in his influential book The Principles of Quantum Mechanics ( see Ref [4]), is not only convenient but insightful and intuitive, what made it readily available to a much larger community. Putting these ideas into practice , we can use the Fourier transform to to connect x and p representations of the quantum state. This is done in the Proposition 1.1. Let the system be in a state ψ, the following holds Z ∂ p = (−i~) ψ(x, t) ψ † (x, t)dx ∂x R Proof. It is a straightforward application of eqs. 1.1.5, 1.1.4 and 1.1.6. In fact, we find that

−i~

R

R

∂ ψ(x, t) ∂x

R ~ ˜ ˜† ′ )e−i(k−k′ )x dkdk ′ dx ψ † (x, t)dx = 2π R Rk ψ(k)ψ (k ˜ ψ˜† (k ′ )δ(k − k ′ )dkdk ′ = ~ k ψ(k) 2 R ˜ = ~ k ψ(k) dk R p 2 ˜ ) dp = p¯ = p ψ( ~

Note that the Proposition 1.1 gives us the expected value of the momentum, free from Fourier transform under the price of the introduction of partial differentials. This gives us an important hint about the nature of quantum mechanical systems: We can define a Momentum Operator Pˆ ~ ∂ ψ Pˆ ψ = i ∂x

(1.1.8)

1.1. QUANTUM STATES

11

For instance, this became clear by a repeated application of the ideias leading to Proposition 1.1, gives us pn

n ˆ = P ψ, ψ ,

where we use the notation (f, g) =

Z

f (x)g † (x)dx.

(1.1.9)

R

Note 1.a. Despite being a well-known fact , it should worth remember , at least to our most inexperienced readers, that usually pn 6= p¯n . For exemple, eq. 1.1.2 can be written as (ψ, ψ) = 1. We will further discuss this notation and the role of operators like Pˆ in quantum mechanics in section . But before that, we want to show an intriguing property of quantum systems: The famous Uncertainty Principle.

1.1.1

Uncertainty Principle

The famous Heisenberg Uncertainty Principle is a theorem about Fourier transforms, once we grant a certain model of quantum mechanics. That is, there is an unavoidable mathematical mechanism that yields an inequality which has an paradigm shit interpretation in physics. This First note that integrating by parts Z Z ∂ ∂ † (ψ, ψ) = − x (ψψ )dx = −2Re xψ(x) ψ † (x)dx. ∂x R R ∂x That is

Z ∂ † (ψ, ψ) 6 2 xψ(x) ψ (x) dx. ∂x R

Furthermore, Cauchy-Schwartz identity applies here implying that Z ∂ ∂ † (ψ, ψ) 6 2 xψ(x) ψ (x) 6 2kxψkk ψk, ∂x ∂x R


12

where kf k2 = (f, f ). Now we use eq. 1.1.4 and Parceval’s identity (eq. 1.1.1) to converts derivatives (in the position) to multiplication (by the momentum p ): ~

g ∂ ∂ ˜ ψk = k ψk = kk ψk. ∂x ∂x Since for any quantum state kψk = (ψ, ψ) = 1, so we obtain the Heisenberg k

inequality (using p = ~k)

˜ > kxψkkpψk

~2 . 4

A similar argument gives, for any x0 , p0 ∈ R 2 ˜ >~ . k(x − x0 )ψkk(p − p0 )ψk 4

(1.1.10)

Put x0 = x¯ and p0 = p¯, then k(x − x0 )ψk = x2 − x¯2 = ∆x ˜ = p2 − p¯2 = ∆p . k(p − p0 )ψk Applying this in Heisenberg’s general inequality (i.e. eq. (1.1.10)) we obtain the famous Heisenberg Principle: ∆x∆p >

~ . 2

Roughly speaking the quantities ∆x and ∆p can interpreted as the error obtained in measurements of position and momentum respectively. That is Heisenberg’s uncertainty principle, is a fundamental limit to the precision with which the pair (x, p) can be known simultaneously, or in another words it gives a lower bound on how spread out the probability distributions of x and p must be. Note that the only relevant Physical assumptions are in Definition 1.A, namely: the probabilistic interpretation of the quantum state and that Fourier transform relates x and p.

1.2

Inner Product and Hilbert Spaces

In this section, we develop the notation introduced in eq. (1.1.9)

1.2. INNER PRODUCT AND HILBERT SPACES

13

Definition 1.B. For every states ψ, φ we define the scalar product on L2 Z (φ, ψ) = φ∗ (x, t)ψ(x, t)dx R

Lemma 1.2.1. Let a, b ∈ C and φ, ψ, ν ∈ L2 . The scalar product on L2 has

the following properties

1. (φ, aψ + bν) = a(φ, ψ) + b(φ, ν); 2. (ψ, φ) = (ψ, φ)∗ , where the asterisk denotes complex conjugation. 3. kψk2 = (ψ, ψ) > 0 unless ψ = 0 Note 1.b. Note that statement c of Lemma 1.2.1 is not quite correct. To be precise, we should say that if kψk2 = 0 then ψ(x) = 0 almost everywhere, i.e.

(in a technical sense) the set for which the property holds takes up nearly all

possibilities. However, to simplify our discussion, we identify two functions that agrees almost everywhere. A scalar product obeying Lemma 1.2.1 is called an inner product, and a vetor space provided with an inner product is called an inner product space. The Lemma 1.2.1 has the following important consequences: Corolary 1.2.1. Let ψ, φ belong to an inner product space, then the following holds i. |(ψ, φ)| 6 kψkkφk (Schwarz inequality). ii. kψ + φk 6 kψk + kφk (Triangle inequality). Proof. It is clear that [i.] holds, when ψ = 0, so assume that both φ and ψ ane non-zero. Also, we assume that (ψ, ϕ) 6= 0 since otherwise the inequality is obviously true. Let σ = ψ −

that

(ψ,ϕ) ϕ, kϕk2

therefore the linearity of the inner product implies

(σ, ϕ) = (ψ, ϕ) −

(ψ, ϕ) (ϕ, ϕ) = 0. kϕk2


14 Now,

kψk2 = kσ + (ψ,ϕ) (ψ,ϕ) 2 = kσk + kϕk2 > kϕk2 .

(ψ,ϕ) ϕk2 kϕk2

To show [ii.] , just expand kψ + ϕk2 = (ψ + ϕ, ψ + ϕ), and apply Schwarz

inequality. We leave the details as an exercise.

We are now capable of addressing an important question, that we are postponing, that is what functions are acceptable as quantum states? Our Remark raise some of the issues. But since we need the Definition 1.A, we must insist that these functions are square integrable on the real line. So, they must belong to the well know space L2 (R). We also desire that these functions satisfies some features: Suppose there is a sequence of functions ψk ∈ L2 such that

kψk − ψj k → 0 as j, k → +∞.

(1.2.11)

A sequence satisfying eq.1.2.11 is called a Cauchy sequence. To have things working properly, we want that all Cauchy sequences converge to some element of L2 . We call this property completeness. More precisely: Given a sequence satisfying eq. 1.2.11 then there is ψ ∈ L2 such that kψ − ψk k → 0 as k → +∞.

(1.2.12)

We say that a sequence satisfying equation 1.2.12 converges strongly (or in the norm) to ψ and write ψk → ψ. We say that a sequence of ψk ’s converges weakly to ψ when

(ψk , ϕ) → (ψ, ϕ) for all ϕ ∈ L2 .

(1.2.13)

We point out that, as a consequence of Schwarz inequality, every strong convergent sequence is also weak convergent, since |(ψk − ψ, ϕ)| 6 kψk − ψkkϕk.

1.3. OBSERVABLES

15

Furthermore, we say that a subset S of L2 is closed if every convergent sequence of ψk ∈ S converge to some ψ ∈ S, (if S is not closed then there

are sequences of elements of S that converge to elements of L2 that are not in S). A subset S of L2 is called dense if for every ψ ∈ L2 and ǫ > 0, there

is an φ ∈ S such that kψ − φk < ǫ. That means that any function in L2 can

be approximated by functions in S. The following property is very useful Lemma 1.2.2. If S is a dense subset of L2 and (ψ, ϕ) = 0 ∀ ϕ ∈ S ⇒ ψ = 0.

(1.2.14)

Proof. Note that (ψ, ϕ) = 0 imply that kψ − ϕk2 = kψk2 + kϕk2 > kψk2 .

Since ϕ ∈ S then it can be taken arbitrarily close to ψ so kψk2 6 0 therefore

ψ = 0.

A vector space with an inner product, that is complete with respect to the norm induced by the inner product is called a Hilbert Space (that is a inner product vector space that satisfies the completeness property). The space L2 is a very important exemple of a Hilbert Space, and many of the statements we made here actually holds for general Hilbert Spaces. However, if you are not familiar with Hilbert Spaces do not despair, you still follow our notes without difficult considering everything in the L2 context.

1.3

Observables

Definition 1.C. A physical quantity that can be measured is called an Observable. Postulate 1. To every observable a there is an corresponding linear operator A with dense domain D(A) ⊂ L2 such that a ¯ the expected value of a for the

system at the state ψ ∈ D(A) is given by

a ¯ = (ψ, Aψ) .


16

Not all operators with the property described in 1 correspond to an observable. First, we need that the observable can only assume real values. Definition 1.D. A linear operator A is called Hermitian if (ψ, Aφ) = (Aψ, φ) for every ψ, φ ∈ D(A). Lemma 1.3.1. A is a Hermitian Operator ⇔ (ψ, Aψ) ∈ R for every ψ ∈

D(A).

Proof. For ψ ∈ D(A), and since A is Hermitian we note that (Aψ, ψ) =

(ψ, Aψ) = (Aψ, ψ)∗ . Conversely, assume that Im(Aψ, ψ) = 0, we have that (A[iϕ + ψ], iϕ + ψ) = (Aφ, φ) + (Aψ, ψ) + i(Aϕ, ψ) − i(Aψ, ϕ), taking imaginary parts we have that Re(Aψ, ϕ) = Re(ψ, Aφ). Finally,

Im(Aψ, ϕ) = Im [−i(Aiψ, ϕ)] = −Re(Aiψ, ϕ) = −Rei(ψ, Aϕ) = Im(ψ, Aϕ). Then (Aψ, ϕ) = (ψ, Aϕ). Now consider, for example, the momentum operator defined on eq.1.1.8. The expression Z

∂ψ (x)ψ † dx ∂x R makes sense only if ψ is differentiable with respect to x and if the integral exists. That means D(Pˆ ), the domain of the momentum operator, can not p¯ = (Pˆ ψ, ψ) = −i~

be the whole L2 . In fact, there is many examples of elements of L2 functions that are not differentiable everywhere. We need ways to extend the domain of an operator to deal with these cases. To do this is enough to require that the domain of the operator to be dense in L2 . Furthermore, we need that the domain of the operator is the largest possible. This motivates the Postulate 2. If A is an operator corresponding to an observable a and B is a Hermitian operator such that D(A) ⊂ D(B), and a ¯ = (ψ, Bψ),

1.3. OBSERVABLES

17

∀ψ ∈ D(A) then B = A (i.e. D(B) = D(A) and Aψ = Bψ). The ideia of this postulate is to give us a way to extend the domain of an operator. For instance, suppose that ψ 6∈ D(A), but there is a sequence

{ψn } ⊂ D(A) such that ψn → ψ and Aψn → f , then we define a Aψ = f .

