CONCEPT GENERAL MECHANICS Classical & Quantum Mechanics

CONCEPT GENERAL MECHANICS Classical & Quantum Mechanics P.J. Mulders and H.G. Raven Nikhef and Department of Physics and Astronomy, Faculty of Sciences, VU University, 1081 HV Amsterdam, the Netherlands E-mail: [email protected] & [email protected]

November 2012 (version 0.0)

1

Introduction

Voorwoord Het college Algemene Mechanica is een combinatie van Klassieke Mechanica en Quantummechanica en wordt verzorgd door Piet Mulders en Gerhard Raven geassisteerd door Maarten Buffing bij de werkcolleges. Het college wordt getentamineerd in twee delen, Deel I: Van klassieke mechanica naar quantummechanica (6 ECTS), Deel II: Quantummechanica (6 ECTS). Bij het college wordt gebruik gemaakt van de boeken ..... en Introduction to Quantum Mechanics, second edition van D.J. Griffiths (Pearson). Dit dictaat beschrijft de context van de colleges. Het gehele vak beslaat 12 studiepunten en wordt gegeven in periodes 2 en 3. In totaal zullen er 28 colleges van 2 uur elk en 28 werkcolleges van 3 uur gegeven worden. In blok 2 zijn er 14 colleges (2 halve dagen per week gedurende 7 weken), afgerond met een tentamen (vaknaam: Van klassieke naar quantummechanica, 6 ECTS vakcode ....) In blok 3 zijn er 12 colleges (4 halve dagen per week gedurende 3 weken), afgerond met een tentamen (vaknaam: Quantummechanica, 6 ECTS, vakcode ...). Bij de aansluitende werkcolleges worden problemen gemaakt, besproken, er worden daarnaast opgaven gegeven die moeten worden ingeleverd. De resultaten van deze opgaven vormen onderdeel van de toetsing.

Piet Mulders en Gerhard Raven November 2012

Contents 1 Introduction 1.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4

2 Let’s go

8

3 Methods in classical mechanics 3.1 Euler-Lagrange equations . . . 3.2 Hamilton equations . . . . . . . 3.3 Conserved quantities (Noether’s 3.4 Poisson brackets . . . . . . . .

. . . .

9 9 10 11 12

4 Methods of Quantum Mechanics 4.1 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Observables in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Compatibility of operators and commutators . . . . . . . . . . . . . . . . . . . . . . . . .

14 14 16 19

5 Symmetries 5.1 Space translations . . . 5.2 Time evolution . . . . . 5.3 Rotational symmetry . . 5.4 Boost invariance . . . . 5.5 The two-particle system 5.6 Discrete symmetries . . 5.7 A relativistic extension∗

22 22 24 25 27 28 29 31

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Introduction

3

Chapter 1

Introduction 1.1

Equations of Motion

A classical system is determined by x(t) from which we get the velocity v(t) = x(t), ˙ etc. The motion is obtained from Newton’s equation, d2 x dV m 2 =− . (1.1) dt dx It determines how we get from x(0) −→ x(t) (given sufficient boundary conditions x(0) and x(0)). ˙ This can be extended to 3 dimensions by looking at three coordinates of an object/particle, r(t) = (x(t), y(t), z(t)). When there are more objects/particles one has many coordinates, r1 (t), r2 (t), . . . . Quantum mechanically, one works with a complex wave function ψ(x, t), which by itself is not physical but contains all relevant information about the state (NL: toestand) referred to as |ψi. For example, the absolute value squared |ψ(x, t)|2 represents a probability. The wave function satisfies the Schrödinger equation, ~2 ∂ 2 ψ ∂ψ =− + V (x, t) ψ(x, t). (1.2) i~ ∂t 2m ∂x2 This determines how the system evolves from ψ(x, 0) −→ ψ(x, t). The constant appearing in this equation is Planck’s constant, h = 6.626 × 10−34 J s. (1.3) or more precise the reduced Planck’s constant ~ = h/2π = 1.055 × 10−34 J s = 6.582 × 10−16 eV s. This also determines the domain where quantum mechanics is needed to get a good description of nature, namely the domain where appropriate quantities are of the order of this constant. In this respect, note that energy × time and/or to momentum × distance or angular momentum have the same dimension as ~. In 3 dimensions one works with wave functions ψ(r, t) and for many particles with wave functions ψ(r 1 , r 2 , . . . , t). What do these equations have to do with each other. Are they the same. The answer is no. The approach in classical mechanics is deterministic. A position is known at a given time and at the same time one can know its velocity or the momentum. From a quantum mechanical wave function, one also can get positions or momenta, but as it turns out not at the same time and in general the outcome of a measurement of position or momenta is not unique. The probabilities of particular outcome for positions or momenta, however, occur with well-determined probabilities. The equations both do have the presence of a potential in common. This is where the dynamics is. In its absence one has a free-moving system. It is interesting to see what this is in both cases, both of them hopefully familiar to you. In classical mechanics the solution with x(0) = x0 and x(0) ˙ = v0 is given by x(t) = x0 + v0 t, 1

(1.4)

2

Introduction

representing the trajectory of an object moving with constant velocity v0 , constant momentum p = m v0 and energy E = 12 mv02 = p2 /2m. The solution of the Schrödinger equation with V = 0 is probably also familiar. The solution for a system moving with a fixed momentum is given by a wave with wave number k and frequency ω, √ (1.5) ψk (x, t) = ρ ei(k x−ω t) . It has a constant probability ρ over space. The momentum is given by p = ~ k and the energy by E = ~ω. As for classical mechanics the energy is given by E = p2 /2m, thus the frequeny is related to the momentum through ω = ~k 2 /2m. It is known as a plane wave solution In contrast to the precisely determined momentum, the position is completely undetermined. We will compare both approaches in much more detail later. The examples are useful to keep in mind as reference examples in various applications. ========================================================== Exercise: Check the solutions in Eqs 1.4 and 1.5. ==========================================================

1.2

Classical Mechanics

Let us first review some of the properties of classical mechanics. As should be clear from the way we wrote Newton’s equation in Eq. 1.1 we consider conservative forces of the type F (x) = −∂V /∂x or in 3-dimensional applications F = −∂V /∂r = −∇V . Note that we in principle could allow even a time-dependence in the potential, i.e. V = V (r, t). We see two interesting quantities popping up, the momentum p = m r˙ and the kinetic energy K = 21 m v 2 satisfying dp ∂V ¨=F =− = mr , dt ∂r d 1 ∂V dr · , mv 2 = F · v = − 2 dt ∂r dt

(1.6) (1.7)

or introducing the energy E = K + V and using that dV /dt = (∂V /∂r) · (dr/dt) + ∂V /∂t, one sees that dE ∂V = . dt ∂t

(1.8)

It shows that in some cases energy and momentum are conserved. If the potential doesn’t depend on particular coordinates, e.g. is independent of z, which means translation invariance in z-direction, the z-component of momentum is conserved. If the potential does not explicitly depend on time (invariance under time translations) one has conservation of energy. There are many other quantities that play a role in classical mechanics, for instance the angular momentum ℓ = r × p. For this quantity one has dp ∂V dℓ =r× = −r × . dt dt ∂r

(1.9)

ˆ ∂V /∂r, thus the right-hand side of the equaFor a central potentials V (r) one has ∇V (r) = ∂V /∂r = r tion vanishes and ℓ is conserved. It shows the link between rotational invariance and conservation of angular momentum. Note that the notation ∂/∂r is somewhat sloppy. It refers to the gradient ∇ with components ∇i = ∂/∂xi .

3

Introduction

========================================================== Exercise: The link with rotational invariance can also be made more explicit by working in polar coordinates. One has x = r sin θ cos ϕ,

y = r sin θ sin ϕ,

z = r cos θ.

One can use this to calculate e.g. ∂V (x, y, z) ∂x ∂V (x, y, z) ∂x ∂V (x, y, z) ∂x ∂V (r, θ, ϕ) = + + . ∂ϕ ∂x ∂ϕ ∂x ∂ϕ ∂x ∂ϕ Use this to show that

∂V ∂V ∂V =x −y , ∂ϕ ∂y ∂x

and using Eq. 1.9 one finds

∂V dℓz =− . dt ∂ϕ This shows the link of conservation of ℓz with invariance under rotations around the z-axis. ==========================================================

Two interacting particles We also recall the situation of two interacting particles with a potential that only depends on the relative distance, i.e. V (r 1 , r 2 ) = V (r 1 − r 2 ). The force on particle 1 and particle 2 are F1 = −

∂V ∂r1

and F 2 = −

∂V = −F 1 , ∂r2

thus action = −reaction and there is no net force on the combined system. The equations of motion (Newton) are two coupled equations m1 r¨1 = F 1 ,

and m2 r¨2 = F 2 = −F 1 .

