Uncertainty relations for angular momentum eigenstates in two and

Uncertainty relations for angular momentum eigenstates in two and three spatial dimensions Christian Brachera兲 Physics Program, Bard College, Annandale-on-Hudson, New York 12504

共Received 17 August 2009; accepted 11 November 2010兲 I reexamine Heisenberg’s uncertainty relation for two- and three-dimensional wave packets with fixed angular momentum quantum numbers m or ᐉ. A simple proof shows that the product of the average extent ⌬r and ⌬p of a two-dimensional wave packet in position and momentum space is bounded from below by ⌬r⌬p ⱖ ប共兩m兩 + 1兲. The minimum uncertainty is attained by modified Gaussian wave packets that are special eigenstates of the two-dimensional isotropic harmonic oscillator, which include the ground states of electrons in a uniform magnetic field. Similarly, the inequality ⌬r⌬p ⱖ ប共ᐉ + 3 / 2兲 holds for three-dimensional wave packets with fixed total angular momentum ᐉ and the equality holds for a Gaussian radial profile. I also discuss some applications of these uncertainty relations. © 2011 American Association of Physics Teachers. 关DOI: 10.1119/1.3534840兴 I. INTRODUCTION A widely known property of wave functions is the complementarity of their position and momentum information content. The Heisenberg uncertainty relation states that the product of the respective uncertainties ⌬x and ⌬px is not less than the minimum value ប / 2 and the equality occurs only for special Gaussian “minimum uncertainty wave packets.” An elegant proof of this statement1,2 is based on the Cauchy– Schwarz inequality, a general property of Hilbert space. In this proof, one first derives an inequality for a pair of operators Aˆ and Bˆ, acting on the state 兩⌿典, 具Aˆ⌿兩Aˆ⌿典具Bˆ⌿兩Bˆ⌿典 ⱖ 兩具Aˆ⌿兩Bˆ⌿典兩2 .

共1兲

The equality occurs only when Aˆ兩⌿典 and Bˆ兩⌿典 are linearly dependent. If Aˆ and Bˆ are self-adjoint operators representing observables and 关Aˆ , Bˆ兴 denotes their commutator, then the additional inequality 兩具Aˆ⌿兩Bˆ⌿典兩2 ⱖ 41 兩具⌿兩关Aˆ,Bˆ兴兩⌿典兩2

共2兲

must hold. Heisenberg’s relation linking the uncertainties in a measurement of the position and momentum component along a single direction in space follows by setting Aˆ = xˆ − 具x典 and Bˆ = pˆx − 具px典. Based on these ideas, I will establish a multidimensional variant of the uncertainty relation for states with a welldefined orbital symmetry. I will extract the wave packets with minimum position-momentum uncertainty and establish a lower bound for the product ⌬r⌬p. Angular momentum eigenstates are centered on the origin of the coordinate system and thus it is reasonable to define the uncertainties as the widths ⌬r = 冑具r2典 and ⌬p = 冑具p2典 of the probability distribution in position and momentum space, a convention I will adopt throughout this paper.3 Two-dimensional states with rotational symmetry may play an important role in the heuristic description of tunneling in three-dimensional systems, where the tunneling particles are expected to follow an “escape path” of minimum classical action as closely as possible in both position and momentum space, a condition that favors minimum uncertainty in the two-dimensional lateral profile of the tunneling 313

Am. J. Phys. 79 共3兲, March 2011

http://aapt.org/ajp

wave function.4–7 Examples include the current flow in the vicinity of a scanning tunneling microscope tip7–9 and belowthreshold field ionization of atoms.10 In the latter case, the electronic angular momentum in the direction of the electric field is conserved and the dependence of tunneling on the value of the magnetic quantum number m is of interest. It will be shown that the minimum uncertainty states represent a set of degenerate ground states of a charge confined to a two-dimensional layer in the presence of a perpendicular, uniform magnetic field 共the Hall configuration兲 and collectively span the lowest Landau level of the system.11–14 The analysis can be extended to three-dimensional wave packets with fixed total angular momentum quantum number ᐉ. As a simple application, I obtain a lower energy bound for the large ᐉ “whispering gallery” eigenmodes15,16 of a spherically confined particle wave. I also use the bound states of the Coulomb problem to illustrate the validity of the generalized uncertainty relation. The minimum uncertainty wave functions within a given angular momentum subspace are identified by modifying the canonical scheme I have sketched above. Because this approach has not been published, I will discuss the technique and give some examples. 共For a very different analysis of position-momentum uncertainty in more than one dimension, see Ref. 17.兲

II. MINIMUM UNCERTAINTY WAVE PACKETS IN TWO DIMENSIONS It is convenient to start with an analysis of the twodimensional problem and consider eigenstates ⌿ of the angular momentum operator Lˆz = xˆ pˆy − yˆ pˆx. In polar coordinates, these functions may be separated into a radial and an angular part characterized by the quantum number m: ⌿共r , ␾兲 = ␺共r兲exp共im␾兲. As an immediate consequence, the mean positions 具x典 and 具y典 and the corresponding average momenta 具px典 and 具py典 all vanish. Hence, the uncertainties of position and momentum are given by the variances 具r2典 = 具x2 + y 2典 and 具p2典 = 具p2x + p2y 典. To introduce the formalism required to determine these expectation values, consider the scalar product 具⌿ 兩 ⌿典. In the © 2011 American Association of Physics Teachers

313

conventional metric space of two-dimensional, squareintegrable functions, this scalar product is given by the integral 具⌿兩⌿典 =

冕冕

dxdy兩⌿共x,y兲兩2 = 2␲

冕

⬁

drr兩␺共r兲兩2 .