This only makes sense if f does not depend on the sequence {ψn }. Now,

suppose we have another sequence φn ∈ D(A) → ψ such that Aφn → f .

Therefore, µn = ψn − φn → 0, and we need that Aµn → 0 for that method

works. An operator with this property is called closable . When it works this ¯ the closure of A. allow us to construct an extended operator denoted by A, Note 1.c. It is also, worth to note that A¯ can not be extended any further using the method outlined above. In fact, suppose that there is a sequence ¯ such that ψk → ψ, Aψ ¯ k → f . So there are sequences ψjk ∈ D(A) ψk ∈ D(A) ¯ k . So we can take one element of each of such that ψjk → ψk and Aψjk → Aψ

this sequences, and construct a new sequence ψkk where ψkk → ψ, Aψkk → f . ¯ and Aψ ¯ = f. Therefore ψ ∈ D(A)

As our previous discussion suggests, not all operators are closeable. Fortunately, for the quantum mechanical theory, we we are dealing with Hermitian operators with dense domain, and we have the Lemma 1.3.2. A Hermitian operator with dense domain is closable. Proof. Let D(A) ∋ ψn → 0 and Aψn → f . For any ϕ ∈ D(A) we have that (Aψn , ϕ) = (ψn , Aϕ). Taking the limits, we have that (f, ϕ) = (0, Aϕ) = 0. Since D(A) is dense Lemma 1.2.2 implies that f = 0. Finally, we can put these results together. This lead us to the


18

Proposition 1.2. If the observable a is real valued then the corresponding operator is a closed Hermitian Operator. Proof. We already prove that A must be Hermitian. Since, we require that its domain is dense, then it is closable. We claim that the closure A¯ is also ¯ there are sequences D(A) ∋ ψn → ψ Hermitian. In fact, for ψ, ϕ ∈ D(A), ¯ and Aϕn → Aϕ, ¯ and therefore and D(A) ∋ ϕn → ϕ, such that Aψn → Aψ ¯ ϕ) = lim(Aψn , ϕk ) = lim(ψn , Aϕk ) = (ψ, Aϕ). ¯ (Aψ, n,k

n,k

A¯ can not be further extended so the proposition follows from Postulate 2.

1.4

Probability and Functions of Observables

It is fundamental in any scientific theory that one can make some sort of prediction of the outcome of an experiment. If the theory is not able to make predictions, then it is not of any use to science (and most important not scientific at all). In particular, in quantum mechanics one need to determine the probability Pψ (a ∈ I) that the real observable a be in the interval

I. Clearly this quantity must depend on the quantum state ψ and the corresponding operator A. We now show how one can obtain this information. Our approach is based on the Note 1.d. A real function of an observable is an observable. The reason we emphasize this point is that usually physical quantities are not measured directly , but calculated from other quantities. So this is a fundamental requirement of the theory. It is also highly likely that a more careful author would prove this statment or include it as an postulate. That means for any observable a and f : R → R, there is an observable f (a). Corre-

sponding to the observable f (a) there is a corresponding Hermitian operator that (despite the abuse of the notation) we denote f (A) (postulate 1).

1.4. PROBABILITY AND FUNCTIONS OF OBSERVABLES

19

Now let χI (λ) be the characteristic function of the interval I, i.e. 1 λ∈I (1.4.15) χI (λ) = 0 λ 6∈ I So let us consider the operator χI (A), where A is the operator corresponding to the observable a. Since that for any state ψ, a real valued observable a can only be or not be in I, we find that χI (a) = Pψ (a ∈ I) × 1 + Pψ (a 6∈ I) × 0 ⇒ (ψ, χI (A)ψ) = Pψ (a ∈ I) Unfortunately, this hardly answers our question. since we still need to understand what is the operator χI (A) and given A how to construct it. We now will try to deal with these problems proving some properties of χI (A). Lemma 1.4.1. The following holds i. kχI (A)ψk 6 kψk, for all ψ ∈ D [χI (A)]2 .

ii. D(χI (A)) = L2

Proof. Since χI (A) is Hermitian, we have that χ2I (a) = (χ2I (A)ψ, ψ) = (χI (A)ψ, χI (A)ψ) = kχI (A)ψk2 , for all normalized ψ in the domain of χ2I (a). Also χ2I (a) = χI (a) = Pψ (a ∈

I) < 1, therefore kχI (A)ψk 6 1. Now, for ϕ 6= 0, write ψ = leads to statement [i.].

ϕ kϕk

and this

Lemma 1.4.2. If A is a Hermitian Operator with dense domain such that (Aψ, ψ) = 0 then A = 0.

∀ψ ∈ D(A)


20

Lemma 1.4.3. For each I, J ⊂ R, the following holds i. χ2I (A) = χ2I (A) ii. χI∪J (A) = χI (A) + χJ (A) − χI∩J (A) iii. χI∩J (A) = χI (A)χJ (A) Proof. Note that χ2I (a) = χI (a) implies that (χ2I (A)ψ, ψ) = (χI (A)ψ, ψ), then ([χ2I (A) − χI (A)]ψ, ψ) = 0, for all ψ ∈ L2 . So statement [i.] follows from lemma 1.4.2.

We leave the demonstration of the remaining statements as an exercise (Hint: Note that χI∪J = χI + χJ − χI∩J and that [χI + χJ ]2 = χI + χJ + 2χI∩J )

We advanced quite a bit, but still we are not able to construct the operator χI (A) given A and not even know if it is defined for every densely defined closed Hermitian operator. To address these issues, in the next section we shall derive more consequences of our postulates.

1.5

Self-adjoint operators

Our main aim in this section is to prove and discuss the Proposition 1.3. If A is the operator corresponding to a real observable, then the operator (1 + A2 ) is onto. The proposition 1.3 is instrumental. It tells us that when dealing with quantum mechanical operator, for each f ∈ L2 the equation (ψ + A2 ψ = f

has a solution ψ ∈ D(A2 ) ⊂ D(A). We should note that given to operators

A and B, D(A + B) = D(A) ∩ D(B), then D(1 + A2) = D(A2 ). Furthermore,

note that (1 + A2) = (A + i)(A − i) = (A − i)(A + i), therefore the proposition 1.3 tells us that for every f ∈ L2 there are ψ, ϕ ∈ D(A) such that (A − i)ψ = (A + i)ϕ = f.

(1.5.16)

1.5. SELF-ADJOINT OPERATORS

21

Once noticed this fact, the proposition 1.3 also suggests that not all densely defined closed Hermitian operator can correspond to an observable. This lead us to the Proposition 1.4. let A be an operator corresponding to a real observable and let ψ, f ∈ D(A) such that for all ϕ ∈ D(A) we have (ψ, Aϕ) = (f, ϕ). Then ψ ∈ D(A) and Aψ = f . Proof. What need to be proved is that ψ ∈ D(A) (if ψ ∈ D(A), we can use

that A is Hermitian then (Aψ − f, ϕ) = 0, and by the lemma 1.2.2 Aψ = f ). Now note that, for every ϕ ∈ D(A), we have

(ψ, [A + i]ϕ) = (ψ, Aϕ) − i(ψ, ϕ) = (f − iψ, ϕ)

(1.5.17)

From eq. 1.5.16, there a w ∈ D(A) such that (A − i)w = f − iψ, which leads to

(ψ, [A + i]ϕ) = (f − iψ, ϕ) = ([A − i]w, ϕ) = (w, [A + i]ϕ). Again by eq. 1.5.16, there is a ϕ ∈ D(A) such that [A + i]ϕ = ψ − w.

Therefore

0 = (ψ − w, [A + i]ϕ) = (ψ − w, ψ − w) = kψ − wk2, which show us that ψ = w ∈ D(A) The property described proposition 1.4 is instrumental to build functions of observables, however (un)fortunately this property is not shared by all densely defined closed Hermitian operators. This is one of the crucial points of this chapter, then take a deep breath and allow yourself a moment of reflection on this issue. For any densely defined operator A with domain D(A) ⊂ L2 we can define A† , the adjoint of A,by A† ψ = f in the proposition


22

1.4, this makes perfectly sense because D(A) is dense. Otherwise there can be more then one function that satisfies proposition 1.4. That means (ψ, Aϕ) = (A† ψ, ϕ).

(1.5.18)

The linearity of the inner product implies that A† is also a linear operator. When A is Hermitian , we can be tempted to swap the position of A in the inner product and say that A† = A, however this is only true if ψ ∈ D(A),

and the proposition 1.4 holds for ψ ∈ L2 . Then it is clear that D(A) ⊂ D(A† ), and restricted to D(A), we have A† = A. For an operator that satisfies the

proposition 1.4, we say that the operator is self-adjoint, i.e. D(A† ) = D(A) and A† = A. Putting all this ideas together we just prove the Proposition 1.5. If a is a real observable then the corresponding observable is self-adjoint.

1.6

Riez Representation

When we are dealing with Hilbert Spaces, Adjoints and more important physicists it is important to have some rigorous mathematical results to guide us. Usually, when dealing with quantum mechanics, physicists use a very particular “language” that they call “Dirac Notation” or “Bra-Ket Notation” to denote abstract vectors and linear functionals (i.e linear operator from the Hilbert space to the scalars), leaving inner products to a some sort of secondary role. The name “Bra-Ket” is so because the inner product of two quantum states is denoted by hψ|ϕi, where hψ|, called a “Bra”, is a linear functional and |ϕi is a vector in the Hilbert space.