(1.10)

It is convenient to introduce new coordinates, know as the center of mass and relative coordinates,       m2     M R = m1 r 1 + m2 r 2   r1 = R + r M ⇐⇒ (1.11)   m1     r = R − r r = r1 − r 2 2     M

where M = m1 + m2 is the total mass. Furthermore it is convenient to introduce the reduced mass m ≡ m1 m2 /(m1 + m2 ). (If there are more particles or confusion may arise, we can use subscripts R12 , etc. for coordinates and masses.) In terms of these new coordinates, the equations of motion decouple, ¨ = 0, MR

¨ = F = −∂V /∂r. and m r

(1.12)

˙ and p = mv = mr˙ The momenta corresponding to the new coordinates are given by P = M V = M R and they satisfy      m1     ˙   p 1 = m1 V + m v = P +p M V = M R = P = p1 + p2 M ⇐⇒ (1.13) p p   m2   v = r˙ = 1 − 2   p = m V − m v = P − p. 2   2 m1 m2  M

4

Introduction

Not only the equations of motion decouple, but it is now also straightforward to show that the kinetic energy K can be written as K

m1 v 21 +

=

1 2

=

p21 + 2m1 2m2

1 2 p22

m2 v 22

=

1 2

MV2+

P2 p2 + . 2M 2m

=

1 2

m v2 (1.14)

The potential energy only depends on the relative position, i.e. V (r 1 − r 2 ) = V (r). With the results of the first part of this section we see immediately that the total energy E = K1 + K2 + V is conserved (the potential does not (explicitly) depend on time). It separates into two parts, p2 P2 + (1.15) E= + V (r), 2M |2m {z } |{z} Ecm

Erel

˙ and the relative (rel) part involves r and r. ˙ FurThe center of mass (cm) part involves R through R thermore the total momentum P = p1 + p2 is conserved (the potential energy does not depend on R). The energies of the particles 1 and 2 separately are not conserved, neither are their momenta p1 and p2 conserved.

1.3

Quantum Mechanics

To go any further with quantum mechanics, one needs to know what to do with ψ(x, t), or in 3 dimensions ψ(r, t). It turns out that its physical significance comes in by doing something with it. In particular |ψ(x, t)|2 dx is the probability to find the system at time t in an interval of size dx around the point x. Note that while the wave function may be complex, the probability ρ(x, t) = |ψ(x, t)|2 = ψ ∗ (x, t)ψ(x, t) 2

(1.16)

3

is real. For three dimensions |ψ(r, t)| d r is the probability to find the system in a small volume d3 r around the point r. The value ψ(x, t) of the function itself is referred to as probability amplitude (NL: waarschijnlijkheidsamplitudo) for the state ψ (or |ψi) to be at place r at time t. Where the system was before time t cannot be answered, one would have to do a measurement for that at the earlier time. But as soon as a measurement is done and the particle is found at a particular position, say the origin, then the state is no longer described by the wave function ψ but by ψ0 , which would be a function peaking around the origin. This sharp function will then evolve (spread) according to the Schrödinger equation. Pretty weird!

Operators and their expectation values Suppose that we have a system described by by the wave function ψ. We measure the position. We repeat the measurement with another system described by the same wave function. And so on. If the possible outcome for the positions would be discrete, i.e. just a collection of possibilities xi we would measure how Pmany times (ni ) particles end up in each of the positions. That probability is Pi = ni /N , where N = i ni is the total number of measurements. One has X Pi , (1.17) 1 = i

hxi

=

hx2 i =

X

xi Pi ,

(1.18)

x2i Pi ,

(1.19)

i

X i

5

Introduction

which states that the probabilities add up to one, and weighing with xi or x2i gives the average of x, x2 , etc., which can be extended to the average of any weight function w(x). The standard deviation (squared) is found by weighing with (xi − hxi)2 . Since hxi is a number, one finds X (x2i − hxi)2 Pi = hx2 i − hxi2 . (1.20) σ 2 = ∆x2 = i

For the quantum mechanical outcome of an x-measurements one has the continuous version of the above with all (real) possibilities for x, with probabilities ρ(x, t) = |ψ(x, t)|2 . This implies the normalization and leads to specific expectation values (NL: verwachtingswaarden), Z 1 = dx ψ ∗ (x, t)ψ(x, t), (1.21) | {z } ρ(x,t)

hˆ xi

=

hˆ x2 i

=

Z

Z

dx x ρ(x, t) =

Z

dx x2 ρ(x, t) =

dx ψ ∗ (x, t) x ψ(x, t),

Z

dx ψ ∗ (x, t) x2 ψ(x, t),

(1.22)

(1.23)

which can be extended to any weight function. For the position one has the standard deviation (NL: standaardafwijking) ∆x of which the square is Z 2 ∆x = dx (x − hˆ xi)2 ρ(x, t) = hˆ x2 i − hˆ xi2 . (1.24) The notation on the right hand side with x between the functions ψ ∗ and ψ can be generalized to any property as well as to three dimensions. For this we first introduce the concept of operator and its expectation value, Z ˆ = d3 r ψ ∗ (x, t) Oψ(x, ˆ hOi t), (1.25)

ˆ is an operator acting on the function ψ and hOi ˆ is referred to as the expectation value of O. ˆ where O In the simple case above we were working with the position and one would have the position operator x ˆ that acts like x ˆψ(x, t) = x ψ(x, t). But after generalizing this to an operator being something that ˆ also the derivative d/dx is an example of an operator working on ψ produces a new function, ψ −→ Oψ, and producing a new function (the derivative function). Also the normalization of the wave function can be considered as an expectation value of an operator. It is in fact just the expectation value of the unit operator, hˆ 1i = 1. Usually the hat on the operators will be omitted. Finally we note that an expectation ˆ ψ or hψ|O|ψi ˆ value does depend on the state ψ, so the better notation would have been hOi instead of ˆ just hOi. Just as one talks about the expectation value of xˆ (in a state ψ), one now can talk about the ˆ (in state ψ). expectation value of O

The momentum operator If we look at the time dependence of hˆ xi, it is (in general) nonzero. We obtain after a few partial integrations Z d i~ hˆ px i ∂ψ(x, t) hˆ xi = − = , (1.26) dx ψ ∗ (x, t) dt m ∂x m with the momentum operator (NL: impulsoperator) pˆ given by pˆx = −i~

∂ . ∂x

(1.27)

6

Introduction

The position and momentum are actually also in quantum mechanics the basic quantities, as they are in classical mechanics. E.g. classically the energy can be written as a function of p and x. That same function is actually the righthand-side of the Schrödinger equation in Eq. 1.2, i~

∂ψ(x, t) ˆ = Hψ(x, t), ∂t

(1.28)

with

2 ~2 ∂ 2 ˆ = pˆ + V (ˆ + V (x, t). (1.29) x, t) = − H 2m 2m ∂x2 This is the energy operator or Hamiltonian, which is the most important operator because its role as time evolution operators. In three dimensions, one has operators for each direction, denoted

ˆ=r r

ˆ = −i~ and p

∂ = −i~ ∇. ∂r

(1.30)

Explicitly the momentum operators are pˆx = −i~

∂ , ∂x

pˆy = −i~

∂ , ∂y

pˆz = −i~

∂ . ∂z

(1.31)

ˆ , which in 3 Other important operators in 3 dimensions are the angular momentum operators ˆℓ = rˆ × p dimensions play a key role. Explicitly, they are ∂ ∂ ∂ ∂ ∂ ∂ ˆ ˆ ˆ , ℓy = −i~ z , ℓz = −i~ x . (1.32) ℓx = −i~ y −z −x −y ∂z ∂y ∂x ∂z ∂y ∂x ==========================================================

Exercise: In the exercise on angular momentum in the section on classical mechanics we have used polar coordinates. Show by acting with the operators on an arbitrary function ψ(x, y, z) that ∂ ℓˆz = −i~ . ∂ϕ ==========================================================

Two interacting particles Let us also for quantum mechanics go to two interacting particles to illustrate some common features of quantum mechanics and classical mechanics. Translating the Hamiltonian for two (interacting) particles from classical mechanics we get the two-particle Hamiltonian, H =−

~2 ~2 ˆ2 ). ∇21 − ∇2 + V (ˆ r1 − r 2m1 2m2 2

(1.33)

An example of this is the interaction for a Hydrogen-like atom with electron and proton in which case the potential is the electron-proton potential. Using total mass M = m1 + m2 and reduced mass m = m1 m2 /M , this Hamiltonian can be rewritten in terms of the center of mass and relative coordinates, although one should realize that these now are position operators, ˆ = m1 rˆ1 + m2 r ˆ2 , MR ˆ2 . rˆ = rˆ1 − r

(1.34) (1.35)

˙ and r˙ to get the momenta. For the momentum operators we now In classical mechanics we looked at R of course have to rewrite the gradients.

7

Introduction

========================================================== Exercise: Show for one dimension that d d d + , = dX dx1 dx2 d m1 d m2 d − . = dx M dx1 M dx2 Start by rewriting d/dx1 and d/dx2 in terms of d/dX and d/dx and then invert the relations. The relations can then trivially be generalized to relations for ∇R = (d/dX, d/dY, d/dZ) and ∇r . ========================================================== The relations show that for the momenta one has ˆ = −i~∇R = p ˆ1 + p ˆ2, P ˆ ˆ ˆ1 p p ~ p − 2, = −i ∇r = m m m1 m2

(1.36) (1.37)

which is idential to the classical relations (but now for quantum operators). One obtains ~2 2 ~2 2 ∇R − ∇r − V (ˆ r) . H =− | 2M {z } | 2m {z } Hcm

(1.38)

Hrel

The Hamiltonian is separable into two parts (like the classical situation). The center of mass (cm) part involves R through ∇R and the relative (rel) part involves r and ∇r . The advantage of this will become clear.