共3兲

0

Thanks to the cylindrical symmetry inherent in ⌿共r , ␾兲, the scalar product 具⌿ 兩 ⌿典 depends on the characteristics of the radial function ␺共r兲 only. This dependence suggests an alternative approach. Instead of working in the original space of two-dimensional wave functions, I introduce a simpler “radial norm” 储␹储r2␾ and associated inner product in the one-dimensional metric space of square-integrable radial functions that vanish for r = 0 and ␹共r = 0兲 = 0, defined as 储␹储r2␾ = 具␹兩␹典r␾ =

冕

⬁

dr兩␹共r兲兩2 .

共4兲

0

By comparison, the original, physically relevant norm 具⌿ 兩 ⌿典 is equivalent to the expectation value of an operator rˆ in the space of radial functions, defined as multiplication of the integrand by the radial distance r 具⌿兩⌿典 = 2␲具␺兩rˆ兩␺典r␾ = 2␲储rˆ1/2␺储r2␾ .

具⌿兩r 兩⌿典 = 2␲具␺兩r 兩␺典r␾ = 2␲储r ˆ3

ˆ 3/2

␺储r2␾ .

共6兲

Equation 共6兲 suggests the definition of a 共self-adjoint兲 “radial distance operator” Rˆ = 共2␲兲1/2rˆ3/2 for the new norm 具⌿兩rˆ2兩⌿典 = 储Rˆ␺储r2␾ .

共7兲

To find the mean square momentum 具p2典, I start with the expression for the Laplacian operator ⵜ2 in polar coordinates18 ⵜ2 =

冉 冊

⳵ 1 ⳵ 1 ⳵2 r + 2 2. r ⳵r ⳵r r ⳵␾

共8兲

For an eigenstate of Lˆz with quantum number m, the last term may be replaced by −m2 / r2. Because pˆ = −iបⵜ, the average 具p2典 is given by 具⌿兩pˆ2兩⌿典 = 2␲ប2

冕

⬁

0

=2␲ប2具␺兩

drr␺共r兲ⴱ

再

冎

m2 1 ⳵ ⳵ r − ␺共r兲 共9a兲 r2 r ⳵ r ⳵ r

m2 ⳵ ⳵ − r 兩 ␺ 典 r␾ . r ⳵r ⳵r

共9b兲

It is convenient to transform Eq. 共9b兲 into a “norm style” expression resembling Eqs. 共3兲–共7兲. The expectation value 具p2典 is a measure of the kinetic energy of the particle and always positive. Hence, pˆ2 is a positive Hermitian operator and can be decomposed into a product of square root operators Pˆ such that pˆ2 = Pˆ+ Pˆ, where Pˆ+ and Pˆ are adjoint to each other.19,20 共Note that a proper root operator is itself self314

Am. J. Phys., Vol. 79, No. 3, March 2011

冉

兩m兩 ⳵ m2 ⳵ ⳵ − r = − 冑r − 冑r r ⳵r ⳵r ⳵r

冊冉冑

r

冊

⳵ 兩m兩 − . ⳵ r 冑r

共10兲

The two operators in brackets form an adjoint pair,24 so I may set Pˆ = 冑2␲ប

冉冑

r

冊

⳵ 兩m兩 − . ⳵ r 冑r

共11兲

With the help of Eq. 共11兲, Eq. 共9b兲 may be rewritten as a norm 具⌿兩pˆ2兩⌿典 = 储Pˆ␺储r2␾ = 2␲ប2

冐冉冑

r

冊冐

⳵ 兩m兩 − ␺ ⳵ r 冑r

2

共12兲

. r␾

An estimate of the uncertainty ⌬r⌬p follows from the Cauchy–Schwarz inequality 共1兲 for the product of the expectation values 共6兲 and 共12兲 共Refs. 1 and 2兲 具⌿兩rˆ2兩⌿典具⌿兩pˆ2兩⌿典 = 具␺兩Rˆ2兩␺典r␾具␺兩Pˆ† Pˆ兩␺典r␾ ⱖ 兩具␺兩Rˆ Pˆ兩␺典r␾兩2 .

共5兲

Because r is non-negative, the expectation value 具␺兩rˆ兩␺典r␾ can be expressed as the radial norm of the function r1/2␺共r兲. In the same spirit, I rewrite the averages 具r2典 and 具p2典 in terms of the metric 储␹储r2␾. By analogy with Eqs. 共3兲–共5兲, 具r2典 has the radial representation ˆ2

adjoint and positive, so that Pˆ+ = Pˆ must additionally hold, but this stronger requirement is not necessary for our purpose.21,22兲 The differential operator in Eq. 共9b兲 factorizes23

共13兲

The equality holds if and only if the states Rˆ兩␺典 and Pˆ兩␺典 are linearly dependent, that is, there exists some 共complex兲 constant ␭ so that Pˆ兩␺典 = ␭Rˆ兩␺典, i.e., if

再

r1/2

冎

⳵ 兩m兩 + ␭r3/2 ␺共r兲 = 0. − ⳵ r r1/2

共14兲

This separable first-order differential equation is solved by a special class of radial functions ␺共r兲 with modified Gaussian shape

␺共r兲 = ␣r兩m兩 exp共− ␭r2/2兲.