Note 1.e. It is curious that (again) Dirac is not the first one to introduce this notation, the notation has its roots on Grassmann’s algebra calculation nearly 80 years before (see [5]). But, again Dirac has all the merit into disseminating the notation that nowadays is widespread in quantum mechanics:

1.6. RIEZ REPRESENTATION

23

almost every phenomenon that is explained using quantum mechanics (including a large portion of what is called modern physics) is usually explained with the help of bra-ket notation . Much beyond of all these bras, part of the (sex)appeal of “Dirac Notation” is the abstract representation-independence it encodes, together with its versatility in producing a specific representation (e.g. x, or p) without much ado, or excessive reliance on the nature of the linear spaces involved. So once again the notation disseminated by Dirac is not only convenient but insightful and intuitive, what made it readily available to a much larger community. The important guide line that we want to discuss is the (Hilbert Space) Riez Representation lemma. This theorem establishes an important connection between a Hilbert space and its linear functionals, that is a connection between “Bra´s” and “ket’s”. Here we will only describe and discuss the theorem, but we will not prove it. For a demonstration, we refer to [6]. First note, that for each ϕ in the Hilbert Space H, we can define a linear functional

Fϕ using the inner product by making, for all ψ ∈ H, Fϕ ψ = (ϕ, ψ).

(1.6.19)

A Bra-Ket enthusiast should say that to every Ket (i.e. an element of the Hilbert space) correspond a unique Bra (i.e. a linear functional). The Riez Representation lemma deals with the converse of this statement (i.e. it answers the question if for every Bra there is a unique Ket). We say that a linear functional F is bounded if there is a C > 0 such that for all ψ ∈ H |Lψ| < Ckψk.

(1.6.20)

The smallest constant C for which eq. 1.6.20 holds is called the norm of F and we denote it by kF k. Note that not every linear functional is bounded, a straightforward exemple in quantum mechanics is the following Fx ψ = ψ(x),


24

for all ψ ∈ L2 , i.e. Fx is the linear functional that associates each state to its value at a given point x.

The Riez Representation lemma states that every bounded linear functional (a “bounded Bra”) correspond to a vector in the Hilbert Space. More precisely, Theorem. [ Riez Representation Lemma] Let F be a bounded linear functional defined everywhere on a Hilbert space H. Then there is ϕ ∈ H such that kϕk = kF k and, for all ψ ∈ H, F ψ = (ϕ, ψ). In other words, Riez Representation lemma say that a bounded linear functional on a Hilbert space is just a inner product.

1.7

Spectral Theorem

At this point, after some simple physical-mathematical requirements, we understand that the operator A corresponding to real observable a is selfadjoint. These requirements are actually enough to caracterize the operators in quantum mechanics, that means if A is self-adjoint we can construct the operators χI (A) that allow us to calculate the probabilities that the observable lies in the interval I. This is due to the Spectral theorem. The Spectral theorem (for self-adjoint operators) basically sates that any self-adjoint operator is unitarily equivalent to a multiplication operator. This result is far from trivial and here we will only state the theorem, explain it and not try to give a proof. For a demonstration we refer to Ref [6]. An orthogonal projection O on a closed subspace S of H is a linear op-

erator such that for all ψ ∈ S, P ψ = ψ and for all ψ⊥ ∈ S ⊥ , P ψ = 0. Any

arbitrary ϕ ∈ H can be written as a unique decomposition ϕ = ψ + ψ⊥ , so that linearity extend O to hole space (and Oϕ = ψ). Clearly, O is idempotent

1.7. SPECTRAL THEOREM

25

(i.e. O 2 = O), since for any ϕ = ψ + ψ⊥ , we have O 2 ϕ = O(Oϕ) = Oψ = ψ. Given any two vector ϕi = ψi + ψi⊥ (i = 1,2), we note that (Oϕ1, ϕ2 ) = (ψ1 , ψ2⊥ ) + (ψ1 , ψ2 ) = (ψ1 , ψ2 ) = (ψ1 , Oϕ2) + (ψ1⊥ , Oϕ2) = (ϕ1 , Oϕ2), which show that O is Hermitian. Moreover, kOϕk = kψk 6 kϕk, then O is bounded. We note that orthogonal projections satisfy many of the properties of the operators χI of section 1.4. For example, suppose that S and S ′ are closed subspaces of H. We want to consider O the orthogonal projection on

C = S ∩ S ′ . Let OS and OS ′ be the orthogonal projections on S and S ′

respectively. Clearly, if ψ ∈ C, then Oψ = OS ψ = OS ′ ψ = ψ. Also C ⊥ is a

subset of both S ⊥ and S ′⊥ , then Oψ⊥ = OS ψ⊥ = OS ′ ψ⊥ = 0. This leads to

the conclusion that O = OS OS ′ , and we suggest the reader to compare this with the lemma 1.4.3. For each self-adjoint operator, the Spectral theorem garantes the existence of a family of orthogonal projection,that provides a canonical decomposition, called the spectral decomposition, of the underlying Hilbert space on which the operator acts. More precisely, Theorem. [Spectral Theorem] Let A be a self-adjoint operator on H.

There is a family of ortogonal projection EA (λ) depending on a real parameter

λ, called a spectral family such that

i. λ1 < λ2 ⇒ EA (λ1 )EA (λ2 ) = EA (λ1 ) ii. For ǫ > 0, EA (λ + ǫ) → EA (λ) as ǫ → 0. iii. For ψ ∈ H

EA (λ)ψ → 0 as λ → −∞ EA (λ)ψ → ψ as λ → +∞

26


iv. ψ ∈ D(A) ⇔

R

R

λ2 dkEA (λ)ψk < +∞

v. For ψ ∈ D(A) and φ ∈ H, (φ, Aψ) =

Z

R

λd (φ, EA (λ)ψ)

vi. If f is a complex valued function, the operator f (A) is given by Z f (λ)dEA (λ), f (A) = R

defined D(f (A)) consisting of all ψ ∈ H such that Z |f (λ)|2 dkEA (λ)ψk < +∞ R

Now back to business (i.e. prove some results). If a is a real valued observable, with corresponding operator A then from the spectral theorem the mathematical expectation a¯ of the observable a for a system at state ψ ∈ D(A), is given by

a ¯=

Z

R

λd(ψ, EA(λ)ψ).

Furthermore, the probability Pψ (a ∈ I) that the observable a lies in the interval I for a system at state ψ ∈ D(A) Z Pψ (a ∈ I) = d(ψ, EA (λ)ψ). I

As we point out, this theorem gives us a prescription to construct the operator χI (A). In fact, using [vi.], we find that Z χI (A) = dEA (λ). I

Note 1.f. For I = (−∞, λ], we have EA (λ) = χI (A). For a concrete example, consider the position of the particle. It is clearly a real valued observable, and therefore there is a corresponding self-adjoint ˆ given by operator X, ˆ [Xψ](x) = xψ(x)

(1.7.21)

1.7. SPECTRAL THEOREM

27

It is worth to remember that an operator is not defined unless its domain is specified. Here, it is of double importance since we the operators of interest arise form real observables, and therefore are self-adjoint. Even a superficial analysis should convince us that a slightest change in the domain can destroy ˆ is self-adjoint. the self-adjointness. So we need to specify a domain where X ˆ we can choose in order to X ˆ be The simplest (an largest) domain D(X), self-adjoint is is the set of those ψ ∈ L2 such that xψ ∈ L2 . In fact , it is ˆ is Hermitian. Or more directly, take an f ∈ L2, we see easy to see that X ˆ and (X ˆ ± i)ψ± = f , which show that (X ˆ ± i) is onto, that ψ± = f ∈ D(X) x±i

ˆ is densely defined, since for any so the proposition 1.4 holds. Moreover, X ˆ But f ∈ L2 we can take ψǫ = ǫx2f+1 ∈ D(X). |ψǫ − f k 6 ǫx2

|f | , +1

ǫx2

ˆ is self-adjoint. which implies that ψǫ → 0 as ǫ → 0. Therefore X

Now, the Spectral theorem states that there is a family of orthogonal

projections E(x0 ) such that ˆ= X

Z

x0 dE(x0 ). R

Our last remark, show us that the orthogonal projections E(x0 ), are actually given by

[E(x0 )](x) = H(x − x0 )ψ(x), where H(x) is the step function (H(x) = 0 for x > 0 and H(x) = 1 for x 6 0). We close this chapter stating a last postulate, necessary to the logical self-consistency of our physical theory. The postulate, usually referred “the wave function colapse” is the following: Postulate 3. If the measurement of the observable a of a system in a state ψ gives the result λ immediately after the measurement the new state of the


28 system became

φ=

EA (λ)ψ kEA (λ)ψk

This is to say that immediately after a measurement of the observable a, we know for sure the result if we perform the same measurement again. That means, that if we find that a ∈ I in our first measurement, the second one (performed immediately after) will also gives that a ∈ I. So the system can

not remain in the same state ψ, otherwise we can only predict the probability of a ∈ I in the second measurement. It is the essence of measurement in

quantum mechanics and connects the wave function with classical observables like position and momentum. The colapse is indeed one of the two process

by which a quantum system evolve in time, the other one is given by the Schrödinger equation that we will discuss in chapter 3. For a more detailed discussion of the “wave function colapse” we refer to Ref. [7].

1.8

Exercises

1. Show that eq. 1.1.3 holds. 2. Prove Lemma 1.1.2 and the corollary 1.1.2. 3. Prove the triangle inequality. 4. Give an example of a sequence in which ψk → ψ weakly, but not strongly.

5. Prove [ii.] and [iii.] of the lemma 1.4.3. 6. Construct the operator χI (Pˆ ). 7. Show that the linear functional Fx ψ = ψ(x) is not bounded.

Chapter 2 The Spectrum Having at our disposal the mathematical formalism capable of describing a quantum system, as discussed in the previous chapter, we are now in a position to aspire a connection between “our” theory and the results obtained in a laboratory. For this purpose it is not enough, altough necessary, to know the quantum state of an experimental object. This is because one hardly measures the state itself but, rather, infers information about it through a set of indirect measurements [8]. Therefore, if we have an electron confined to live in a certain region, a particle in a box, so to speak, in order to extract information about its condition we must design experiments to specify, for instance -and if possibleits position, its momentum, its energy, its (spin) angular momentum, etc. We have seen so far that there exist some restrictions on the outcomes of a measurement, be it due to the uncertainty principle, quantization conditions or the self-adjointness postulate. It becomes, then, paramount to investigate the possible results measured by an experimentalist. This set of allowed outcomes of an observable measurement consists what is called the spectrum of the associated operator. In this chapter we would like to shed light in this question. 29

CHAPTER 2. THE SPECTRUM

30

2.1

Spectrum and Resolvent

We have seen how in quantum mechanics, we can predict the probability that a real observable a lies in some interval I. We can now ask what values an observable can assume or if it can attain any value. The set of possible values depends on the corresponding self-adjoint operator A, and we call it σ(A) the spectrum of A. To properly answer to this question, we shall apply the spectral theorem to our observables. To do this, we give the Definition 2.A. We say that a scalar λ is in the resolvent set ρ(A) of a closed operator A on H if there is a bounded operator Rλ on H, called the

resolvent of A, such that

Rλ (λ − A)ψ = ψ (λ − A)Rλ ϕ = ϕ,

ψ ∈ D(A) ϕ ∈ H.