Chapter 2

Let’s go Following the starting points of classical mechanics and quantum mechanics in the previous chapter, one needs to get familar with the concepts of quantum mechanics and classical mechanics. These will be treated in two separate series of lectures. We suggest that you occasionally have a look at these notes, in particular the chapters three and four, which you may consider a summary of concepts.

Topics in classical mechanics 1. D’Alembert’s principle 2. Variational approach 3. Principle of least action; Euler-Lagrange equations 4. Symmetries and conserved quantities (Noether’s theorem) 5. Phase space 6. Hamilton equations 7. Canonical transformations 8. Poisson brackets 9. Non-inertial systems

Topics in quantum mechanics 1. Time dependence: stationary solutions, oscillations, probability density and probability current. 2. Schrödinger equation in one dimension: spectrum, bound states, scattering states. 3. Examples, a.o. square well, the harmonic oscillator, delta potential. 4. Exercises emphasizing probabilities, measurements, two/few level systems. 5. Momentum operator and plane waves. Wave packets. 6. Angular momentum operators and spherical harmonics. 7. Three dimensional Schrödinger equation and radial Schrödinger equation. 8. Hydrogen atom.

8

Chapter 3

Methods in classical mechanics 3.1

Euler-Lagrange equations

Starting with positions and velocities, Newton’s equations in the form of a force that gives a change of ˙ momentum, F = dp/dt = d(mr)/dt can be used to solve many problems. Here momentum including the R mass m shows up. One can extend it to several masses or mass distributions in which m = d3 x ρ(r), where density and mass might itself be time-dependent. In several cases the problem is made easier by introducing other coordinates, such as center of mass and relative coordinates, or using polar coordinates. Furthermore many forces are conservative, in which case they can be expressed as F = −∇V . In a system withPseveral degrees of freedom r i , the forces in F i −m¨ ri may contain internal constraining · δr = 0 (no work). This leads to D’alembert’s principle, forces that satisfy i F int i i X (3.1) − m¨ ri · δri = 0. F ext i i

This requires the sum, because the variations δri are not necessarily independent. Identifying the truly independent variables, ri = ri (qα , t), (3.2) with i being any of 3N coordinates of N particles and α running up to 3N − k where k is the number of constraints. The true path can be written as a variational principle, Z t2 X F ext dt (3.3) − m¨ r i · δr i = 0, i t1

i

or in terms of the independent variables as a variation of the action between fixed initial and final times, Z t2 dt L(qα , q˙α , t) = 0. (3.4) δS(t1 , t2 ) = δ t1

The solution for variations in qα and q˙α between endpoints is Z t2 X ∂L ∂L δqα + δ q˙α dt δS = ∂qα ∂ q˙α t1 α t2 Z t2 X X ∂L ∂L d ∂L dt = δqα + − δqα ∂q dt ∂ q ˙ ∂ q ˙ α α α t1 α α t1

9

(3.5)

10

Methods in classical mechanics

which because of the independene of the δqα ’s gives the Euler-Lagrange equations ∂L d ∂L = . dt ∂ q˙α ∂qα

(3.6)

For a simple unconstrained system of particles in an external potential, one has X 1 ˙2 L(ri , r˙ i , t) = 2 mi r i − V (r i , t),

(3.7)

i

giving Newton’s equations. The Euler-Lagrange equations lead usually to second order differential equations. With the introduction of the canonical momentum for each degree of freedom, pα ≡

∂L , ∂ q˙α

(3.8)

∂L , ∂qα

(3.9)

one can rewrite the Euler-Lagrange equations as p˙ α = which is a first order differential equation for pα .

3.2

Hamilton equations

The action principle also allows for the identification of quantities that are conserved in time, two examples of which where already mentioned in our first chapter. Consider the effect of changes of coordinates and time. Since the coordinates are themselves functions of time, we write t qα (t)

t′ = t + δt, qα′ (t) = qα (t) + δqα ,

−→ −→

(3.10) (3.11)

and the full variation qα (t) −→ qα′ (t′ ) = qα (t) + δqα + q˙α (t) δt . {z } |

(3.12)

∆qα (t)

The Euler-Lagrange equations remain valid (obtained by considering any variation), but considering the effect on the surface term, we get in terms of ∆qα and δt the surface term ! X t2 δS = . . . + pα ∆qα − H δt , (3.13) α

t1

where the quantity H is known as the Hamiltonian X H(qα , pα , t) = pα q˙α − L(qα , q˙α , t). α

The full variation of this quantity is δH =

X α

q˙α δpα − p˙ α δqα −

∂L δt, ∂t

(3.14)

11


which shows that it is conserved if L does not explicitly depend on time, while one furthermore can work with q and p as the independent variables (the dependence on δ q˙ drops out). One thus finds the Hamilton equations, ∂H dH ∂H ∂L ∂H and p˙ α = − with = =− . (3.15) q˙α = ∂pα ∂qα dt ∂t ∂t The (q, p) space defines the phase space for the classical problem. For the unconstrained system, we find pi = m r˙ i and H(pi , ri , t) = and r˙ i =

3.3

pi mi

X p2 i + V (r i , t), 2m i i

and p˙i = −

∂V ∂ri

with

dH ∂V = . dt ∂t

(3.16)

(3.17)

Conserved quantities (Noether’s theorem)

Next we generalize the transformations to any continuous transformation. In general, one needs to realize that a transformation may also imply a change of the Lagrangian, L(qα , q˙α , t) → L(qα , q˙α , t) +

dΛ(qα , q˙α , t) , dt

(3.18)

that doesn’t affect the Euler-Lagrange equations, because it changes the action with a boundary term, t2 (3.19) S(t1 , t2 ) → S(t1 , t2 ) + Λ(qα , q˙α , t) . t1

Let λ be the continuous parameter (e.g. time shift τ , translation ax , rotation angle φ, . . . ), then ∆qα = (dqα′ /dλ)λ=0 δλ, δt = (dt′ /dλ)λ=0 δλ, and ∆Λ = (dΛ/dλ)λ=0 δλ. This gives rise to a surface term of the form (Q(t2 ) − Q(t1 ))δλ, or a conserved quantity ′ ′ X dt dΛ dqα +H − . (3.20) Q(qα , pα , t) = pα dλ λ=0 dλ λ=0 dλ λ=0 α ==========================================================

Exercise: Get the conserved quantities for time translation (t′ = t + τ and r ′ = r) for a one-particle system with V (r, t) = V (r). ========================================================== Exercise: Get the conserved quantity for space translations (t′ = t and r ′ = r + a) for a system of two particles with a potential of the form V (r 1 , r 2 , t) = V (r 1 − r 2 ). ========================================================== Exercise: Show for a one-particle system with potential V (r, t) = V (|r|) that for rotations around the z-axis (coordinates that change are x′ = cos(α) x − sin(α) y and y ′ = sin(α) x + cos(α) y) the conserved quantity is ℓz = xpy − ypz . ==========================================================

12


Exercise: Consider the special Galilean transformation or boosts in one dimension. The change of coordinates corresponds to looking at the system from a moving frame with velocity u. Thus t′ = t and x′ = x − ut. Get the conserved quantity for a one-particle system with V (x, t) = 0. ========================================================== Exercise: Show that in the situation that the Lagrangian is not invariant but changes according to L′ = L +

dΛ + ∆LSB dt

(that means a change that is ’more’ than just a full time derivative, referred to as the symmetry breaking part ∆LSB ), one does not have a conserved quantity. One finds d ∆LSB dQ . = dt dλ λ=0 Apply this to the previous exercises by relaxing the conditions on the potential. ==========================================================

3.4

Poisson brackets

If we have an quantity A(qi , pi , t) depending on (generalized) coordinates q and the canonical momenta pi and possible explicit time-dependence, we can write X ∂A dA ∂A ∂A q˙i + p˙i + = dt ∂q ∂p ∂t i i i X ∂A ∂H ∂A ∂A ∂H ∂A + − = {A, H}P + . (3.21) = ∂qi ∂pi ∂pi ∂qi ∂t ∂t i The quantity

X ∂A ∂B ∂A ∂B − {A, B}P ≡ ∂qi ∂pi ∂pi ∂qi i

(3.22)

is the Poisson bracket of the quantities A and B. It is a bilinear product which has the following properties (omitting subscript P), (1) {A, A} = 0 or {A, B} = −{B, A}, (2) {A, BC} = {A, B}C + B{A, C} Leibniz identity,

(3) {A, {B, C}} + {B, {C, A}} + {C, {A, G}} (Jacobi identity).