共15兲

Here, ␣ is a constant of proportionality and Re关␭兴 ⬎ 0 must hold for reasons of normalizability. For real ␭, the functions 共15兲 become the ground state of a two-dimensional harmonic oscillator in the subspaces of angular momentum eigenstates with quantum number m.11,25

III. UNCERTAINTY RELATION IN TWO DIMENSIONS For the subsequent analysis, I split the operator occurring in the expectation value in Eq. 共13兲 into a Hermitian and an anti-Hermitian contribution

冉

冊 冉

冊

⳵ ⳵ Rˆ Pˆ = 2␲ប rˆ2 − 兩m兩rˆ = 2␲ប − 共兩m兩 + 1兲rˆ + rˆ rˆ . ⳵r ⳵r 共16兲 共Some important caveats in applying this step are pointed out by Chisholm.26 Fortunately, they do not affect the problem under consideration.兲 Because rˆ is a self-adjoint operator, the expectation value of the Hermitian contribution −共兩m兩 + 1兲 ⫻具␺兩rˆ兩␺典r␾ is real and the anti-Hermitian27 part rˆ⳵rrˆ has a purely imaginary expectation value 具␺兩rˆ⳵rrˆ兩␺典r␾. Therefore Christian Bracher

314

兩具␺兩Rˆ Pˆ兩␺典r␾兩2 = 共2␲ប兲2兵共兩m兩 + 1兲2具␺兩rˆ兩␺典r2␾ + 兩具␺兩rˆ⳵rrˆ兩␺典r␾兩 其 2

共17兲

According to Eq. 共5兲, the norm of ⌿共r , ␾兲 is given by 具⌿ 兩 ⌿典 = 2␲具␺兩rˆ兩␺典r␾. Hence, the inequality 共13兲 may be rewritten as: 具⌿兩rˆ2兩⌿典具⌿兩pˆ2兩⌿典 ⱖ ប2共兩m兩 + 1兲2具⌿兩⌿典2

具⌿兩⌿典 =

=

具⌿兩r 兩⌿典具⌿兩p 兩⌿典 ⱖ ប2共兩m兩 + 1兲2 . 具r 典具p 典 = 具⌿兩⌿典具⌿兩⌿典 2

2

ˆ2

共19兲

This inequality improves the less stringent bound obtained from the standard Heisenberg uncertainty relation 共21兲

共Note that for a rotationally symmetric state, 具x2典 = 具y 2典 = 21 具r2典 and 具p2x 典 = 21 具p2典 must hold.兲 The results 共19兲 and 共21兲 coincide only in the special case m = 0, where the minimum uncertainty state is an isotropic Gaussian wave packet, thus confirming Eq. 共15兲. For states of minimum uncertainty, the equality must hold both in Eqs. 共13兲 and 共18兲. The equality in Eq. 共13兲 requires wave packets with the modified Gaussian profile defined in Eq. 共15兲. The left-hand and right-hand statements in Eq. 共18兲 coincide if the expectation value of the anti-Hermitian operator rˆ⳵rrˆ vanishes 具␺兩rˆ⳵rrˆ兩␺典r␾ =

冕

⬁

drr␺ⴱ共r兲

0

⳵ 关r␺共r兲兴 = 0. ⳵r

共22兲

This expectation value is purely imaginary and thus becomes zero for any choice of a real wave function ␺共r兲 in Eq. 共22兲. For a minimum uncertainty wave function, the parameter ␭ in Eq. 共15兲 must be chosen to be real and positive. IV. UNCERTAINTY RELATION IN THREE DIMENSIONS

共23兲

I can always assume that the orbital part Y ᐉ共␪ , ␾兲 is normalized. The norm of the wave packet 储⌿储2 can be expressed as a radial integral 315

Am. J. Phys., Vol. 79, No. 3, March 2011

冕

␲

sin ␪d␪

冕

2␲

0

0

冕

⬁

0

d␾Y ⴱᐉ共␪, ␾兲Y ᐉ共␪, ␾兲

drr2兩␺共r兲兩2 = 具␺兩rˆ2兩␺典r␪␾ = 储rˆ␺储r2␪␾ .

共24c兲

Thanks to spherical symmetry, the scalar product 具⌿ 兩 ⌿典 in three-dimensional space can be mapped to the simpler norm 储rˆ␺储r␪␾ in the space of square-integrable radial functions that vanish for r = 0. Both the center-of-mass 具r典 of the wave packet ⌿ and its average momentum vector 具p典 must be zero. 关The wave packet ⌿ has well-defined parity 共−1兲ᐉ under reflections in space, which implies that the probability distribution 兩⌿共r , ␪ , ␾兲兩2 and its center-of-mass 具r典 does not change under reflection; a similar observation holds for the momentum 具p典.兴 Hence, the average width of the wave packet in position and momentum space is given by the variances 具r2典 and 具p2典 and the task is to express them in terms of the new metric 储 . . . 储r␪␾. Finding 具r2典 is straightforward 具⌿兩rˆ2兩⌿典 = 具␺兩rˆ4兩␺典r␪␾ = 储rˆ2␺储r2␪␾ = 储Rˆ3D␺储r2␪␾ ,