The definition say that if exists then Rλ = (λ−A)−1 . With this definition we have the Proposition 2.1. If A is a self-adjoint operator then all non-real numbers are in ρ(A) Proof. For every non-real λ, f (x) =

1 , λ−x

real line. So by the spectral theorem Z f (A) = R

is bounded and continuous over the

1 dEλ (A) λ−x

is a bounded operator defined everywhere. More over f (A) satisfies the requirements on definition 2.A. The key point here is that if an real observable can not attain (in some sense) a certain value λ0 then λ0 lies in the resolvent of the corresponding operator. More precisely, we have the Proposition 2.2. let A be a self-adjoint operator and I an open interval of the real line. If χI (A) = 0 then I ⊂ ρ(A).

2.1. SPECTRUM AND RESOLVENT

31

Proof. Let λ0 ∈ I, since I is open, δ the minimum distance between λ0 and the ends of I is positive. So define   f (x) = 

1 λ0 −x

x 6∈ λ

0

x∈λ

Clearly, in the real line, f (x) is piecewise continuous and bounded by 1δ . Thus by the spectral theorem f (A) =

Z

R−I

1 dEλ(A) λ0 − x

is a bounded operator defined everywhere. Furthermore, f (A)(λ0 − A)ψ = (1 − χI (A))ψ = ψ, ∀ψ ∈ D(A) (λ0 − A)f (A)ϕ = (1 − χI (A))ϕ = ϕ, ∀ϕ ∈ H. Therefore λ0 ∈ ρ(A). Since this holds for all λ ∈ I, we that I ⊂ ρ(A). The converse of this theorem is also true. This leads us to the Proposition 2.3. For any real λ0 ∈ ρ(A) there is an open interval I ⊂ R such that λ0 ∈ I and χI (A) = 0.

Proof. Suppose there is no such interval. Therefore there must be a sequence of open intervals In → such that λ0 ∈ In and χIn (A) 6= 0. So there is a nonzero ψn ∈ H, such that χIn (A)ψn = ψn . So the Spectral theorem implies k(λ0 − A)ψn k = k(λ0 − A)χIn (A)ψn k 6 sup |λ0 − λ| → 0. λ∈In

But since λ0 ∈ ρ(A) this is an absurd, because kψn k = k(λ0 − A)−1 (λ0 − A)ψn k = k(λ0 − A)−1 kk(λ0 − A)ψn k → 0, which contradicts the fact that kψn k > 0. Corolary 2.1.1. If A is self-adjoint, then ρ(A) is an open set.


32

Proof. Let λ ∈ ρ(A). If λ is not real, then there is a small open disk Dλ with center in λ which contain no real numbers. So by the proposition ??, we have

that Dλ ⊂ ρ(A). Now, if z are real then the proposition 2.3 say that there

is an open interval I ⊂ ρ(A) with center in λ. The length of this interval is

the diameter of the open disk DI centered in λ, and clearly DI ⊂ ρ(A).

The points that are not in the resolvent set are in the spectrum. In other words: Definition 2.B. The set σ(A) , called the spectrum of A is the set C − ρ(A). That means that σ(A) consist of those points λ in that the operator (λ − A) is not invertible. So a straightforward consequence of corollary is the Corolary 2.1.2. If A is a self-adjoint operator, then the spectrum σ(A) is a closed set. The main reason we go through all this mess is that spectrum is a fundamental object in quantum theory. Its role becomes clear in the Proposition 2.4. An observable can only assume values in the spectrum of the corresponding operator. Proof. Let A be the operator corresponding to the observable a. For λ0 ∈

ρ(A), by Proposition 2.3 there is an interval I such that χi (A) = 0. Following section 1.4, we have Pψ (a ∈ I) = (χI (A)ψ, ψ) = 0,

for any ψ ∈ H. Therefore, a can not assume values in I for any state function.

So when faced with an observable in quantum mechanics (e.g. we wish to make a measurement on a physical system) , we must consider the corresponding operator to this observable and (possibly ) the first thing we should

2.1. SPECTRUM AND RESOLVENT

33

look at is the spectrum of this operator. In fact, as the proposition above states, we can only measure values in the spectrum of the operator! Thus, a criteria for finding the spectrum of an self-adjoint operator is essential to quantum mechanics. To develop this criteria, we will need some preliminary results: Lemma 2.1.1. A closed vector subspace S of a Hilbert Space H is it self a Hilbert Space.

Proof. We just net to prove that S ⊂ H is complete. Let ψk ∈ S be a

Cauchy sequence. Since H is complete ψk → ψ ∈ H. But, since S is closed

then it contains all limit points, therefore ψ ∈ S. So all Cauchy sequences of

elements of S converges to an element in S. So S is a complete inner product space, i.e. a Hilbert space.. We now state a very useful theorem for determine the sepctrum of a self-adjoint operator: Proposition 2.5. Let A be a self-adjoint operator. λ ∈ R lies in σ(A) if and only if there is a sequence ψk ∈ D(A) such that, kψk k = 1 and k(λ − A)ψk k → 0.

(2.1.1)

Proof. Suppose λ 6∈ σ(A) and eq. 2.1.1 holds. Then λ ∈ ρ(A) and therefore Rλ (λ − A)ψk = ψk , a contradiction with eq. 2.1.1. Now suppose that eq. 2.1.1 does not hold. Our strategy is to show that (λ−A) is invertible, so λ ∈ ρ(A). To do this we claim that there is a constant C such that

kψk 6 Ck(λ − A)ψk. If not, there must be a sequence ϕk ∈ D(A) such that kϕk k → ∞. k(λ − A)ϕk k

(2.1.2)


34 So we can take ψk = sumption).

ϕk , kϕk k

and eq. 2.1.1 holds (which contradicts our as-

Form eq. 2.1.2, we have that (λ − A) is injective. If it was not injective

then there must be ϕ1 6= ϕ2 such that

(λ − A)(ϕ1 − ϕ2 ) = 0, which contradicts eq. 2.1.2. Furthermore, consider a sequence ψn ∈ R(λ − A), such that ψn → ψ ∈ H.

Since (λ − A) is injective, we call ϕn the unique solution of (λ − A)ϕn = ψn .

Now, eq. 2.1.2 shows that the sequence ϕn is a Cauchy sequence, in fact, kϕk − ϕn k 6 Ckψk − ψn k → 0. Thus ϕn → ϕ ∈ H. So for f ∈ D(A), we have (ϕ, (λ − A)f ) = lim(ϕn , (λ − A)f ) = lim(ψn , f ) = (ψ, f ).

Therefore, ϕ ∈ D(A) and (λ − A)ϕ = ψ. Then we conclude that R(λ − A)

is closed, and by proposition 2.1.1 it is a Hilbert Space.

Let f be an arbitrary vector of H and w ∈ R(λ − A). We take v to be

the (unique) solution of (λ − A)v = w, and construct the linear Functional

F on R(λ − A) by defining

F w = (v, f ).

This is a bounded linear functional, since by eq. 2.1.2 , we have |F w| 6 kvkkf k 6 Ckf kkwk. So by Riez representation lemma, there is a u ∈ R(λ − A) such that F w =

(u, w), ∀w ∈ R(λ − A). Thus, we find taking vinD(A), ((λ − A)v), u) = (f, v).

The self-adjointness of A implies that u ∈ D(A) and (λ − A)v = f. Therefore f ∈ R(λ − A) and H = R(λ − A). So (λ − A) is one-to-one (injective) and surjective, therefore invertible, which implies λ 6∈ σ(A).

2.2. FINDING THE SPECTRUM

35

As a consequence we have Corolary 2.1.3. Self-Ajoint operators are closed. Proof. We left this proof as an exercise. Corolary 2.1.4. If A is a closed operator and kvk 6 CkAvk then A is injective and R(A) is closed.

Proof. Do λ = 0 in the proof of the proposition 2.5. There are some special elements in the subset of σ(A) called the eigenvalues. Definition 2.C. λ is said to be an eigenvalue of A if there is a non-vanishing solution for Aψ = λψ. When this non-vanishing solution ψ exist, we call it an eigenvector of the operator A corresponding to the eigenvalue λ Note 2.a. The prefix eigen- is adopted from the German word eigen for “own”, “unique to”, “peculiar to”, or “belonging to” in the sense of “idiosyncratic” in relation to the originating object(operator). Therefore it is usual to carry it to more specific objects, for example: eigenfunctions in L2 spaces or eigenstate in Quantum mechanics ... With this definition it is clear that Corolary 2.1.5. If λ is an eigenvalue of A then λ ∈ σ(A). Proof. It is straightforward, if λ is an eigenvalue then there is ψ 6= 0, such that (λ − A)ψ = 0. So (λ − A) is not injective, and λ ∈ σ(A).

2.2

Finding the spectrum

We now want to apply these ideias to find the spectrum of some operators of interest. we consider as an initial example the position operator, discussed


36

in eq. 1.7.21. The easiest elements are the eigenvalues. Thus we should look for solution of

ˆ Xψ = λψ . ⇒ xψ(x) = λψ(x)

Thus, there are no eigenvalues, since the only solution of the equation above is ψ = 0 for x 6= λ. That means ψ(x) almost everywhere, and to us

this means ψ = 0! So we need to use proposition 2.5. In order to do this,

it worth to have some sort of motivation to construct the sequence in the statement of proposition 2.5. An object that can satisfies ψ = 0 for x 6= λ, is the (infamous) Dirac’s Delta function. So we can think in a sequence such

that ψk (x) → δ(x − λ), and kψk k = 1. These sequences are some times called a delta sequence. For example, we can take ψk (x) =

1 − k2 (x−λ)2 e 2 , N(k) √

where N(k) is such that kψk k = 1, that means N 2 (k) = kπ . Clearly, Z 1 2 ˆ k(λ − X)ψk k = 2 y 2 e−y dy → 0, 3 N (k)k R ˆ as k → ∞. So for any λ real we have that λ ∈ σ(X). That means the spectrum of the position operator is the whole real line.

The argument for the momentum operator is similar. But we shall use the Fourier transform and the identity of eq.1.1.6. We leave this as an exercise. You should find that the spectrum of the momentum operator is the whole real line.

2.2.1

The spectrum of the Hamiltonian

We are now, interested in the spectrum of the Energy operator or the Hamiltonian H=

P2 ˆ + V (X). 2m

To consider this operator we will need the following lemma.