You may remember these properties for commutators of two operators [A, B] in linear algebra or quantum mechanics. One has the basic brackets, {qi , qj }P = {pi , pj } = 0

and {qi , pj }P = δij .

(3.23)

Many of our previous relations may now be written with the help of Poisson brackets, such as q˙i = {qi , H}P

and p˙ i = {pi , H}P .

(3.24)

Furthermore making a canonical transformation in phase space, going from (q, p) → (˜ q , p˜) such that ˜ p, q˜) the Poisson brackets {A(p, q), B(p, q)}qp = {˜ q (q, p), p˜(q, p)} = 1, one finds that for A(p, q) = A(˜ ˜ p, q˜), B(˜ ˜ p, q˜)}q˜p˜, i.e. they remain the same taken with respect to (q, p) or (˜ {A(˜ q , p˜).

13


A particular set of canonical transformations, among them very important space-time transformations such as translations and rotations, are those of the type ∂G δλ, ∂pi ∂G p′i = pi + δpi = pi − δλ. ∂qi qi′ = qi + δqi = qi +

(3.25) (3.26)

This implies that δA(pi , qi ) =

X ∂A i

∂qi

δqi +

∂A δpi ∂pi

=

X ∂A ∂G ∂A ∂G δλ = {A, G}P δλ + ∂qi ∂pi ∂pi ∂qi i

(3.27)

Quantities G of this type are called generators of symmetries. For the Hamiltonian (omitting explicit time dependence) one has δH(pi , qi ) = {H, G}P δλ. Looking at the constants of motion Q in Eq. 3.20 one sees that those that do not have explicit time dependence leave the Hamiltonian invariant and have vanishing Poisson brackets {H, Q}P = 0. These constants of motion thus generate the symmetry transformations of the Hamiltonian. ========================================================== Exercise: Study the infinitesimal transformations for time and space translations, rotations and boosts. Show that they are generated by the conserved quantities that you have found in the previous exercises (that means, check the Poisson brackets of the quantities with coordinates and momenta). ========================================================== Exercise: Show that the Poisson bracket of the components of the angular momentum vector ℓ = r × p satisfy {ℓx , ℓy }P = ℓz . ==========================================================

Chapter 4

Methods of Quantum Mechanics 4.1

Hilbert space

In quantum mechanics the degrees of freedom of classical mechanics become operators acting in a Hilbert space H , which is a linear space of quantum states, denoted as kets |ui. These form a linear vector space over the complex numbers (C), thus a combination |ui = c1 |u1 i + c2 |u2 i also satisfies |ui ∈ H . Having a linear space we can work with a complete basis of linearly independent kets, {|u1 i, . . . , |u1 i} for an N -dimensional Hilbert-space, although N in quantum mechanical applications certainly can be infinite! An operator A acts as a mapping in the Hilbert-space H , i.e. |vi = A|ui = |Aui ∈ H . Most operators are linear, if |ui = c1 |u1 i + c2 |u2 i then A|ui = c1 A|u1 i + c2 A|u2 i. Depending on the nature of the operators there is often a natural representation of the states. The most well-known is that in which the states |ψi are functions ψ(r, t) over space and time and the operators produce new functions, such as the position operators x î or momentum operators pî , the latter acting as differential operator, pî = −i~ ∂/∂xi . It is important to realize that operators in general do not commute. One defines the commutator of two operators as [A, B] ≡ AB − BA. The commutator of position and momentumoperators, satisfy [ˆ xi , x ˆj ] = [ˆ pi , pˆj ] = 0

and [ˆ xi , pˆj ] = i~ δij ,

(4.1)

resembling the Poisson brackets in Eq. 3.23, which is also reflected in its properties. It is bilinear and • [A, B] = −[B, A] • [A, BC] = [A, B]C + B[A, C]

(Leibniz law)

• [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0

(Jacobi identity),

• [f (A), A] = 0.

The last property follows because a function of operators is defined via a Taylor expansion in terms of powers of A, f (A) = c0 1 + c1 A + c2 A2 + . . . and the fact that [An , A] = An A − AAn = 0. ==========================================================

Exercise: Use the above relations to show some or all examples below of commutators [ℓi , ℓj ] = i~ ǫijk ℓk

and [ℓ2 , ℓi ] = 0,

[ℓi , rj ] = i~ ǫijk rk and [ℓi , pj ] = i~ ǫijk pk , [ℓi , r 2 ] = 0, [ℓi , p2 ] = 0 and [ℓi , p · r] = 0,

[pi , r 2 ] = −2 i~ ri ,

[ri , p2 ] = 2 i~ pi ,

and [pi , V (r)] = −i~ ∇i V.

========================================================== 14

15

Methods of Quantum Mechanics

Quantum states Within the Hilbert space, one constructs the inner product of two states. For elements |ui, |vi ∈ H the inner product is defined as the complex number hu|vi ∈ C, for which • hu|vi∗ = hv|ui, • If |ui = c1 |u1 i + c2 |u2 i then hv|ui = c1 hv|u1 i + c2 hv|u2 i, • hu|ui ≥ 0. The second property implies hu|vi = hv|ui∗ = c∗1 hv|u1 i∗ + c∗2 hv|u2 i∗ = c∗1 hu1 |vi + c∗2 hu2 |vi. Beside the ket-space we can also introduce the dual bra-space, H ∗ = {hu|}, which is anti-linear meaning that |ui = c1 |u1 i + c2 |u2 i ←→ hu| = c∗1 hu1 | + c∗2 hu2 |. The scalar product is constructed from a bra-vector and a ket-vector (”bra(c)ket”). Frequently used in quantum mechanics is the Hilbert space of square integrable functions on the real axis (for one dimension) or on R3 (in three dimensions), with the scalar product Z ∞ hφ1 |φ2 i ≡ dx φ∗1 (x)φ2 (x). (4.2) −∞

It is straightforward to check all properties of a scalar product. With the help of the inner product, we can define normalized states, hu|ui = 1 and orthogonal states satisfying hu|vi = 0. In a linear vector space an orthonormal basis {|u1 i, |u2 i, . . .} with hum |un i = δmn can be constructed, in which every state can be expanded, We have X |ui = cn |un i with cn = hun |ui (4.3) n

and we can write

|ui = Note that hu|ui = 1 implies

P

n

X

2

n

 c1   |un i hun |ui ≡  c2  . | {z } .. . c 

(4.4)

n

|cn | = 1, hence the name probability amplitude for cn .

Operators and their expectation values The link to measured quantities in quantum mechanics involves the matrix elements of (usually linear) operators hu|A|vi. which is referred to as the matrix element of A in states |ui and |vi. It is the inner product of the states A|vi and |ui. If |ui = |vi we call this the expectation value of A, if |ui 6= |vi we refer to it as transition P matrix element. P Using an orthonormal basis of states, one can write the operators as matrices, If |ui = n cn |un i = n |un ihun |ui then we can write for A|ui X A|ui = A|un ihun |ui n

=

X

m,n

=

|um i hum |A|un i hun |ui = | {z } | {z } Amn

cn

X m

|um i 

X

A11  A21  .. .

Amn cn

n

A12 A22 .. .

!

  c1 ...   ...    c2  . .. .

(4.5)

16


With |vi =

P

n

dn |un i, matrix element of A is given by hv|A|ui

=

X

m,n

=

hv|um i hum |A|un i hun |ui | {z } | {z } | {z } cn

Amn

d∗ m



A11  A21 ∗ ∗ (d1 d2 . . .)  .. .

A12 A22 .. .

  ... c1   ...    c2  .. .

(4.6)

The matrix elements between basis states thus are precisely the entries in the matrix, of which each column gives the image of a basis state. The unit operator acts as I|ui = |ui and can with the help of a complete orthornormal basis {|un i} be written as X I= |un ihun |, (4.7) n

directly following from Eq. 4.4 and known as completeness relation

4.2

Observables in quantum mechanics

One of the postulates of quantum mechanics is on the measurements of observables in quantum mechanics. Measurements yield as results the (real) eigenvalues of the (hermitean) operators corresponding to a particular observable. For an operator A we write A|an i = an |an i,

(4.8)

Here the |an i is referred to as the eigenstate belonging to the eigenvalue an . After the measurement (yielding e.g. a1 the system is in the eigenstate |a1 i belonging to that eigenvalue (or a linear combination of states if there is degeneracy, i.e. if there are more states with the same eigenvalue). This postulate has wide implications. As mentioned already we usually want operators with real eigenvalues, which implies hermitean operators. We will further study these operators. The collection of eigenvalues {an } is referred to as the spectrum of an operator. It can bediscrete or continuous. For example for the momentum operator pˆx the eigenfunctions are φk (x) = exp(i kx) with eigenvalues ~k taking any real value. For the angular momentum operator ℓˆz the eigenfunctions are φm (ϕ) = exp(i mϕ) with eigenvalues m~ with m ∈ Z.

Stationary states A special role is played by the eigenvalues and eigenstates of a time-independent Hamiltonian, Hφn = En φn .