共25兲

where the radial distance operator is given by Rˆ3D = rˆ2. To adapt the argument presented in Eqs. 共8兲–共13兲 to evaluate the mean square momentum 具p2典, I first express the Laplacian in spherical coordinates18 ⵜ2 =

冉

冉 冊

⳵ ⳵ 1 ⳵ 2⳵ 1 r + 2 sin ␪ 2 ⳵␪ r ⳵r ⳵r r sin ␪ ⳵ ␪ +

=

The argument presented for two-dimensional wave packets can be extended to three-dimensional space. Again, I assume that the wave packet ⌿ possesses rotational symmetry and is an eigenstate of the total angular momentum operator Lˆ2 with eigenvalue ប2ᐉ共ᐉ + 1兲, that is, has a well-defined total angular momentum quantum number ᐉ. 共No particular assumption is made about the magnetic quantum number m and thus, any superposition of different m states is permissible.28兲 In spherical coordinates, the wave function separates into a product of a radial function ␺共r兲 and a spherical harmonic function Y ᐉ共␪ , ␾兲 representing the angular dependence ⌿共r, ␪, ␾兲 = ␺共r兲Y ᐉ共␪, ␾兲.

drr2兩␺共r兲兩2

共24b兲 =

共20兲

⌬r⌬p = 冑具r2典具p2典 = 2冑具x2典具p2x 典 = 2⌬x⌬px ⱖ ប.

⬁

共18兲

Hence, the product ⌬r⌬p for a two-dimensional angular momentum eigenstate with quantum number m is bounded from below. ⌬r⌬p ⱖ ប共兩m兩 + 1兲.

冕

共24a兲

dxdydz兩⌿共x,y,z兲兩2

0

I reorder Eq. 共18兲 and immediately obtain ˆ2

冕冕冕

冊

⳵2 1 2 r sin ␪ ⳵ ␾2

共26a兲

冉 冊

共26b兲

2

1 1 ⳵ 2⳵ r − 2 2 Lˆ2 . 2 r ⳵r ⳵r បr

Because ⌿ is an eigenstate of Lˆ2 and pˆ2 = −ប2ⵜ2, the expectation value 具p2典 has the radial representation 关see Eq. 共9b兲兴 具⌿兩pˆ2兩⌿典 = ប2

冕

⬁

drr2␺共r兲ⴱ

0

再

冎

ᐉ共ᐉ + 1兲 1 ⳵ 2 ⳵ − 2 r ␺共r兲 r2 r ⳵r ⳵r 共27a兲

=ប2具␺兩ᐉ共ᐉ + 1兲 −

⳵ 2⳵ rˆ 兩␺典r␪␾ . ⳵r ⳵r

共27b兲

To rewrite this 共positive兲 expression as a radial norm, I need to identify a square root operator Pˆ3D so that 具p2典 = 具␺兩Pˆ† Pˆ 兩␺典 .22 The operator in Eq. 共27b兲 can be factor3D 3D

r␪␾

ized into a pair of adjoint operators similar to the twodimensional case23 Christian Bracher

315

ᐉ共ᐉ + 1兲 −

冉

⳵ 2⳵ ⳵ rˆ = − rˆ − ᐉ ⳵r ⳵r ⳵r

冊冉

rˆ

冊

⳵ −ᐉ , ⳵r

共28兲

which suggests the definition

⳵ Pˆ3D = rˆ − ᐉ. ⳵r

共29兲

Therefore, the equivalent to Eq. 共12兲 in three dimensions reads

冐冉

具⌿兩pˆ2兩⌿典 = 储Pˆ3D␺储r2␪␾ = ប2 rˆ

冊冐

⳵ −ᐉ ␺ ⳵r

2

共31兲

The equality holds for minimum uncertainty wave packets, for which Pˆ3D兩␺典 = ␭Rˆ3D兩␺典, where ␭ is a complex parameter

册

⳵ − ᐉ + ␭r2 ␺共r兲 = 0. ⳵r

共32兲

This differential equation is 共apart from the replacement of 兩m兩 by ᐉ兲 identical to Eq. 共14兲. Therefore, all candidates for minimum uncertainty wave packets in two and three dimensions possess the same Gaussian radial profile 共15兲

␺共r兲 = ␣rᐉ exp共− ␭r2/2兲.

共33兲

By using an argument akin to Eq. 共22兲, a straightforward proof shows that minimum uncertainty is attained for those wave packets for which ␭ is real, just as in the twodimensional case. These states may be interpreted as eigenstates of the isotropic harmonic oscillator.11,25,29 It remains to establish the lower bound for the uncertainty product ⌬r⌬p in three dimensions, following the argument given in Sec. III. In the first step, I identify the Hermitian and anti-Hermitian contributions to the operator Rˆ3D Pˆ3D in the inequality 共31兲

冉 冊

3 ⳵ ⳵ Rˆ3D Pˆ3D = rˆ3 − ᐉrˆ2 = − ᐉ + rˆ2 + rˆ3/2 rˆ3/2 . 2 ⳵r ⳵r

共34兲

The first Hermitian term in the sum is responsible for the real contribution to the expectation value 具␺兩Rˆ3D Pˆ3D兩␺典, whereas the anti-Hermitian operator rˆ3/2⳵rrˆ3/2 delivers a purely imaginary value. I evaluate the inequality 共31兲 further and obtain 具⌿兩rˆ2兩⌿典具⌿兩pˆ2兩⌿典 ⱖ ប2

再冉 冊 ᐉ+

3 2

冏 冉 冊

+ 具␺兩rˆ3/2

ⱖប2 ᐉ +

3 2

2

具␺兩rˆ2兩␺典r2␪␾

⳵ 3/2 rˆ 兩␺典r␪␾ ⳵r

冏冎 2

共35a兲

2

具⌿兩⌿典2 .