37

Lemma 2.2.1. If A is a self-adjoint operator, then A2 is a self-adjoint operator. Proof. We have that A2 + 1 = (A + i)(A − i), here we stress that the domain of both operator are D(A2 ). If A is self-adjoint, then for each f ∈ H there is u ∈ D(A2 ) such that (A2 + 1)u = f . Furthermore since ±i ∈ ρ(A) for each

f there are v± ∈ D(A) such that (A − i)v− = f and (A + i)v+ = v− . Since

both v± are in D(A), it is also true for Av− . Therefore, R(A2 + 1) is the whole space and therefore A2 is self-adjoint. This show that

P2 2m is self-adjoint. H0 is the (Hamiltonian) energy operator of a free particle. H0 =

That is a particle that is not subject to a external potential. To study its spectrum, we write √ 1 √ (λ − H0 ) = ( 2mλ − P )( 2mλ − P ), 2m √ √ and observe that 2mλ ∈ R ⇒ 2mλ 6∈ ρ(P ). So for λ < 0, we have that

λ ∈ ρ(H0 ). Now, consider the case λ > 0. We write λ =

~k 2 , 2m

again we search

for eigenvalues so

2

H0 ψ = ~k ψ 2m dψ 2 ⇒ dx = −k ψ 2 +ikx ⇒ ψ(x) = c+ e + c− e−ikx , The only way this ψ(x) can lie in L2 is that c+ = c− = 0! Thus again, there are no eigenvalues! Again we employ can employ proposition 2.5 to find the spectral points. Not surprisingly, the method is quite analogue the one you probably use to find the spectrum of the momentum operator. Let 1 x ikx e , ψn (x) == √ ϕ n n

where ϕ(x) =

1 −x2 e , N

and N is such that Z φ(x)2 dx = 1. R


38 So

1 kψn k = n

Moreover, d ψ (x) dx n d2 ψ (x) dx2 n

Z 2 Z x ϕ dx = |ϕ(y)| dy = 1. n

1 ϕ′ ( nx )eikx = ikψn (x) + n−3/2 2ik = −k 2 ψn (x) + n−3/2 ϕ′ ( nx )eikx +

1 ϕ′′ ( nx )eikx n−5/2

So we have that kψn′′ + k 2 ψn k = k

x 1 2ik kϕ′ ( )k + −3/2 kϕ′′ . −1/2 n n n

And therefore, as n → ∞, kψn′′ + k 2 ψn k → 0. Thus, λ > 0 lies in σ(H0 ). Therefore the positive real axis is contained in the energy spectrum of the

free particle. Since the spectrum is a closed set, 0 is also an element of σ(H0 ). This proves the Proposition 2.6. σ(H0 ) = [0, +∞) To consider the case of a particle subject to a potential, must consider ˆ Again, D(V ) the simplest and largest domain of V is the operator V (X). the subset of H such that ψ ∈ D(V ) and V ψ ∈ H. Once we fix this domain, we know the domain of H, which is given by D(H0 ) ∩ D(V ). And here comes the problem. Is H self-adjoint ?

In fact, there are many examples that show us that with no further restrictions on V , it is not true that (in general) H is self-adjoint. At first this seems to be a terrible failure to our theory. But actually, this only raise questions and should be taken as an opportunity for research. This is one of the central problems of modern mathematical physics. One could fill libraries with the mathematics generated trying to answers this question. Mathematically, this is the question of when the sum of the Laplace operator and a multiplication operator on L2 is self-adjoint (more precisely, essentially self-adjoint). A more particular and immediate question is the following: Is there some potential such that H is self-adjoint?


39

(Un)Fortunately, the proposition above describes a large class of potentials such that H is self-adjoint. Proposition 2.7. If for all ψ ∈ D(H0 ) ∩ D(V ), there are constants a < 1 and b such that,

kV ψk 6 akH0 ψk + bkψk, then H = H0 + V is self-adjoint. To prove this theorem we use the two following results of general interest. Lemma 2.2.2. If B is a bounded linear operator on H and kBk < 1, then there is a bounded linear operator N on H such that

N(1 − B) = (1 − B)N = 1 kNk 6 k1 − Bk−1 Proof. The operator N is the inverse of the operator (1 − B), an is given by

the Neumann series

N = (1 − B)−1 = 1 + B + B 2 + B 3 + ....

Lemma 2.2.3. Let A be a densely defined Hermitian operator on H. Suppose

there is a complex number z such that R(z − A) = R(z † − A). Then A is

self-adjoint.

Proof. The hypotheses implies that (1 + A2 ) be onto, therefore A is selfadjoint. We leave the details as an exercise. Now, back to proposition 2.7. Proof. Follows from the Lemmas 2.2.2 and 2.2.3. In fact, for λ ∈ R and ψ ∈ D(H0 ) ∩ D(V ), we have

k(H0 − iλ)ψk2 = kH0 ψk2 + λ2 kψk2 .


40

The self-adjointness of H0 and proposition 2.1 implies that if λ 6= 0, for every ϕ ∈ H, we can find ψ ∈ D(H0 ) such that (H0 − iλ)ψ = ϕ. Thus kϕk2 = kH0 (H0 − iλ)−1 ϕk2 + λ2 k(H0 − iλ)−1 ϕk2 . We can see that

kH0 (H0 − iλ)−1 ϕk 6 kϕk λk(H0 − iλ)−1 ϕk 6 kϕk.

If V satisfies the hypotheses of proposition 2.7 then kV (H0 − iλ)−1 ϕk 6 akH0 (H0 − iλ)−1 ϕk + bk(H0 − iλ)−1 ϕk 6 (a +

b )kϕk. |λ|

Since a < 1, we can find a sufficiently large λ such that we have kV (H0 − iλ)−1 k 6 (a +

b )k 6 1. |λ|

So we can write H − iλ = (1 + V (H0 − iλ)−1 )−1 (H0 − iλ). The right hand side is onto,hence R(H − iλ) = H for |λ| sufficiently large.

The adjointness of H follows from lemma 2.2.3.

Proposition 2.7 gives a useful way to answer the question on if H is selfadjoint. To do this, we must know when the potential satisfies the hypotheses of the proposition. This can be done with the following criteria, which we state without demonstration. Theorem. [A Criteria for self-Adjointness] The following statements are equivalent i. D(H0) ⊂ D(V ). ii. kV ψk2 6 C(kH0 ψk2 + kψk2 ) iii. C0 = supx

R x+1 x

|V (x)|2 dx < +∞

2.3. EXERCISES

41

iv. For ǫ > 0, there is a constant K such that kV ψk2 6 ǫkH0 ψk2 + Kkψk2 v. For ǫ > 0, there is a constant C such that kV ψk 6 ǫkH0 ψk + Ckψk So if any of this statements holds then H is self-adjoint. In any case, the spectrum of H will depend on a fundamental way on the potential V .

2.3

Exercises

1. Prove Corollary 2.1.3 2. Show that the spectrum of the momentum operator is the real line. 3. Give a detailed proof of Lemma 2.2.3. 4. Show that λ < 0 lies in ρ(H0 ). 5. Prove that 0 is not an eigenvalue of H0 .

Chapter 3 Quantum Dynamics One important point we should stress is that time is just a parameter in quantum mechanics, that is it is not an operator. That means time is not an observable in the language of the previous chapters. So it makes no sense to treat time in the same way as we treat the position or momentum operators. In this chapter, we analyse how a quantum state ψ change in time, and how the probabilities and expected values depend on the time parameter.

3.1

Time evolution and Schrödinger Equation

So we need to look for a time evolution operator U such that U(t − t0 )ψ(t0 ) = ψ(t) We expect that time is a continous parameter so that for a quantum state ψ lim ψ(t) = ψ(t0 )

t→t0

Therefore the time evolution operator U must be continous on t and lim U(t − t0 ) = U(0) = I,

t→t0

Also, a quantum state reamains a quantum state as time goes by so kψ(t)k2 = ψ(t), ψ(t) = U(t)ψ(0), U(t)ψ(0) = kU(t)ψ(0)k2 = kψ(0)k2 43

CHAPTER 3. QUANTUM DYNAMICS

44 Therefore

U † (t)U(t) = I

(3.1.1)

An operator obeying eq.3.1.1 is called a unitary operator . Another necessary feature is that if we are interested in obtaining time evolution from t0 to t2 , then we can obtain the same result by considering time evolution first from t0 to t1 and then from t1 to t2 , ψ(t2 ) = U(t2 − t1 )ψ(t1 ) ψ(t2 ) = U(t2 − t1 )U(t1 − t0 )ψ(t0 ) Therefore, for t2 > t1 > t0 U(t2 − t0 ) = U(t2 − t1 )U(t1 − t0 ). An operator with this property is said to be strongly continous. Another physical requirement concerns energy conservation that if the potential V does not explicitly depend on the time then the total energy must be conserved. In our quantum mechanical setting this means that for a given state ψ the mathematical expectation of the energy and the probability that the energy lies in an interval I does not depend on time. Note 3.a. A final requirement is the inertia principle, which means that if the potential V is invariant under translations then the momentum must be conserved. In our quantum mechanical setting this means that for a given state ψ the mathematical expectation of the energy and the probability that the momentum lies in an interval I does not depend on time. We sumarize our demands on the Postulate 4. The time evolution of a 1-D quantum system from the time 0 to a time t is given by an operator U(t) such that

3.1. TIME EVOLUTION AND SCHRÖDINGER EQUATION

45

i. U(t) is continous on t. ii. U(t1 + t2 ) = U(t1 )U(t2 ) iii. U † (t)U(t) = I iv. If the potential does not depend explicitly on t then ψ, U † H U ψ = ψ, H ψ , Now we can ask about U Lemma 3.1.1. Let A be a self-adjoint operator and let ft (A) = exp(−itA) =

∞ X (−it)n n=0

n!

An

then D (ft (A)) = L2 and the following holds i. ft1 +t2 (A) = ft2 (A)ft1 (A) ii. f (t)ψ → ψ as t → 0. iii.

U (t)ψ−ψ t

→ −iAψ as t → 0, for all ψ ∈ D(A).

iv. if limt→0

U (t)ψ−ψ t

exists then ψ ∈ D(A).

Proof. Follows directly from the spectral theorem. Our next result (also know as the Stone’s theorem on one-parameter unitary groups) establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operator that are strongly continous. To do this we need the converse of Lemma

3.1.1, this result is due to Stone and we state it here without demonstration (for a proof see [9]).


46

Theorem. [Stone’s theorem] Let f (t)t∈R be a strongly continuous oneparameter unitary group on a Hilbert space H. Then there exists a unique self-adjoint operator A on H such that f (t) = eiAt .