(4.9)

The eigenvalues define the energy spectrum of the system. The eigenstates are referred to as stationary states, We can use the fact that the Hamiltonian also describes the time evolution, H = i~ ∂/∂t to obtain the time-dependent solution, ψn (x, t) = φn e−i En t/~ , (4.10) Depending on the starting point at time t = 0 a non-disturbed system evolves as a stationary state or a superposition of stationary states.

17


Hermitean operators The Hamiltonian and many other operators (like position and momentum operators) used in quantum mechanics are hermitean operators. Besides real eigenvalues, the eigenstates form a complete set of eigenstates. To formalize some of the aspects of hermitean operators, first look at the adjoint operator A† , which is defined by giving its matrix elements in terms of those of the operator A, hu|A† |vi ≡ hv|A|ui∗ = hv|Aui∗ = hAu|vi.

(4.11)

Thus note that de bra-state hAu| = hu|A† . In matrix language one thus has that A† = AT ∗ . An operator A is hermitean when expectation values are real, thus hu|Aui = hu|Aui∗ = hAu|ui.

(4.12)

By applying this to a state c1 |ui + c2 |vi with arbitrary coefficients one sees that for a linear operator this definition is equivalent with hu|Avi = hAu|vi which can be also written as hu|A|vi = hu|A† |vi, thus A = A† , i.e. A is self-adjoint. Now consider the eigenstates |an i of a hermitean operator A, i.e. A|an i = an |an i. For the eigenvalues (an ) and eigenstates (|an i), of a hermitean operator we have some important properties, • Normalizing the eigenstates, one sees that han |A|an i = an are the (real) eigenvalues. • Eigenstates corresponding with nondegenerate eigenvalues are orthogonal, If A|an i = an |an i and A|am i = am |am i and am 6= an then ham |an i = 0. If eigenvalues are degenerate, we can construct orthogonal eigenstates within the subspace of degenerate eigenstates using straightforward orthogonalization known from linear algebra. • Thus, eigenstates can be choosen as an orthonormal basis, ham |an i = δmn . Using this basis A is diagonal,   a1 0 . . . X   (4.13) A= |an ian han | =  0 a2 . . .  . . . . .. .. .. n

We can now express the expectation value of a hermitean operator as X X hu|A|ui = hu|an i an han |ui = an |cn |2 . | {z } {z } | n n c∗ n

(4.14)

cn

This confirms the interpretation of the expectation value of an operator in a state |ui as the average outcome of measuring the observable A. The quantity |cn |2 is the P probability to find the state |an i and obtain the result an in a measurement. Schematically for |ui = n cn |an i state before → measurement → outcome and probability → state after.  .    → .. → an Prob = |cn |2 → |an i Aˆ → |ui →   .  → ..

In general the resulting state |an i after the measurement is not an eigenstate of H, and one must re-expand it in energy eigenstates to find its further time evolution.

18


Unitary operators Useful operators in quantum mechanics are unitary operators. The definition states that an operator U is unitary when U −1 = U † , or U U † = U † U = I. It is easy to prove that a unitary operator conserves scalar products, hU v|U wi = hv|wi (4.15) With a unitary matrix we can transform an orthonormal basis {|ui i} in another such basis {U |ui i}. As a matrix the states U |ui i are the columns of a matrix, which are normalized and orthogonal to each other. Since also U −1 = U † is unitary, also the rows are normalized and orthogonal to each other. In general, we can diagonalize a hermitean matrix with unitary matrices, X hui |A|uj i = hui |an ian han |uj i or A = S Adiag S † . (4.16) n

where S is the matrix that has the eigenstates |an i as columns and Adiag is the matrix with eigenvalues on the diagonal (Eq. 4.13).

Coordinate and momentum representation The formal treatment of coordinate and momentum operators along the lines above, also provides the link between wave quantum mechanics (Schrödinger) and matrix quantum mechanics (Heisenberg). With the Dirac notation at hand, we can formalize some issues on wave mechanics. We have used |ψi and ψ(r) ˆ|ri = r|ri one writes operators more or less interchangeable. Using position eigenstates |ri satisfying r Z Z I = d 3 r |rihr| and rˆ = d 3 r |rirhr|. (4.17) and expands any state as |ψi =

Z

3

d r |rihr|ψi ≡

Z

d 3 r |ri ψ(r)

(4.18)

in which ψ(r) ∈ C is the coordinate space wave function. The form of the unit operator also fixes the normalization, hr|r ′ i = δ 3 (r − r ′ ), (4.19) which is also the coordinate space wave function for |r ′ i.

========================================================== Exercise: Check that rˆψ(r) = r ψ(r) with the above definition of ψ(r). ========================================================== Exercise: Similarly use the unit operator to express hψ|ψi and hφ|psii in terms of ψ(r) and φ(r). ========================================================== In coordinate representation the momentum operator is given by Z ˆ = d 3 r |ri (−i~∇) hr|, p

(4.20)

which can be viewed as the implimentation of [ˆ ri , pˆj ] = i~ δij . The eigenfunctions denoted as |pi are hr|pi =

√ ρ exp (i p · r/~)

(4.21)

19


This defines ρ, e.g. in a box ρ = 1/L3 , which is in principle arbitrary. Quite common choices for the normalization of plane waves are ρ = 1 or ρ = (2π~)−3 (non-relativistic) or ρ = 2E (relativistic). The quantity ρ also appears in the normalization of the momentum eigenstates, hp|p′ i = ρ (2π~)3 δ 3 (p − p′ ).

(4.22)

ˆ |pi = p|pi one has with the above normalization, Using for momentum eigenstates p Z Z d 3p d 3p ˆ |pihp| and p = |piphp|. I= 3 (2π~) ρ (2π~)3 ρ

(4.23)

The expansion of a state |ψi =

Z

d 3p |pi hp|ψi ≡ (2π~)3 ρ

Z

d 3p ˜ |pi ψ(p) (2π~)3 ρ

(4.24)

defines the momentum space wave function, which is the Fourier transform of the coordinate space wave function.

4.3

Compatibility of operators and commutators

Two operators A and B are compatible if they have a common (complete) orthonormal set of eigenfunctions. For compatible operators we know after a measurement of A followed by a measurement of B both eigenvalues and we can confirm this by performing again a measurement of A. Suppose we have a complete common set ψabr , labeled by the eigenvalues of A, B and possibly an index r inP case of degeneracy. Thus A ψabr = a ψabr and B ψabr = b ψabr . Suppose we have an arbitrary state ψ = abr cabr ψabr , then we see that measurements of A and B or those in reverse order yield similar results, X X cabk ψabk → B → b → cabk ψabk , ψ→ A →a→ ψ→ B →b→

b,k

k

X

X

a,k

cabk ψabk → A → a →

cabk ψabk .

k

Compatible operators can be readily identified using the following theorem: A and B are compatible ⇐⇒ [A, B] = 0.

(4.25)

Proof (⇒): There exists a complete common set ψn of eigenfunctions for which one thus has [A, B]ψn = (AB − BA)ψn = (an bn − bn an )ψn = 0. Proof (⇐): Suppose ψa eigenfunction of A. Then A(Bψa ) = ABψa = BAψa = B aψa = a Bψa . Thus Bψa is also an eigenfunction of A. Then one can distinguish (i) If a is nondegenerate, then Bψa ∝ ψa , say Bψa = bψa which implies that ψa is also an eigenfunction of B. (ii) If a is degenerate (degeneracy s), consider that part of the Hilbert space that is spanned by the functions ψar (r = 1, . . . , s). For a given ψap (eigenfunction of A) Bψap also can be written in terms of the ψar . Thus we have an hermitean operator B in the subspace of the functions ψar . In this subspace B can be diagonalized, and we can use the eigenvalues b1 , . . . bs as second label, which leads to a common set of eigenfunctions. We have seen the case of degeneracy for the spherical harmonics. The operators ℓ2 and ℓz commute and the spherical harmonics Ymℓ (θ, ϕ) are the common set of eigenfunctions.