共35b兲

The average of rˆ2 in the radial norm is the same as the original norm of the wave packet 具⌿ 兩 ⌿典 关see Eq. 共24c兲兴. The latter inequality becomes an equality only if the expectation value of rˆ3/2⳵rrˆ3/2 vanishes, as it must for purely real radial wave functions ␺共r兲. 316

Am. J. Phys., Vol. 79, No. 3, March 2011

3 . 2

共36兲

As mentioned, the equality is attained only for Gaussian-type wave packets of the form 共33兲 with ␭ real. V. EXAMPLES

具⌿兩rˆ2兩⌿典具⌿兩pˆ2兩⌿典 ⱖ 兩具␺兩Rˆ3D Pˆ3D兩␺典兩2 .

r

冉 冊

⌬r⌬p ⱖ ប ᐉ +

共30兲

. r␪␾

Now, I again employ the Cauchy–Schwarz inequality as in Eq. 共13兲

冋

When expressed in terms of the averages 具r2典 and 具p2典, Eq. 共35b兲 presents a lower bound for the product of the uncertainties ⌬r = 冑具r2典 and ⌬p = 冑具p2典 for three-dimensional wave packets with rotational symmetry characterized by the orbital quantum number ᐉ

To illustrate the application of these uncertainty relations, I discuss three model problems. I first demonstrate that the lowest-energy states with definite radial symmetry of a charge in a homogeneous magnetic field are minimum uncertainty states in two dimensions.11–14 In the second example, the uncertainty products for bound states in the Coulomb problem are evaluated and are found to obey the uncertainty relation 共36兲. I then employ this relation to establish a lower bound for the eigenenergies of a particle wave with angular momentum quantum number ᐉ confined to a spherical well by infinitely high potential walls and show that the ground state energy must grow faster than 共ᐉ + 3 / 2兲2. A. Charge in a homogeneous magnetic field Consider the motion of an otherwise free particle 共charge q and mass ␮兲 in a two-dimensional layer subject to a perpendicular, uniform magnetic field of strength B, that is, the dynamics of the charge in the Hall configuration. If the magnetic field points along the z-axis, a suitable choice for the vector potential A共r兲 representing the homogeneous field is 1 B A共r兲 = 共B ⫻ r兲 = 共− y,x,0兲 2 2

共37兲

because this form treats the x- and y-coordinates on an equal footing. The standard minimal-coupling Hamiltonian for the motion of the charge becomes ˆ 共rˆ,pˆ兲 = 1 关pˆ − qA共rˆ兲兴2 H 2␮ =

1 2 q q 2B 2 2 2 共pˆx + pˆ2y 兲 − B · 共rˆ ⫻ pˆ兲 + 共xˆ + yˆ 兲. 2␮ 2␮ 8␮ 共38兲

In terms of the Larmor frequency ␻L = qB / 共2␮兲, it is apparent that the dynamics of the charge in the magnetic field is equivalent to the motion of the same particle in a twodimensional, isotropic harmonic oscillator with frequency ␻L, but with an additional angular momentum-dependent energy shift caused by the Zeeman term14 ˆ =H ˆ +H ˆ H osc Zeeman ,

共39兲

where the two partial Hamiltonians are given by 2 ˆ 共rˆ,pˆ兲 = pˆ + 1 ␮␻2 rˆ2, H osc 2␮ 2 L

ˆ ˆ H Zeeman = − ␻LLz .

共40兲

ˆ and H ˆ Because H Zeeman commute, I can find energy eigenosc states of the particle with definite angular momentum quanChristian Bracher

316

tum number m. Following the argument in Sec. II, the wave function separates in polar coordinates ⌿共r , ␾兲 = ␺共r兲exp共im␾兲. If I express the momentum operator pˆ2 = −ប2ⵜ2 in polar coordinates using Eq. 共8兲, Eq. 共40兲 shows that the radial wave function ␺共r兲 obeys the simplified Schrödinger equation

冋

E␺共r兲 = −

册

冉 冊

⳵ ប2 ⳵ m2ប2 ␮␻L2 r2 r + + − mប␻L ␺共r兲. 2 2␮r ⳵ r ⳵ r 2␮r2 共41兲 11–13

The This eigenvalue problem can be solved analytically. similarity to the quantum harmonic oscillator indicates that the eigenenergies are spaced in multiples of the characteristic frequency ប␻L, En = 共2n + 1兲ប␻L, and 共n = 0 , 1 , 2 , . . .兲. Finding a complete basis set of eigenfunctions is more involved.12,13 It is readily shown that modified Gaussian functions of the form 共15兲 m −␮␻Lr2/共2ប兲