We know that the time evolution operator U(t) must have the form eiAt for some self-adjoint operator A. The energy conservation requiriment will gives us who is A. Lemma 3.1.2. If the potential does not explicitly depend on t, then H = U † (t)HU(t). Proof. From Stone’s Theorem, U(t) = eiAt where A is a self-adjoint operator. Now let H ′ = U † (t)HU(t), it easy to see that the self-adjointness of H implies that H ′ is self-adjoint. The energy conservation requairiment of Postulate 4 implies that the expected value E¯ does not depend on t, that is ¯ E(t) = (ψ(t), Hψ(t)) = (U(t)ψ(0), HU(t)ψ(0)) = (ψ(0), U(t)† HU(t)ψ(0)) = (ψ(0), H ′ψ(0)) must coincide with ¯ E(0) = (ψ(0), Hψ(0)). Thus (U(t)ψ, HU(t)ψ) = (ψ, U † (t)HU(t)ψ) = (ψ, H ′ ψ) = (ψ, Hψ), so for all ψ ∈ D(H) we find that (ψ, (H ′ − H)ψ) = 0 therefore H ′ = H. In fact, taking ψ(0) as an eigenstate of the Hamiltonian, Hψ(0) = Eψ(0), then as we let it evolve in time, U(t) Hψ(0) = E U(t)ψ(0) = E ψ(t),

3.1. TIME EVOLUTION AND SCHRÖDINGER EQUATION

47

it should still remain an eigenstate of the system at a later time, if the energy is to be conserved, Hψ(t) = Eψ(t). Comparing these expressions we get U(t) Hψ(0) = Hψ(t) = H U(t)ψ(0), that is, the object U(t) must commute with the Hamiltonian, [H, U(t)] = H U(t) − U(t) H = 0. Having in mind all these restrictions one has to impose on the evolution operator, it must be of a general form i

U(t) = e− ~ A t , with A being a symmetry of the problem, [H, A] = HA − AH = 0. The latter can, in principle, be considered a function of the Hamiltonian H and be expanded as A = A(H) =

X

cn H n .

n

For simplicity, we make use of the correspondence principle and argue that, similarly to what happens in classical mechanics where the evolution is governed by the Hamiltonian function itself, the quantum evolution operator can be given by i

Proposition 3.1. U(t) = e− ~ Ht , where H is the Hamiltonian. It acts as a group element on the initial state and the Hamiltonian operator belongs to an algebra. Once the Hamiltonian is determined, from a physical point of view, in terms of relevant operators - such as those representing the observables for position and momentum, X and P , for instance


48

- there is still freedom in the choice of representation. The evolution of the evolution operator itself is then given by i~

∂ U(t) = H U(t). ∂t

On the other hand, i~

∂ ∂U(t) ∂ U(t)ψ(0) = i~ ψ(t) = i~ ψ(0) = H U(t) ψ(0) ∂t ∂t ∂t

so a quantum state evolves deterministically according to the linear Schrödinger equation. Hψ = i~

∂ ψ, ∂t

(3.1.2)

If the Hamiltonian is time-dependent one has to consider the following formal solution in terms of a time-ordered exponential Z H ψ(t) = p exp −i dt |ψi(0) ~ ! Z Z ′ Z t 1 t H(t′ ) ′ t H(t′′ ) ′′ H(t) dt − dt dt + · · · ψ(0). = 1−i ~ 2 0 ~ ~ 0 0 In situations where the Hamiltonian is time-independent, we have a reduction to the simpler expression discussed above, so that the state evolves according to i

ψ(t) = e− ~ Ht ψ(0). Note 3.b. It worth to note that any quantum-mechanical object is completely characterized by the state function and the time-evolution of state function is completely deterministic. Everything, the system, the equipment, the environment, and the observer are part of a (quite complex) state vector of “universe”. The measurements with different results are part of state functions at different points of the spacetime, furthermore the measurement is a complicated process involving interactions between the system and equipment. And the equipment alone has something of order of 1023 degrees of freedom!

3.2. APPLICATIONS TO TWO-LEVEL SYSTEMS

49

That means we neither know nor are able to compute the states of the equipment that we use to make the measurement. That is why some people say that quantum mechanics is deterministic, but it is also probabilistic: that means you can deterministically calculate the probability of the outcome of an experiment. This is to distinguish it from non-deterministic (i.e. stochastic) systems where you do not generally have “one” solution but an entire family of solutions depending on random variables. Historically, two equivalent representations were introduced at the dawn of the quantum theory. The first was Heisenberg’s attempt to reproduce experiments with the launch of infinite dimensional matrices to describe coordinates, momenta and energies. Independently, Schrödinger proposed a formalism based on wave equations and differential operators to represent some observables. Later it became clear that Schrödinger’s differential operators correspond to a neat way of representing Heisenberg’s more cumbersome matrices of infinite dimensions. There is also a difference in the perspective between both approaches, which will become more evident in what follows, making that of Schrödinger more closely related to equation (3.1.2). We start by presenting this formulation. Note 3.c. In Heisenberg’s formalism the observables vary in time whereas the vectors remain fixed.

3.2

Applications to Two-Level Systems

Now that we have set out the basic framework to describe quantum phenomena and explored some relevant consequences of the theory, it is time to discuss in some detail a number of important and elementary physical systems. From a mathematical point of view, it is simpler to tackle a quantum problem which admits a finite dimensional representation. The first example


50

to investigate is, thus, a two level problem for which the quantum state can be in only two states. A 2×2 Hilbert space can be used, for example, to reproduce the behaviour of a quantum spin 21 , which can be in the states up and/or down. Nonetheless it can also be used to describe, approximately only, a larger system but for which there are two states that are almost decoupled from the remaining states. For a finite dimensional vector, charactering a particular quantum state, the normalization of the probability modifies according to Z X |ψ(x, t)|2 dx = 1 −→ |ψi (t)|2 = 1. R

i

A suitably normalized basis to describe vectors in this Hilbert space is, for example, ψ↑ =

1 0

ψ↓ =

,

0 1

,

† † with ψ↑,↓ ψ↑,↓ = 1 and ψ↓,↑ ψ↑,↓ = 0. If, in this orthonormal basis, the energy

observable, specified by a Hamiltonian operator, has a general Hermitian form, (φ, Hψ) = (Hφ, ψ), H=

α β eiδ −iδ βe γ

then the evolution of a given state ψ is given by the Schrödinger equation ∂ i~ ∂t ψ = Hψ. Taking the following combinations δ

δ

ψ+ = cos 2θ ei 2 ψ↑ + sin 2θ e−i 2 ψ↓ , δ δ ψ− = sin 2θ ei 2 ψ↑ − cos 2θ e−i 2 ψ↓ , 2β with the convenient reparametrization θ = arctan α−γ , the equation of

motion can be written as

∂ ψ± = E± ψ± . ∂t The advantadge of introducing these states above, denoted eigenvectors, is i~

that they evolve in a very simple way, ψ± (t) = e−i

E± t ~

ψ± (0),

3.2. APPLICATIONS TO TWO-LEVEL SYSTEMS

51

governed by E± , the associated eigenvalues, E± =

α+γ α−γ ± sec θ. 2 2

Therefore, if one performs a measurement the particle can only be in either of the two possible states, ψ+ with energy E+ , or ψ− with energy E− . But if one does not measure its state it can be in linear combination of both eigenstates. The particle can evolve from one state to another and we can define the transition frequency between the states as ω=

E+ − E− . ~

If we let these up and down states evolve, it is convenient to express it in terms of the eigenvectors, iδ ψ↑ = e− 2 cos θ2 ψ+ + sin θ2 ψ− , iδ ψ↓ = e+ 2 sin 2θ ψ+ − cos θ2 ψ− ,

so the action of the evolution operator becomes almost trivialized, θ ~i E+ t θ ~i E− t − iδ 2 cos 2 e ψ+ + sin 2 e ψ− , U(t) ψ↑ = e i i iδ U(t) ψ↓ = e+ 2 sin 2θ e ~ E+ t ψ+ − cos 2θ e ~ E− t ψ− .

Now we are in a position to compute the probability of a initial state ψ↑

being in a state ψ+ , or ψ− , after 2 † ψ+ U(t) ψ↑ = 2 † ψ− U(t) ψ↑ =

an interval t, 2 i δ cos 2θ ei 2 e ~ E+ t = cos2 θ2 , θ i δ2 ~i E+ t 2 = sin2 2θ , sin 2 e e

respectively, as well as the probability of a initial state ψ↓ transitioning into

a state ψ+ , or ψ− , 2 2 i δ † ψ+ U(t) ψ↓ = sin 2θ e−i 2 e ~ E− t = sin2 θ2 , 2 2 i δ † ψ− U(t) ψ↓ = cos 2θ e−i 2 e ~ E− t = cos2 2θ .

In both situations, once we have a initial state prepared, be it an up or a down state, and let it evolve, after a while any energy measurement can


52

only produce the values E± , associated to states ψ± , respectively. Thus the probability of being in either of these eigenstates sum to unity, as is clear from the expressions above. Note that the probability of having an up state or a down state transtioning to an eigenstate does not vary with time and depends only on how much how close the up and down states are close to being an eigenstate, or ultimately on α, β, γ. Moreover we can calculate the transition amplitudes of starting with up and down states, suitably prepared, letting them evolve under the current Hamiltonian operator and measuring, after some time has elapsed, if the particles are again in one of the up and down states, respectively, i ω ω ω † ψ↑,↓ U(t) ψ↑,↓ = cos2 2θ ei 2 t + sin2 θ2 e−i 2 t ei 2 t e ~ E∓ t , i ω † ψ↓,↑ U(t) ψ↑,↓ = 2i sin 2θ cos θ2 sin ω2 t ei 2 t e ~ E− t ,

to show that the probability of being at the initial state after a while varies with time, P↑↑ (t) = 1 − sin2 θ sin2

ωt 2

∈

(0, 1),

and similarly for the probability of transmutation between states, P↑↓ (t) = sin2 θ sin2

ω 2

t

∈

(0, 1).

As expected, since there is no further degree of freedom in the system, the total probability of transition is conserved, P↑↑ (t) + P↑↓ (t) = 1, at any instante, but the interference pattern between initial and final states varies harmonically with time.