20


The uncertainty relations For measurements of observables A and B (operators) we have [A, B] = 0 ⇒ A and B have common set of eigenstates. For these common eigenstates one has ∆A = ∆B = 0 and hence ∆A ∆B = 0. [A, B] 6= 0 ⇒ A and B are not simultaneously measurable. The product of the standard deviations is bounded, depending on the commutator. The precise bound on the product is given by ∆A ∆B ≥

1 |h [A, B] i|, 2

(4.26)

known as the uncertainty relation. The proof of the uncertainty relation is in essence a triangle relation for inner products. Define for two hermitean operators A and B, the (also hermitean) operators α = A − hAi and β = B − hBi. We have [α, β] = [A, B]. Using positivity for any state, in particular |(α + iλ β)ψi with λ arbitrary, one has 0 ≤ h(α + iλ β)ψ|(α + iλ β)ψi = hψ|(α − iλ β)(α + iλ β)|ψi = hα2 i + λ2 hβ 2 i + λ h i[α, β] i . | {z } hγi

Since i [α, β] is a hermitean operator (why), hγi is real. Positivity of the quadratic equation hα2 i + λ2 hβ 2 i + λ hγi ≥ 0 for all λ gives 4 hα2 i hβ 2 i ≥ hγi2 . Taking the square root then gives the desired result. The most well-known example of the uncertainty relation is the one originating from the noncompatibility of positionn and momentum operator, specifically from [x, px ] = i~ one gets ∆x ∆px ≥

1 ~. 2

(4.27)

Constants of motion The time-dependence of an expectation value and the correspondence with classical mechanics has been used to find the candidate momentum operator (see Eq. 1.26). We now take another look at this and write ∗ Z Z dhAi d ∂ψ ∂A ∂ψ = Aψ + ψ ∗ A + ψ∗ ψ d3 r ψ ∗ (r, t)Aψ(r, t) = d3 r dt dt ∂t ∂t ∂t ∂ 1 h [A, H] i + h Ai. (4.28) = i~ ∂t Examples of this relation are the Ehrenfest relations d hri = dt d hpi = dt

1 h[r, H]i = i~ 1 h[p, H]i = i~

1 1 hpi h[r, p2 ]i = , i~ 2m m 1 h[p, V (r)]i = h−∇V (r)i. i~

(4.29) (4.30)

An hermitean operator A that is compatible with the Hamiltonian, i.e. [A, H] = 0 and that does not have explicit time dependence, i.e. ∂A ∂t = 0 is referred to as a constant of motion. From Eq. 4.28 it is clear that its expectation value is time-independent.

21


========================================================== Exercise: Check the following examples of constants of motion, compatible with the Hamiltonian and thus providing eigenvalues that can be used to label eigenfunctions of the Hamiltonian. (a) The hamiltonian for a free particle H=

p2 2M

Compatible set: H, p

The plane waves φk (r) = exp(i k · r) form a common set of eigenfunctions. (b) The hamiltonian with a central potential: H=

p2 + V (|r|) 2M

Compatible set: H, ℓ2 , ℓz

This allows writing the eigenfunctions of this hamiltonian as φnℓm (r) = (u(r)/r) Ymℓ (θ, ϕ). ==========================================================

Chapter 5

Symmetries In this chapter, we want to complete the circle by discussing symmetries in all of their glory, in particular the space-time symmetries, translations, rotations and boosts. They are important, because they change time, coordinates and momenta, but in such a way that they leave basic quantities invariant (like Lagrangian and/or Hamiltonian) or at least they leave them in essence invariant (up to an irrelevant change that can be expressed as a total time derivative). Furthermore, they modify coordinates and momenta in such a way that these retain their significance as canonical variables. We first consider the full set of (non-relativistic) space-time transformations known as the Galilean group. These include ten transformations, each of them governed by a (real) parameter. They are one time translation, three space translations (one for each direction in space), three rotations (one for each plane in space) and three boosts (one for each direction in space). They change the coordinates t r r r

→

t′ = t + τ,

→

→ →

one time translation,

(5.1)

r =r+a r ′ = R(ˆ n, α)r

three translations, three rotations,

(5.2) (5.3)

r ′ = r − ut

three boosts.

(5.4)

′

with parameters being (τ , a, αˆ n, u). The Lagrangian and Hamiltonian that respect this symmetry are the ones for a free particle or in the case of a many particle system, those for the center of mass system, L=

1 mr˙ 2 2

and H =

p2 . 2m

˙ with p = mr.

5.1

Space translations

Let’s look at the translations T (a) in space. It is clear what is happening with positions and momenta, r → r ′ = r + a and p → p′ = p.

(5.5)

δr = δa and δp = 0.

(5.6)

and infinitesimally, This is through Noether’s theorem consistent with ’conserved quantities’ Q·δa = p·δa, thus the momenta p, which also generate the translations (see Eq. 3.27), δxi = {xi , p}P ·δa = δai

and δpi = −{pi , p}P ·δa = 0. 22

(5.7)

23

Symmetries

Translations in Hilbert space Let’s start with the Hilbert space of functions and look at ways to ’translate a function’ and then at ways to ’translate an operator’. Let us work in one dimension. For continuous transformations, it turns out to be extremely useful to look at the infinitesimal problem (in general true for so-called Lie transformations). We get for small a a ’shifted’ function i dφ + . . . = 1 + a px + . . . φ(x), φ′ (x) = φ(x + a) = φ(x) + a (5.8) dx ~ | {z } U(a)

which defines the shift operator U (a) of which the momentum operator pˆx = −i~ (d/dx) is referred to as the generator. One can extend the above to higher orders, φ′ (x) = φ(x + a) = φ(x) + a

1 d d φ + a2 2 φ + . . . , dx 2! dx

Using the (operator) definition

1 2 A + ..., 2! i i U (a) = exp + a pˆx = I + a pˆx + . . . . ~ ~ eA ≡ 1 + A +

one finds

(5.9)

In general, if G is a hermitean operator (G† = G), then eiλG is a unitary operator (U −1 = U † ). Thus the shift operator produces new wavefunctions, preserving orthonormality. To see how a translation affects an operator, we look at Oφ, which is a function changing as −1 U φ (x), (Oφ)′ (x) = Oφ(x + a) = U Oφ(x) = U | O{zU } |{z}

(5.10)

O −→ O′ = U O U −1 = eiλG O e−iλG .

(5.11)

O′

φ′

thus for operators in Hilbert space

which expanded gives O′ = (1 + iλG + . . .)O(1 − iλG + . . .) = O + iλ[G, O] + . . ., thus δO = −i[O, G]δλ.

(5.12)

One has for the translation operator O′ = ei apˆx /~ O(0) e−i apˆx /~

and δO = −(i/~) [O, pˆx ] δa,

(5.13)

i and δ pˆx = − [ˆ px , pˆx ] δa = 0. ~

(5.14)

which for position and momentum operator gives i δˆ x = − [ˆ x, pˆx ] δa = δa ~

Note that to show the full transformations for x ˆ and pˆx one can use the exact relations dO′ dO′ i = ei λG O, G e−i λG and i = O, G . dλ dλ λ=0

(5.15)

==========================================================

Exercise: Show that for the ket state one has U (a)|xi = |x − ai. An active translation of a localized state with respect to a fixed frame, thus is given by |x + ai = U −1 (a)|xi = U † (a)|xi = e−i apx /~ |xi . ==========================================================

24

Symmetries

5.2

Time evolution

Time plays a special role, both in classical mechanics and quantum mechanics. We have seen the central role of the Hamiltonian in classical mechanics (conserved energy) and the dual role as time evolution and energy operator in quantum mechanics. Actively describing time evolution, i dψ + . . . = 1 − τ H + . . . ψ(t), ψ(t + τ ) = ψ(t) + τ (5.16) dt ~ | {z } U(τ )

we see that evolution is generated by the Hamiltonian H = i~ d/dt and given by the (unitary) operator U (τ ) = exp (−i τ H/~) .

(5.17)

Since time evolution is usually our aim in solving problems, it is necessary to know the Hamiltonian, usually in terms of the positions, momenta (and spins) of the particles involved. If this Hamiltonian is time-independent, we can solve for its eigenfunctions, Hφn (x) = En φn (x) and use completeness to get the general time-dependence (see Eq. 4.10).

Schr¨ odinger and Heisenberg picture The time evolution from t0 → t of a quantum mechanical system thus is generated by the Hamiltonian, U (t, t0 ) = exp (−i(t − t0 )H/~) ,

satisfying i~

∂ U (t, t0 ) = H U (t, t0 ). ∂t

(5.18)

Two situations can be distinguished: (i) Schr¨ odinger picture, in which the operators are time-independent, AS (t) = AS and the states are time dependent, |ψS (t)i = U (t, t0 )|ψS (t0 )i, ∂ |ψS i = ∂t ∂ i~ AS ≡ ∂t

i~

H |ψS i,

(5.19)

0.

(5.20)

(ii) Heisenberg picture, in which the states are time-independent, |ψH (t)i = |ψH i, and the operators are time-dependent, AH (t) = U −1 (t, t0 ) AH (t0 ) U (t, t0 ), ∂ |ψH i ≡ ∂t ∂ i~ AH = ∂t

i~

0,

(5.21)

[AH , H].

(5.22)

We note that in the Heisenberg picture one has the equivalence with classical mechanics, because the time-dependent classical quantities are considered as time-dependent operators. In particular we have d d ˆ (t) = [r, H] and i~ p ˆ (t) = [p, H], r dt dt to be compared with the classical Hamilton equations. i~

(5.23)

========================================================== Exercise: Show that the time dependence of expectation values is the same in the two pictures, i.e. ′ hψS′ (t)|AS |ψS (t)i = hψH |AH (t)|ψH i.