␺m共r兲 = r e

m −qBr2/共4ប兲

=r e

共42兲

solve the Schrödinger Eq. 共41兲 for any value of the magnetic quantum number m ⱖ 0, where the corresponding energy of the particle E0 = ប␻L is independent of m. This amount represents its ground state energy in the magnetic field. The corresponding minimum uncertainty wave packets ⌿共r , ␾兲 = ␺m共r兲eim␾ form a basis for this energy subspace, corresponding to the lowest Landau level of the system.13 B. Uncertainty product for bound states of hydrogen Next, I will establish the validity of the uncertainty limit 共36兲 for all bound states of the Coulomb problem with spherical symmetry.30 For a particle with mass ␮ and charge q, the relevant quantities can be expressed in terms of a single parameter, the Bohr radius a = 4␲⑀0ប2 / 共␮q2兲. Consider the bound state ⌿nᐉ with principal quantum number n and angular momentum quantum number ᐉ.28 The energy En of such a state is given by En = −

␮q4 1 ប2 1 =− . 2 2 2 2 2␮a2 n2 32␲ ⑀0ប n

共43兲

According to the virial theorem, the average kinetic energy of the charge in the Coulomb field is half as big in magnitude as its potential energy: 具Ekin典 = − 21 具Epot典. Both contributions add up to the total energy 共43兲, so 具Ekin典 = −En. Hence, the mean square of the momentum 具p2典 is a function of the quantum number n only 具p2典 = 2␮具Ekin典 =

ប2 . n 2a 2

共44兲

The average square distance 具r2典 can be obtained from recursion relations31 and in terms of n and ᐉ I have 1 具r2典 = 关5n2 + 1 − 3ᐉ共ᐉ + 1兲兴n2a2 . 2

共45兲

Therefore, the uncertainty product for the state ⌿nᐉ is 1 具r2典具p2典 = 关5n2 + 1 − 3ᐉ共ᐉ + 1兲兴ប2 . 2

Am. J. Phys., Vol. 79, No. 3, March 2011

⌬r⌬p =

冑冉 冊 ᐉ+

3 共ᐉ + 2兲ប. 2

共46兲

共47兲

Hence, ⌬r⌬p ⬎ 共ᐉ + 3 / 2兲ប holds for all orbital wave functions ⌿nᐉ of hydrogen. 共In particular, the uncertainty product for the ground state of hydrogen is ⌿10 = 冑3ប.兲

C. Whispering gallery modes in a spherical box Finding the eigenstates and eigenenergies of a free particle with mass ␮ that is confined to an impenetrable spherical well of radius a is a common textbook problem.32 For fixed angular momentum quantum number ᐉ, the energy eigenfunctions of the particle are related to the spherical Bessel functions jᐉ共u兲 ⌿nᐉ共r, ␪, ␾兲 = jᐉ共knᐉr兲Y ᐉ共␪, ␾兲.

共48兲

Here, Y ᐉ共␪ , ␾兲 is a spherical harmonic function and the wave numbers knᐉ 共n = 0 , 1 , 2 , . . .兲 are selected by the boundary condition ⌿nᐉ = 0 at the surface of the spherical box r = a. Therefore, the allowed values of knᐉ and the corresponding eigenenergies Enᐉ are determined by the zeroes znᐉ of the spherical Bessel functions jᐉ共u兲 共Ref. 33兲 Enᐉ =

2 ប2knᐉ ប2 2 = z . 2␮ 2␮a2 nᐉ

共49兲

The difficult part is the determination of the constants znᐉ, in particular, for large values of ᐉ. The eigenstates with large ᐉ represent free-particle waves hugging the surface of the spherical box, which are analogous to the acoustical “whispering gallery modes” that propagate along the walls of circular domes.15,16,34 As a function of the quantum number ᐉ, the smallest zero z0ᐉ 共representing a wave without radial nodes兲 always slightly exceeds ᐉ,33 implying that the ground state energy E0ᐉ for a free particle in the ᐉth orbital grows a bit faster than the square of ᐉ. By using the uncertainty relation derived in Sec. IV, it is easy to set up a lower bound for Enᐉ that corroborates this observation. First, note that the particle is localized within the spherical box and therefore its average square distance 具r2典 from the center is bounded by the radius of the box 具r2典 ⬍ a2 .

共50兲

共This simple estimate is particularly good for large ᐉ, as the wave then skims along the surface of the box.兲 This inequality can be combined with the uncertainty relation 共36兲

冉 冊

具r2典具p2典 ⱖ ប2 ᐉ +

To prove that this expression is in accordance with the inequality 共36兲, note that for fixed angular momentum ᐉ, the 317

uncertainty product grows with increasing principal quantum number n. For a given orbital subspace, it thus becomes minimal for the smallest possible value n = ᐉ + 1, that is, for a radial wave function without nodes. Upon substitution into Eq. 共45兲, I find 具r2典具p2典 = 21 共2ᐉ2 + 7ᐉ + 6兲ប2. The quadratic term can be factorized to obtain

3 2

2

共51兲

to establish a lower bound for the energy Eᐉ. Because the energy of a free particle is purely kinetic 关Eᐉ = 具p2典 / 共2␮兲兴, the inequalities follow from Eqs. 共50兲 and 共51兲 共Ref. 35兲 Christian Bracher

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Table I. Comparison of the exact ground state energy E0ᐉ in Eq. 共49兲 of a free particle 共mass ␮兲 in a spherical box of radius a with the lower bound Eᐉ,min = 共ᐉ + 3 / 2兲2 given in Eq. 共52兲 for various values of the orbital quantum number ᐉ. 共All energies are given in units of ប2 / 2␮a2.兲 ᐉ E0ᐉ Eᐉ,min

具p 典 ⱖ 2␮ 2

Eᐉ =

0 9.869604 2.25

冉 冊 3 2 2␮具r2典

2

ប2 ᐉ +

⬎

1 20.19073 6.25

冉 冊

3 ប2 2 ᐉ+ 2 2␮a

2 33.21746 12.25

5 87.53122 42.25

2

.