3.3

Schrödinger’s wave equation

In the previous section we discussed one of the simplest quantum problems, for which the states are described by vectors living in a two-dimensional

3.3. SCHRÖDINGER’S WAVE EQUATION

53

Hilbert space. Since the particles can be in only two linearly independent states, the physical observables correspond to 2 × 2 matrices. on the other

hand, the particle has infinitely many degrees of freedom, matrix representations are not the most convenient and vectors are replaced by functions. For particles restricted to move in one dimension and the equation govern-

ing the quantum phenomena is the Schrödinger equation given in eq. 3.1.2. Using the momentum operator of chapter 1, this can be written as a linear partial differential equation ~2 ∂ 2 ∂ + V (x, t) ψ(x, t), i~ ψ(x, t) = − ∂t 2m ∂x2 which simply corresponds to a representation of the fundamental equation (3.1.2) in terms of space-time coordinates. The problem of determining the evolution of the probability distribution for a quantum particle is equivalent to solving a PDE with suitable boundary conditions. This is a fairly nontrivial task unless the interaction has a simple form. Our first result in this context is the Lemma 3.3.1. If the potential V is time-independent, the state function evolve as a (infinite dimensional) linear combination of E

ψ(x, t) = ϕ(x)e−i ~ t , where ϕ and E are given by −

~2 ∂ 2 ϕ + V (x)ϕ = Eϕ. 2m ∂x2

Proof. If the potentials are time-independent, V (x, t) = V (x), we can separate variables as ψ(x, t) = ϕ(x)φ(t) so that we have 1 ∂φ ~2 1 ∂ 2 ϕ + V (x) = i~ = E. − 2m ϕ ∂x2 φ ∂t


54

The time-dependence can be immediately obtained, E

φ(t) = φ(0)e−i ~ t , and the complete solution has the general form E

ψ(x, t) = ϕ(x)e−i ~ t .

Note 3.d. We can see that the probability amplitude is not affected by the flow of time, since |ψ(x, t)| = |ϕ(x, 0)|. The solution of Schrödinger’s time-independent second order differential equation,

~2 ∂ 2 ϕ + V (x)ϕ = Eϕ, − 2m ∂x2 can be expanded in terms of two linearly independent functions, ϕ(x) = c1 ϕ1 (x) + c2 ϕ2 (x),

(3.3.3)

(3.3.4)

complemented by appropriate boundary conditions, and the following requirements on the wave function: • ϕ(x) and ϕ′ (x) must be finite • ϕ(x) and ϕ′ (x) must be single-valued • ϕ(x) and ϕ′ (x) must be continuous Note 3.e. As we discuss in chapter 1, to lie in L2 the state functions must vanish both at x → +∞ and x → −∞ (or in general, it must satisfy appro-

priate boundary conditions), therefore not any value of E is allowed. Instead, the boundary conditions usually impose restrictions on the energy eigenvalues, denoted quantization conditions. In what follows we will present some properties which are consequences of Schrödinger’s wave equation.

3.3. SCHRÖDINGER’S WAVE EQUATION

55

Lemma 3.3.2. If the potential V (x) is even, then the solutions of the timeindependent Schrödinger equation have a definite parity Proof. Consider the time-independent Schrödinger equation, −

~2 ∂ 2 ϕ(x) + V (x)ϕ(x) = Eϕ(x), 2m ∂x2

we can see that applying a parity transformation on it, we have −

~2 ∂ 2 ϕ(−x) + V (−x)ϕ(−x) = Eϕ(−x). 2m ∂x2

If the potential is invariant under parity, then V (−x) = V (x), ϕ(−x) satisfies the same equation as ψ(x), so both ϕ(x) and ϕ(−x) are admissible solutions to the same problem, up to boundary conditions. Thus, the solutions of a parity-symmetric problem can be expressed in terms of symmetric and antisymmetric combinations, ϕ± (x) =

ϕ(x) ± ϕ(−x) √ , 2

leading to solutions with a definite parity: either even or odd. Now, we would like to show that the structure of the Schrödinger equation implies the existence of an important conservation law, that associated to the probabilistic interpretation of the quantum theory. In fact, there would be an unconciliable problem with Max Born’s probabilisitic interpreation, the so called Copenhagen interpretation, if it was not compatible with Schródinger’s evolution equation (see Ref. [10]). Proposition 3.2. If the potential is a real function and ψ ∈ D(H), then 2 ~2 ∂2 ψ∗ ∂ ∗∂ ψ − ψ − ψ = +i~ (ψ ∗ ψ) . (3.3.5) 2 2 2m ∂x ∂x ∂t Proof. Taking the fundamental equations for ψ(x, t) and ψ † (x, t) ∂ ψ, H ψ = +i~ ∂t ∂ † ψ, ψ † H † = −i~ ∂t


56 We see that, for ψ ∈ D(H),

(ψ, H ψ) − (Hψ, ψ) = +i~

∂ (ψ, ψ), ∂t

so the total probability (ψ, ψ) is preserved if the Hamiltonian is Hermitian. On the other hand, apply a complex conjugation transformation to Schrödinger’s time-dependent equation and we obtain the equation satisfied by ψ ∗ , −

~2 ∂ 2 ψ ∗ ∂ψ ∗ ∗ ∗ + V (x) ψ = −i~ . 2m ∂x2 ∂t

Then multiplying them from the left and right by ψ ∗ and ψ, respectively, and subtracting one from the other we get 2 ∂2 ψ∗ ∂ψ ∗ ~2 ∗∂ ψ ∗ ∗ ∗ ∂ψ ψ − ψ + ψ (V (x) − V (x) ) ψ = +i~ ψ +ψ − 2m ∂x2 ∂x2 ∂t ∂t and if the potential is a real function, V (x) = V (x)∗ , it simplifies to 2 ~2 ∂2 ψ∗ ∂ ∗∂ ψ − ψ − ψ = +i~ (ψ ∗ ψ) . 2 2 2m ∂x ∂x ∂t

Notice that the expression above corresponds to a conservation law, since it is of the form of a continuity equation, ∂ρ ∂J + = 0, ∂t ∂x

(3.3.6)

associated to the conservation of the total charge, the total quantum probability, Q=

Z

ρ dx =

Z

ψ ∗ ψ dx = (ψ, ψ)

(3.3.7)

since its variation vanishes,

if the associated current,

+∞ ∂Q = 0, +J ∂t −∞

i~ J = 2m

∂ ∗ ∗ ∂ ψ ψ −ψ ψ , ∂x ∂x

(3.3.8)

(3.3.9)

3.4. TIME DEPENDENCE OF EXPECTED VALUES

57

is localized, vanishing at the faraway boundaries. The continuity equation above shows that there is a flow of the object J (inwards or outwards) if the density varies (increases or decreases). Notice that if the wave function is real, as is the case of a decaying exponential solution, there is no flux of probability, meaning the probability distribution remains unchanged with time. Note 3.f. In problems where an incident particles scatters, due to the presence of a potential, for instance, one can define the fraction of the wave which is transmitted and that which is reflected. The quantities are given by the so called transmission and reflection coefficients, T =

3.4

Jt , Ji

R=

Jr . Ji

Time dependence of Expected Values

It is time now to use our knowledge about the Schrödinger equation to extract information about the evolution of the expectation values associated to a certain observable. Proposition 3.3. Let O to be an operator corresponding to a real valuedobservable, then

d ∂O 1 , O = [O, H] + dt i~ ∂t where [A, B] = AB − BA. Proof. Given a certain operator O, its expected value on a certain state may

in principle be time-dependent if the reference state is evolving, Z O(t) = (ψ, O ψ) = ψ ∗ (x, t) O ψ(x, t) dx

(3.4.10)

Thus we can write Z Z Z ∂ψ d ∂ψ ∗ ∗ ∂O O ψ + dx ψ ψ + dx ψ ∗ O + O = dx dt ∂t ∂t ∂t Z Z Z 1 ∗ 1 ∗ ∂O ∗ = dx − ψ H O ψ + dx ψ ψ + dx ψ O Hψ i~ ∂t i~ Z Z ∂O 1 dx ψ ∗ (OH − HO) ψ + dx ψ ∗ ψ. = i~ ∂t


58

It can then be cast in the more compact form, ∂O 1 d . O = [O, H] + dt i~ ∂t

Note 3.g. This is the quantum analogue of the the Poisson brackets in classical mechanics, suggesting one to interpret the quantum expectation value as the classical function itself in the classical limit, when ~ → 0 and

1 i~

{·, ·}P B .

[·, ·] →

It becomes evident that if the observables do not depend explicitly on time, we have the Corolary 3.4.1. If O does not depend explicitly on time 1 d O = [O, H]. dt i~

(3.4.11)

So if the observable is not depende explicitly on time the expectation values of the observables are preserved in time if and only if [O, H] = 0. These objects are considered constants of motion if this happens. An important message to take from this point is that the Hamiltonian determines not only the evolution of the quatum states, as prescribed by Schrödinger’s equation, but it also dictates the time evolution of the expectation values of observables, as can be seen from the relation above.

3.4.1

Newton’s 2nd Law and Quantum Mechanics

A remarkable consequence of the general relation between the expectation of any quantum mechanical operator and the expectation of the commutator of that operator with the Hamiltonian of the system, as seen in the Proposition 3.3, is the so called Ehrenfest Theorem, which we now discuss.

3.4. TIME DEPENDENCE OF EXPECTED VALUES

59

The Proposition 3.3 is particularly useful when we evaluate it for simple Hamiltonian of the form H =

P2 2m

+ V (X). It can be shown that 1 [X, P 2 ] 2m

[X, H] =

=

i~ P, m

∂V , [P, H] = [P, V (X)] = −i~ ∂X

and since nor X or P depends explicitly on time, we have dX dt dP dt

=

1 [X, H] i~

=

1 [P, H] i~

=

1 P, m

=

∂V . − ∂X

(3.4.12)

Note 3.h. The relations expressed in Eq. 3.4.12 are equivalent to Hamilton’s equations of motion (for a conservative potential) [11], dX dt

=

dP dt

∂V = − ∂X = − ∂H , ∂X

1 P m

=

∂H , ∂P

A simple way to interpret Eq. 3.4.12 is given by Newton’s second law for mechanics, ¯ M ddtX = P¯ , ∂V dP¯ = − ∂X , dt

leading to M

d2 ∂V = F. X=− 2 dt ∂X

The content of the last expression is that Newton’s second law, F = ma, is after all, still valid at the quantum level in some sense: it holds true if one is interested only in expectation values. The same results can also be constructed from a slightly different approach, starting from the definition of the expectation value of the position observable, x˙ = =

d x dt

R

R

d dt ∂ψ(x)∗ ∂t

=

ψ(x)∗ x ψ(x) dx x ψ(x) + ψ(x)∗ x

∂ψ(x) ∂t

dx.