This is important to get the Ehrenfest relations in Eqs 4.29 and 4.30 from the Eqs 5.23. ==========================================================

25

Symmetries

5.3

Rotational symmetry

Rotations are characterized by a rotation axis (ˆ n) and an angle (0 ≤ α ≤ 2π), r −→ r ′ = R(ˆ n, α) r where the latter refers to the polar angle around is given by   ′   x  x              = y  y′  −→      ′    z z

or

ϕ −→ ϕ′ = ϕ + α,

(5.24)

the n ˆ -direction. The rotation R(ˆ z , α) around the z-axis    cos α − sin α 0   x           y    sin α cos α 0        . 0 0 1 z

(5.25)

==========================================================

Exercise: Check that for polar coordinates (defined with respect to the z-axis), x = r sin θ cos ϕ,

y = r sin θ sin ϕ,

z = r cos θ,

the rotations around the z-axis only change the azimuthal angle, ϕ′ = ϕ + α. ==========================================================

Conserved quantities and generator in classical mechanics Using Noether’s theorem, we construct the conserved quantity for a rotation around the z-axis. One has the infinitesimal changes δx = −y δα and δy = x δα, thus Qz δα = px δx + py δy + pz δz = (xpy + ypz )δα, thus Qz = ℓz and in general for all rotations all components of the angular momentum ℓ are conserved, at least if {H, ℓ}P = 0. The angular momenta indeed generate the symmetry, leading to δx = {x, ℓz }P δα = −yδα and δy = {y, ℓz }P δα = xδα, δpx = {px , ℓz }P δα = −py δα

and δpy = {py , ℓz }P δα = px δα.

(5.26) (5.27)

with as general expressions for the Poisson brackets {ℓi , xj }P = ǫijk xk

and {ℓi , pj }P = ǫijk pk .

(5.28)

The result found for the {ℓi , ℓj }P = ǫijk ℓk bracket indicates that angular momenta also change like vectors under rotations.

Rotation operators in Hilbert space Rotations also gives rise to transformations in the Hilbert space of wave functions. Using polar coordinates and a rotation around the z-axis, we find i ∂ φ + . . . = 1 + α ℓz + . . . φ, (5.29) φ(r, θ, ϕ + α) = φ(r, θ, ϕ) + α ∂ϕ ~ | {z } U(ˆ z ,α)

from which one concludes that ℓz = −i~(∂/∂ϕ) is the generator of rotations around the z-axis in Hilbert space. As we have seen, in Cartesian coordinates this operator is ℓz = −i~(x ∂/∂y − y ∂/∂x) = xpy − ypx , the z-component of the (orbital) angular momentum operator ℓ = r × p. The full rotation operator in the Hilbert space is i i U (ˆ z , α) = exp + α ℓz = 1 + α ℓz + . . . . (5.30) ~ ~

26

Symmetries

The behavior of the various quantum operators under rotations is given by O′ = eiαℓz /~ O e−iαℓz /~

i or δO = − [O, ℓz ] δα. ~

(5.31)

Using the various commutators calculated for quantum operators in Hilbert space (starting from the basic [xi , pj ] = i~ δi j commutator, [ℓi , rj ] = i~ ǫijk rk ,

[ℓi , pj ] = i~ ǫijk pk ,

and [ℓi , ℓj ] = i~ ǫijk ℓk ,

(5.32)

(note again their full equivalence with Poisson brackets) one sees that the behavior under rotations of the quantum operators r, p and ℓ is identical to that of the classical quantities. This is true infinitesimally, but also for finite rotations one has for operators r → r′ = R(ˆ n, α)r and p → p′ = R(ˆ n, α)p. Finally the rotational invariance of the Hamiltonian corresponds to [H, ℓ] = 0 and in that case it also implies time independence of the Heisenberg operator or the the expectation values hℓi.

Generators of rotations in Euclidean space A characteristic difference between rotations and translations is the importance of the order. The order in which two consecutive translations are performed does not matter T (a) T (b) = T (b) T (a). This is also true for the Hilbert space operators U (a) U (b) = U (b) U (a). The order does matter for rotations. This is so in coordinate space as well as Hilbert space, R(ˆ x, α) R(ˆ y , β) 6= R(ˆ y , β) R(ˆ x, α) and U (ˆ x, α) U (ˆ y , β) 6= U (ˆ y , β) U (ˆ x, α). Going back to Euclidean space and looking at the infinitesimal form of rotations around the z-axis, R(ˆ z , δα) = 1 − i δα Jz

(5.33)

one also can identify here the generator  0   ˆ ) 1 ∂R(α, z   i = Jz =  −i ∂ϕ α=0  0

 −i 0    0 0   . 0 0

In the same way we can consider rotations around the x- and y-axes that are generated by     0 0 0  0 0 i            0 0 −i  0 0 0  Jx =  Jy =        , , 0 i 0 −i 0 0

(5.34)

(5.35)

The generators in Euclidean space do not commute. Rather they satisfy [Ji , Jj ] = i ǫijk Jk .

(5.36)

The same non-commutativity of generators is exhibited in Hilbert space by the commutator of the corresponding quantum operators ˆℓ/~ and by the Poisson brackets for the conserved quantities in classical mechanics. But realize an important point. Although identical, the commutation relations for ˆℓ are in Hilbert space, found starting from the basic (canonical) commutation relations between r and p operators! The consistence of commutation relations in Hilbert space with the requirements of symmetries is a prerequisite for achieving a consistent quantization of theories. ==========================================================

27

Symmetries

Exercise: For rotation operations, we have seen that the commutation relations for differential operators ℓ/~ and for Euclidean rotation matrices J are identical. It is also possible to get a representation in a matrix space for the translations. We’ll do that here for two dimensions. Embedding the two-dimensional space in a 3-dimensional one, (x, y, z) → (x, y, z, 1), the rotations and translations can be described by  cos α     sin α  Rz (α) =   0    0

− sin α cos α 0 0

0 0 1 0

0 0 0 1

       ,   

 0     0  T (a) =   0    0

0 0 0 0

0 ax 0 ay 0 az 0 1

       .   

Check this and find the generators Jz , Px , Py and Pz . The latter are found as T (δa) = 1 + i δa·P . Calculate the commutation relations between the generators in this (extended) Euclidean space. You will find [Ji , Pj ] = i ǫijk Pk (5.37) (at least for i = 3). Compare these relations with those for the (quantum mechanical) differential operators and the classical Poisson brackets. ==========================================================

5.4

Boost invariance

For the free particle lagrangian, we consider the boost transformation (going to a frame moving with velocity u), governed by real parameters u, t′ = t

and r ′ = r − u t,

and p′ = p − m u.

(5.38)

while also the Lagrangian and Hamiltonian change but with a total derivative, L′ = L +

d m r · u + 12 m u2 t . dt | {z }

(5.39)

Λ

The (classical) conserved quantities from Noether’s theorem become K = t p − m r.

(5.40)

which is conserved because one has {K, H}P = p

and

∂K = p, ∂t

thus

dK = 0. dt

(5.41)

The nature of K is seen in {Ki , Kj }P = 0,

{ℓi , Kj }P = ǫijk Kk .

(5.42)

The way it changes the coordinates, momenta is consistent with {Ki , rj }P = −t δij ,

{Ki , pj }P = m δij .

(5.43)

In the Hilbert space of quantum operators the basic commutator [ri , pj ] = i~ δij is sufficient to reproduce all the above Poisson brackets as commutators in Hilbert space, where the boost operator is given by U (u) = exp iu · K/~.

(5.44)

28

Symmetries

Explicitly we have for the quantum operators ∂K = p, thus ∂t [ℓi , Kj ] = i~ǫijk Kk .

]K, H] = i~p and [Ki , Kj ] = 0,

[Ki , rj ] = −i~ t δij ,

dhKi = 0, dt

[Ki , pj ] = i~ m δij .

(5.45) (5.46) (5.47)

Finally if one implements the transformations in space-time, e.g. writing them as matrices as done for rotations and translations, one gets a set of ten generators of the Gallilei transformations, which are denoted as H (time translation generator), P (three generators of translations), J (three generators of rotations) and K (three generators for boosts). They satisfy [Pi , Pj ] = [Pi , H] = [Ji , H] = 0, [Ji , Jj ] = i ǫijk Jk , [Ji , Pj ] = i ǫijk Pk , [Ji , Kj ] = i ǫijk Kk , [Ki , H] = i Pi , [Ki , Kj ] = 0, [Ki , Pj ] = i m δij .

(5.48)

This structure is indeed realized in the description of the quantum world, by implementing the above commutator structure of the symmetry group (Lie algebra of generators of the Gallilei group) by the commutator structure in the Hilbert space or the Poisson bracket structure in the phase space of classical mechanics. To be precise [ˆ u, vˆ] i{u, v}P ⇐⇒ . (5.49) ~ are homomorphic to the commutation relations of the generators in the symmetry group, while the correspondence between Poisson brackets and quantum commutation relations is further extended to coordinates and momenta.

5.5

The two-particle system

We ended the previous section with the commutation relations that should be valid for a free-moving system to which the invariance under Gallilei transformations applies. It is easy to check that for a single (free) particle the classically conserved quantities and the quantum mechanical set of operators H = mc2 +

p2 , 2m

P = p, J =ℓ+s=r×p+s K = mr − tp.