共52兲

Table I provides a comparison of the exact ground state energy 共49兲 with the bound 共52兲 for selected values of ᐉ. The approximation is poor for small ᐉ, but improves with increasing values of ᐉ, just as for the hydrogen problem. VI. SUMMARY I have shown that all two-dimensional wave packets ⌿共r , ␾兲 with well-defined angular momentum m must obey the generalized Heisenberg uncertainty relation ⌬r⌬p = 冑具r2典具p2典 ⱖ 共兩m兩 + 1兲ប.

共53兲

Similarly, all three-dimensional wave packets ⌿共r , ␪ , ␾兲 with orbital symmetry 共total angular momentum ᐉ兲 satisfy the related inequality

冉 冊

⌬r⌬p ⱖ ᐉ +

3 ប. 2

共54兲

In both cases, the equality is attained only for a special class of minimum uncertainty wave packets that are given by the modified Gaussian functions ⌿共r, ␾兲 = ␣r兩m兩e−␭r

2/2+im␾

共55兲

and 2

⌿共r, ␪, ␾兲 = ␣rᐉe−␭r /2Y ᐉ共␪, ␾兲,

共56兲

respectively. Here, ␭ ⬎ 0 is a positive real constant, ␣ is a normalization factor, and Y ᐉ共␪ , ␾兲 is a spherical harmonic of order ᐉ. Together with Heisenberg’s traditional uncertainty relation ⌬x⌬px ⱖ ប / 2 for one spatial dimension, the results 共53兲–共55兲 suggest a more general inequality for the product of the position and momentum uncertainties, valid for states with definite rotational symmetry 共total angular momentum quantum number j兲 in n dimensions36–38

冉 冊

⌬r⌬p ⱖ j +

n ប, 2

共57兲

with the minimum uncertainty states being modified Gaussians of the form 共55兲. A proof of this conjecture is outside the scope of this article.39 ACKNOWLEDGMENTS The author would like to thank Manfred Kleber for stimulating discussions on the problem of multidimensional tunneling and the anonymous referees for valuable suggestions that led to a more comprehensive paper. 318

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10 226.0052 132.25

20 673.6974 462.25

50 3322.218 2652.25

100 11957.45 10302.25

a兲

Electronic mail: [email protected] D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed. 共Pearson Prentice Hall, Upper Saddle River, 2005兲, pp. 110–114. 2 R. Shankar, Principles of Quantum Mechanics, 2nd ed. 共Springer, New York, 1994兲, pp. 237–241. 3 Alternatively, I could examine the variance in the radial distance 冑具r2典 − 具r典2 and a complementary “radial momentum.” This approach is briefly considered in A. Messiah, Quantum Mechanics 共North-Holland, Amsterdam, 1965兲, Vol. I, pp. 417–420. 4 P. L. Kapur and R. Peierls, “Penetration into potential barriers in several dimensions,” Proc. R. Soc. London, Ser. A 163, 606–610 共1937兲. 5 H. M. Van Horn and E. E. Salpeter, “WKB approximation in three dimensions,” Phys. Rev. 157, 751–758 共1967兲. 6 J. Summhammer and J. Schmiedmayer, “Wave aspects of electron and ion emission from point sources,” Phys. Scr. 42, 124–128 共1990兲. 7 C. Bracher, W. Becker, S. A. Gurvitz, M. Kleber, and M. S. Marinov, “Three-dimensional tunneling in quantum ballistic motion,” Am. J. Phys. 66, 38–48 共1998兲. 8 T. Kramer, C. Bracher, and M. Kleber, “Four-path interference and uncertainty principle in photodetachment microscopy,” Europhys. Lett. 56, 471–477 共2001兲. 9 B. Donner, M. Kleber, C. Bracher, and H. J. Kreuzer, “A simple method for simulating scanning tunneling images,” Am. J. Phys. 73, 690–700 共2005兲. 10 C. Nicole, H. L. Offerhaus, M. J. J. Vrakking, F. Lépine, and C. Bordas, “Photoionization microscopy,” Phys. Rev. Lett. 88, 133001 共2002兲. 11 C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics 共Wiley Interscience, New York, 1977兲, Vol. I, pp. 727–764. 12 L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory), 2nd ed. 共Pergamon, Oxford, 1965兲, pp. 424–427. 13 R. R. Moore, “Landau level problem using analytic function theory,” Am. J. Phys. 45, 589 共1977兲. 14 B. L. Johnson, “Understanding the Laughlin wave function for the fractional quantum Hall effect,” Am. J. Phys. 70, 401–405 共2002兲. 15 Lord Rayleigh, The Theory of Sound 共Macmillan, London, 1926兲, Vol. II, pp. 126–128. 16 C. V. Raman and G. A. Sutherland, “On the whispering-gallery phenomenon,” Proc. R. Soc. London, Ser. A 100, 424–428 共1922兲. 17 D. N. Williams, “A new proof of the uncertainty relation,” Am. J. Phys. 47, 606–607 共1979兲. 18 G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 3rd ed. 共Academic, San Diego, 1985兲, p. 97 and p.104. 19 M. Reed and B. Simon, Functional Analysis (Methods of Modern Mathematical Physics) 共Academic, New York, 1980兲, Vol. I, pp. 195–198. 20 An analogous decomposition used in matrix theory is the Cholesky factorization of a positive definite Hermitian matrix A into a product of a lower triangular matrix L and its upper triangular adjoint A = L†L. See, for example, R. A. Horn and C. R. Johnson, Matrix Analysis 共Cambridge U. P., Cambridge, 1985兲, pp. 405–407. 21 This decomposition is not unique. If Pˆ is a square root of pˆ2, then ˜P ˆ Pˆ is also a square root of pˆ2 for any unitary operator U ˆ : ˜P†˜P =U †ˆ†ˆ ˆ † ˆ ˆ2 ˆ ˆ = P U UP = P P = p . 22 A discussion of the proper square root operator for s-wave states in the three-dimensional case can be found in S. N. Mosley, “The positive radial momentum operator,” e-print arXiv:math-ph/0309055v1. 23 Attempts to introduce a “radial momentum operator” pˆr as a square root of the kinetic energy operator in polar or spherical coordinates without redefining the norm along the lines of Eqs. 共3兲 and 共9b兲 are fraught with mathematical difficulties and physical paradoxes. For an illuminating discussion, see R. L. Liboff, I. Nebenzahl, and H. H. Fleischmann, “On the 1