60

From Schrödinger’s wave equation R ∂ 2 ψ(x)∗ ~ x˙ = 2mi x ψ(x) − ψ(x)∗ x ∂x2 R ∂ ~ ψ(x)∗ ∂x ψ(x) dx = mp , = mi

∂ 2 ψ(x) ∂x2

dx

as expected. Similar calculations provide us with the corresponding equation for p. ˙ In the end, Newton’s second law can again be stated: F¯ = m¨ x.

3.5

Quantum Pictures

We have seen that according to Schrödinger’s description the observables are described by constant operators acting vectors which vary in time according to ΨS (t) = U(t)ΨS (0) = U(t)ΨH whereas in Heisenberg’s framework the states are fixed. We can choose Heisenberg states to coincide with Schrödinger’s at the initial time, ΨH = ΨS (0) = U(t)† ΨS (t). Not surprisingly we have that in either picture the time dependent expectation value of an observable in a given state coincide, O = (ΨS (t), OS ΨS (t)) = (ΨH , OH (t) ΨH ). This requirement imposes restrictions also on the time dependence of the operators and we can relate them via OH (t) = U(t)† OS U(t), so that the operator has a constant form according to Schrödinger, OS = U(t)OH (t)U(t)† , but has a dynamical nature in Heisenberg’s picture. The time evolution given by Schrödinger’s equation has the following counterpart in Heisenberg’s Picture, i~

dOH (t) = [H, OH (t)], dt

dictating the dynamics of Heisenberg’s operators.

(3.5.13)

3.5. QUANTUM PICTURES

61

There is, however, a further useful formulation, due to Dirac. To explore it one should be able to decompose the Hamiltonian in a time-independent term H0 and an interacting time-dependent part V (t), H = H0 + V (t). We then introduce the simpler evolution operator, i

U0 (t) = e− ~ H0 t , together with a transformed quantum state, ΨI (t) = U0 (t)† ΨS (t), or ΨS (t) = U0 (t)ΨI (t). This new vector evolves in time according to the interaction potential i~

∂ ΨI (t) = VI ΨI (t), ∂t

where we have defined VI (t) = U0 (t)† V (t)U0 (t). Moreover, also the transformed observable, OI (t) = U0 (t)† OS (t)U0 (t), evolves in time, but governed by the noninteracting part of the Hamiltonian H0 , i~

dOI (t) = [H0 , OI (t)]. dt

The latter formalism is also called the interaction picture for a quantum description, since the states evolution rely essentially on the interaction potentical VI . In the following table we summarize how states and observables evolve in time under the different pictures we have presented.


62

State Vector Observable Operator

Heisenberg Invariant Evolves with H

Dirac Evolves with VI Evolves with H0

Schrödinger Evolves with H Invariant

Table 3.1: Quantum pictures

3.6

Exercises

1. Show that, for a discrete spectra En , the general solution of the timedependent Schrödinger equation is the linear superposition of the stationary states ψ(x, t) =

∞ X

i

an ψn (x)e− ~ En t ,

n=1

where ψn (x) are eigenfunctions of H with eigenvalues En . 2. Use the spectral theorem to obtain a continuous version of exercise 1. 3. Show that when an particle is confined to a box, its energy levels are quantized. 4. Show that when an electron is confined to a square well, even though there is a nonvanishing probability of finding it outside the box, there is no flux of probability to the outer region. 5. Show (and interprete) that when a Gaussian wave packet is left to evolve according to Schrödinger’s equation, one has the following relation

~2 (∆x) (∆p) = 4 2

2

~2 t2 1+ 2 2 . mσ

Chapter 4 Approximation Methods So far we have seen some examples of quantum problems which can be completely solved in closed form. Overall we can mention the problems of a free particle, of a particle in a box, of particles scattering on simple onedimensional barriers, the harmonic oscillator, the problem of central forces, like the Hydrogen atom, just to name a few. In order to overcome the difficulties in “diagonalizing” a problem we present in this chapter some useful techniques to help us investigate real problems. The methods can be divided in two categories: the perturbative and the non-perturbative approaches.

4.1

Non-perturbative methods

Here we start the description of problems for which a closed form solution is either very difficult to find and unknown or - more commonly - not possible. In what follows we present some convenient methods which can be applied to various problems, but whose validity is restricted.

4.1.1

Variational Methods

A useful method and of simple implementation is the one that permits us to determine the ground state energy of a quantum problem, that is the minimal value of the energy E0 , given a specific dynamics, characterized by 63

CHAPTER 4. APPROXIMATION METHODS

64 a Hamiltonian H.

Proposition 4.1. The energy expectation value of a system is never less than its ground state energy, H ≥ E0 . Proof. We start by denoting ϕn the eigenstates of the system, H ϕn (x) = En ϕn (x), so that any state can be written as an expansion in this eigenbasis, Z X X ϕ(x) = ϕn (x) dx ϕ∗n (x)ϕ(x) = ϕn (x) (ϕn , ϕ). n

n

This general state can be used to evaluate the expectation value of a certain observable. In the case of the energy operator, we have H =

(ϕ, Hϕ) (ϕ, ϕ)

which can be rewritten in the following form P (En − E0 )|(ϕn , ϕ)|2 H = E0 + n=1P . 2 n=0 |(ϕn , ϕ)|

Since we are considering E0 to be associated to the ground state, all other energy levels are higher and all terms in the summation above are non negative [12]. Note 4.a. If one uses the theorem stating that the ground state of a quantum problem is nondegenerate, then all terms in the summation can be show to be positive. You can find a proof in the book of Simon/Reed, "Methods of Mathematical Physics", vol.4 "Analysis of Operators". Therefore, we have a general result that the expectation value for an energy measurement is greater than, or equal, the ground state energy, H ≥

E0 .

4.1. NON-PERTURBATIVE METHODS

65

Alternatively, we can expand the generic state as a combination of eigenstates but with unknown coefficients, cn (α), ϕα (x) =

X

cn (α) ϕn (x).

n

In this case, we can show that the expression H(α) = reduces to, using

P

n

(ϕα , H ϕα ) (ϕα , ϕα )

|cn (α)|2 = 1, H(α) =

X n

|cn (α)|2En .

Once again we can argue that its minimum value is indeed E0 , occurring when the expectation value is evaluated on the ground state ϕ0 (x), H(α) ≥ E0 . These results seem dull at first sight but they serve to a useful purpose, namely of determming the ground state of a certain system about which we do not know much. Complicated system tend to occur in physical situations more frequently than the idealized settings commonly found in textbooks. In general one is a priori not convinced about the best tools to solve a quantum problem and here we present a fairly universal technique which has also a simple implementation. The knowledge of a single energy eigenvalue in the whole spectrum might sound insignificant in principle. However, in many physical applications, especially when one is concerned about statitic properties of quantum systems, the ground state energy contains sufficient information to allow a characterization of the system. As we minimize the expectation value of a Hamiltonian operator whose spectrum is bounded from bellow, we are capable of finding the value of the


66

newly introduced parameter α = α0 which satisfies the following minimum condition, ∂H(α) = 0. ∂α α0

We can then estimate the ground state energy of a system as E0 . H(α0 ). Although very general, this approach presents the disadvantadge that it relies on one’s ability to construct the most effective form for the coefficients cn (α), or ultimately for ϕα (x) itself, as some choices will lead to better results than others. In order to optimize the method it is advisable to make use of symmetriy properties and physical reasoning to start with a convenient ansatz for these objects. In some cases it is possible to find the exact ground state ϕ0 (x) = ϕα0 (x). Next we investigate an alternative non-perturbative method which is more compatible with the other end of the spectrum, that is it more suited for high eigenvalues, being a complementaray approach.

4.1.2

Extension to Excited States

Proposition 4.2. If you have an enumerable spectrum which can be ordered, E0 < E1 ≤ E2 ≤ · · · then the upper bound on the n-th energy level is given

by

En ≤ ψ, H ψ if the state ψ used is orthogonal to all previous n − 1 states. The proof is left as an exercise.


4.1.3

67

A 2 × 2 example H=

√1 2

√ 2 0

1 |φ(α)i = √ 1 + α2

1 α

√ 1 + 2 2α E(α) = hφ(α)|H|φ(α)i = 1 + α2 dE =0 dα α0 √ α0 = − 2 1 |φ(α0 )i = √ 3

1 √ − 2

E0 ≤ E(α0 ) = −1 Suppose we know the ground state of the system 1 1 √ |1i = √ 3 − 2 The orthogonal state is 1 |φi = √ 3

√ 2 1

E1 ≤ hφ|H|φi = 2


68

4.1.4

Method os Successive Powers

Proposition 4.3. Assuming the spectrum is bounded, E0 ≤ E1 ≤ E2 ≤ · · · ≤ EM , the largest eigenvalue reads hφ|H m+1|ψi , m→∞ hφ|H m |ψi

EM = lim

where the state φ may be any, as long as it is orthogonal to all eigenfucntions of H, φ, ψn 6= 0. Note 4.b. If this projection φ, ψn vanishes we have to take another φ state.

Proof. Projecting on a test function φ the m-th action of the Hamiltonian operator on a (unknown) combination of its eigenfucntions X φ, H m ψ = cn Enm φ, ψn . n

Thus, in the limit of large m → ∞ the largest eigenvalue EM contributions will dominate and

m+1 φ, ψM cM EM φ, H m+1 ψ = lim m→∞ m φ, H m ψ cM EM φ, ψM is a well defined ratio if φ, ψM 6= 0, as assumed, and provides the estimative for EM above.

4.1.5

WKB - Semiclassical approximation

The method developed by Wentzel, Kramers and Brillouin - from which the acronym stems - considers the Schrödinger equation, ~2 ′′ ψ (x) + (V (x) − E) ψ(x) = 0, − 2m in a semiclassical approach.


69

Proposition 4.4. A good approximation for the eigenfunctions of a Schrödinger problem in the semicassical limit is given by plane waves of the form R √ i 2m(E−V (x))dx e± ~ ψ(x) ∼ 41 . 2m(E − V (x))

Proof. For that it considers a representation similar to the one found in Hamilton-Jacobi’s formulation of classical mechanics, reexpressing the wavefunction in terms of Hamilton’s principal (action) funcion, i

ψ(x) = e ~ S(x) , so the equation of motion becomes −i~ S ′′ (x) + S ′ (x)2 = 2m(E − V (x)) = p(x)2 , where we have introduced a position-dependent momentum function p(x) as an analogy with classical mechanics. Here we are more interested in the excited states so that the quantum systems behaves more similarly to classical mechanics. In this so called semiclassical limit the wave properties are less important and -similarly to what is done in the geometric limit of physical optics- we can restrain to the optical approximation, ~ S ′′ (x)

An Introduction to the Mathematical Aspects of Quantum Mechanics:

An Introduction to the Mathematical Aspects of Quantum Mechanics:

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