(5.50) (5.51) (5.52) (5.53)

satisfy the required Poisson brackets in classical phase space and the required commutation relations in Hilbert space, in the latter case starting with the canonical commutation relations [ri , pj ] = i~ δij . This is true even if we allow for a set of spin operators [si , sj ] = i~ ǫijk sk as long as these satisfy [ri , sj ] = [pi , sj ] = 0. The latter two commutators imply that spin decouples from the spatial part of the wave function. In classical language one would phrase this as that the spins do not depend on position or velocity/momentum. A constant contribution to the energy doesn’t matter either. Upon adding a potential V (r) to the Hamiltonian, the symmetry requirements would fail and we do not have Gallilei invariance. A potential (e.g. centered around an origin) breaks translation invariance, the specific r-dependence might break rotational invariance, etc. As we have seen, for two particles it is convenient to change to CM and relative coordinates R and r with dorresponding conjugate momenta P and p. and introduce the sum mass M and the reduced

29

Symmetries

mass m. The center of mass system should reflect again a free particle. On the other hand, the behavior under Gallilei transformations, implies applying the transformation to both coordinates. The sum of the generators is given by H = H1 + H2 =

P2 + Hint , 2M

P = p1 + p2 , J = J 1 + J 2 = ℓ1 + s1 + ℓ2 + s2 = R × P + S, K = K1 + K2 = M R − t P ,

(5.54)

with S = r × p + s1 + s2 , p2 + V (r, p, s1 , s2 ), Hint = M c2 + 2m

(5.55) (5.56)

only involving relative coordinates or spins (commuting with CM operators). These center of mass generators then satisfy the classical Poisson brackets or quantum commutation relations for the Gallilei group, starting simply from the canonical relations for each of the particles. The CM system behaves as a free (composite) system with constant energy and momentum and a spin determined by the ’relative’ orbital angular momentum and the spins of the constituents. The example also shows that even without spins of the constituents (s1 = s2 = 0) a composite system has an intrinsic angular momentum showing up as its spin. ========================================================== Exercise: Check that the canonical Poisson brackets or commutation relations for r1 and p1 and those for r2 and p2 imply the canonical commutation relations for R and P as well as for r and p. The explicit check of this for momenta expressed as time derivatives in classical mechanics or as derivative operators was what we did in Chapter 1, e.g. when proving Eqs 1.36 and 1.37. ==========================================================

5.6

Discrete symmetries

Three important discrete symmetries that we will be discuss are space inversion, time reversal and (complex) conjugation.

Space inversion and Parity Starting with space inversion operation, we consider its implication for coordinates, r −→ −r

and

t −→ t,

(5.57)

implying for instance that classically for p = mr˙ and ℓ = r × p one has p −→ −p

and

ℓ −→ ℓ.

(5.58)

The same is true for the explicit quantummechanical operators, e.g. p = −i~ ∇. In quantum mechanics the states |ψi correspond (in coordinate representation) with functions ψ(r, t). In the configuration space we know the result of inversion, r → −r and t → t, in the case of more

30

Symmetries

particles generalized to ri → −r i and t → t. What is happening in the Hilbert space of wave functions. We can just define the action on functions, ψ → ψ ′ ≡ P ψ in such a way that P φ(r) ≡ φ(−r).

(5.59)

The function P φ is a new wave function obtained by the action of the parity operator P . It is a hermitian operator (convince yourself). The eigenvalues and eigenfunctions of the parity operator, P φπ (r) = π φπ (r),

(5.60)

are π = ±1, both eigenvalues infinitely degenerate. The eigenfunctions corresponding to π = +1 are the even functions, those corresponding to π = −1 are the odd functions. ========================================================== Exercise: Proof that the eigenvalues of P are π = ±1. Although this looks evident, think carefully about the proof, which requires comparing P 2 φ using Eqs 5.59 and 5.60. ========================================================== The action of parity on the operators is as for any operator in the Hilbert space given by −1 Aφ −→ P Aφ = |P AP Pφ , {z } |{z} A′

thus

A −→ P AP −1 .

(5.61)

φ′

(Note that for the parity operator actually P −1 = P = P † ). Examples are r −→ P rP −1 = −ˆ r, p −→ P pP −1 = −ˆ p, −1 ℓ −→ P ℓP = +ˆℓ,

H(r, p) −→ P H(r, p)P

(5.62) (5.63) −1

= H(−r, −p).

(5.64) (5.65)

If H is invariant under inversion, one has P HP −1 = H

⇐⇒

[P, H] = 0.

(5.66)

This implies that eigenfunctions of H are also eigenfunctions of P , i.e. they are even or odd. Although P does not commute with r or p (classical quantities are not invariant), the specific behavior P OP −1 = −O often also is very useful, e.g. in discussing selection rules. The operators are referred to as P -odd operators. ========================================================== Exercise: Show that the parity operation leaves ℓ invariant and show that the parity operator commutes with ℓ2 and ℓz . The eigenfunctions of the latter operators (spherical harmonics) indeed are eigenfunctions of P . What is the parity of the Yℓm ’s. ========================================================== Exercise: Show that for a P -even operator (satisfying P OP −1 = +O or [P, O] = P O − OP = 0) the transition probability Probα→β = |hβ|O|αi|2 for parity eigenstates is only nonzero if πα = πβ . What is the selection rule for a P -odd operator (satisfying P OP −1 = −O or {P, O} = P O + OP = 0).

==========================================================

31

Symmetries

Time reversal In classical mechanics with second order differential equations, one has for time-independent forces automatically time reversal invariance, i.e. invariance under t → −t and r → r. There seems an inconsistency with quantum mechanics for the momentum p and energy E. Classically it equals mr˙ which changes sign, while ∇ → ∇. Similarly one has classically E → E, while H = i~(∂/∂t) appears to change sign. The problem can be solved by requiring time reversal to be accompagnied by a complex conjugation, in which case one consistently has p = −i~∇ → i~∇ = −p and H → H. Furthermore a stationary state ψ(t) ∼ exp(−iEt) now nicely remains invariant, ψ ∗ (−t) = ψ(t). Such a consistent description of the time reversal operator in Hilbert space is straightforward. For unitary operators one has (mathematically) also the anti-linear option, where an anti-linear operator satisfies T (c1 |φ1 i + c2 |φ2 i) = c∗1 T |φ1 i + c∗2 T |φ2 i. It is easily implemented as T |φi = hT φ|,

(5.67)

which for matrix elements implies hφ|ψi = hφ|T † T |ψi = hT ψ|T φi = hT φ|T ψi∗ hφ|A|ψi = hφ|T † T A T † T |ψi = hT φ|T A T † |T ψi∗ .

(5.68) (5.69)

Operators satisfying T T † = T † T = 1, but swapping bra and ket space (being anti-linear) are known as anti-unitary operators. Together with conjugation C, which for spinless systems is just complex conjugation, one can look at CP T -invariance by combining the here discussed discrete symmetries. For all known interactions in the world the combined CP T transformation appears to be a good symmetry. The separate discrete symmetries are violated, however, e.g. space inversion is broken by the weak force that causes decays of elementary particles with clear left-right asymmetries. Also T and CP have been found to be broken.

5.7

A relativistic extension∗

The symmetry group for relativistic systems is the Poincaré group, with as essential difference that boosts (with parameter u still representing a moving frame, are (for a boost in the x-direction given by ct′ = γ ct − βγ x ′

x = x − βγ ct,

(5.70) (5.71)

while y ′ = y and z ′ = z. The quantities β and γ are given by β=

u c

1 1 and γ = p . = p 2 1 − u2 /c2 1−β

(5.72)

These transformations guarantee that the velocity of light is constant in any frame, satisfying c 2 t2 − r 2 = c 2 t′ 2 − u ′ 2 = c 2 τ 2 ,

(5.73)

where τ is referred to as eigentime (time in rest-frame of system). Mathematically one can look at this structure as we did for rotations and find the commutation relations of the Poincaré group [P i , P j ] = [P i , H] = [J i , H] = 0, [J i , J j ] = i ǫijk J k , [J i , P j ] = i ǫijk P k , [J i , K j ] = i ǫijk K k , [K i , H] = i P i , [K i , K j ] = −i ǫijk J k /c2 , [K i , P j ] = i δ ij H/c2 .

(5.74)

32

Symmetries

The action of a relativistic free particle is actually extremely simple, Z Z p S = mc2 dτ = mc2 dt 1 − v 2 /c2 .

One can find the Lagrangian and Hamiltonian, p L = mc2 1 − v 2 /c2

and H =

p m2 c4 + p 2 c2 ,

(5.75)

(5.76)

where we have p = mv γ. Note that for E = mc2 dt/dτ and p = mc dr/dτ , so (E/c, p) transform among themselves exactly as (ct, r). For a single free particle or for the CM coordinates, the classically conserved quantities or generators of the Poincaré group can be found using Noether’s theorem. They are p p 2 c2 + m2 c4 , H = P = p, J K

= r × p + s, p×s 1 (rH + Hr) − tp + . = 2c2 H + mc2

(5.77)

A consistent relativistic quantum mechanical treatment in terms of relative coordinates, however, requires great care [See e.g. L.L. Foldy, Phys. Rev. 122 (1961) 275 and H. Osborn, Phys. Rev. 176 (1968) 1514] and is possible in an expansion in 1/c. Interaction terms, however, enter not only in the Hamiltonian, but also in the boost operators.