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radial momentum operator,” Am. J. Phys. 41, 976–980 共1973兲. The adjoint of the anti-Hermitian operator ⳵ / ⳵r is −⳵ / ⳵r 共as integration by parts shows兲 and the adjoint of a product of operators is the reverse product of their adjoints: 共AˆBˆ兲† = Bˆ†Aˆ†. 25 See Ref. 2, pp. 316–317. 26 The generalized uncertainty relation in Eqs. 共1兲 and 共2兲 may lead to paradoxical results if the domains of the operators Aˆ and Bˆ are incompatible with each other, an issue that does not affect the operators Rˆ and Pˆ. See E. D. Chisolm, “Generalizing the Heisenberg uncertainty relation,” Am. J. Phys. 69, 368–371 共2001兲. 27 The adjoint of an anti-Hermitian operator Bˆ is its negative Bˆ† = −Bˆ. It follows that iBˆ is a Hermitian operator with real expectation values, implying that 具B典 must always be an imaginary number. 28 Because both rˆ2 and pˆ2 are scalar operators, the uncertainty relation cannot depend on the choice of magnetic quantum number m. See also A. Messiah, Quantum Mechanics 共North-Holland, Amsterdam, 1965兲, Vol. II, pp. 569–570. 29 Note that the particular state ␺共r兲Y ᐉᐉ共␪ , ␾兲 is the product of the twodimensional minimum uncertainty state 共15兲 for m = ᐉ in the x-y plane, with the usual oscillator ground state along the perpendicular direction 2 e−␭z /2 and therefore an eigenstate of the isotropic harmonic oscillator in three dimensions. 30 The hydrogen problem possesses energy eigenstates that are superposi24

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tions of contributions with different quantum number ᐉ and thus lack spherical symmetry. The results in this paper do not apply to them. 31 A. Messiah, Quantum Mechanics 共North-Holland, Amsterdam, 1965兲, Vol. I, p. 484. 32 See, for example, Ref. 1, pp. 141–144. 33 Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun 共Dover, New York, 1965兲, pp. 437–442, 467. 34 A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32, 377–400 共2002兲. 35 Equations 共49兲 and 共52兲 furnish a proof of the inequality z0ᐉ ⬎ ᐉ + 3 / 2 for the smallest zero of the spherical Bessel function jᐉ共u兲. 36 H. Narumi and T. Nakau, “On the quantization of angular momentum in multi-dimensional space 共I兲,” Bull. Inst. Chem. Res., Kyoto Univ. 26, 26–35 共1951兲; “On the quantization of angular momentum in multidimensional space 共II兲,” 27, 29–41 共1952兲. 37 H. Narumi, “On the eigenvalue problem in terms of a complete set of the Casimir operators,” J. Phys. Soc. Jpn. 11, 786–792 共1956兲. 38 G. A. Gallup, “Angular momentum in n-dimensional spaces,” J. Mol. Spectrosc. 3, 673–682 共1959兲. 39 The less restrictive uncertainty relation ⌬r⌬p ⱖ nប / 2, valid for arbitrary wave packets ⌿共r兲 in n-dimensional space, follows directly from the Cauchy–Schwarz inequality, together with the original Heisenberg relation 共具x21典 + . . . +具x2n典兲共具p21典 + . . . +具p2n典兲 ⱖ 共冑具x21典具p21典 + . . . +冑具x2n典具p2n典兲2 ⱖ 共nប / 2兲2.

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