Reality of linear and angular momentum expectation values in bound
  states
Reality of linear and angular momentum expectation values in bound states
Utpal Roy,∗ Suranjana Ghosh,† and T. Shreecharan‡
Physical Research Laboratory, Ahmedabad 380009, India
Kaushik Bhattacharya§
Insituto de Ciencias Nucleares, Universidad Nacional Autónoma de Mexico,
Circuito Exterior, C.U., A. Postal 70-543, C. Postal 04510, Mexico DF, Mexico
ABSTRACT
In quantummechanics textbooks the momentum operator is defined in the Cartesian coordinates and
rarely the form of the momentum operator in spherical polar coordinates is discussed. Consequently
one always generalizes the Cartesian prescription to other coordinates and falls in a trap. In this work
we introduce the difficulties one faces when the question of the momentum operator in spherical polar
coordinate comes. We have tried to point out most of the elementary quantum mechanical results,
related to the momentum operator, which has coordinate dependence. We explicitly calculate the
momentum expectation values in various bound states and show that the expectation value really
turns out to be zero, a consequence of the fact that the momentum expectation value is real. We
comment briefly on the status of the angular variables in quantum mechanics and the problems
related in interpreting them as dynamical variables. At the end, we calculate the Heisenberg’s
equation of motion for the radial component of the momentum for the Hydrogen atom.
I. INTRODUCTION
Quantum mechanics is a treasure house of peculiar and interesting things. Elementary textbooks of quantum
mechanics [1, 2, 3] generally start with the postulates which are required to define the nature of the dynamical
variables in the theory and their commutation relations. The choice of the dynamical variables is not that clear, as
the coordinates in Cartesian system are all elevated to the status of operators where as time remains a parameter.
More over in spherical polar coordinates only the radial component can be represented as an operator while the angles
still remain as a problem. The difficulty of giving different status to the spatial coordinates and time is bypassed in
quantum field theories where all the coordinates and time become parameters of the theory. But the problem with
angles still remain a puzzle which requires to be understood in future.
When we start to learn quantum mechanics, most of the time we begin with elementary calculations relating to the
particle in a one dimensional infinite well, particle in a finite potential well, linear harmonic oscillator and so on. The
main aim of these calculations is to solve the Schrödinger equation in the specific cases and find out the bound state
energies and the energy eigenfunctions in coordinate space representation. While solving these problems we overlook
the subtleties of other quantum mechanical objects as the definition of the momentum operator in various coordinates,
the reality of its expectation value, etc.. In the last one or two decades there has been a number of studies regarding
the self-adjointness of various operators [4]. The aim of these studies has been to analyze the self-adjointness of
various operators like momentum, Hamiltonian etc. and find out whether these operators are really self-adjoint in
some interval of space where the theory is defined, if not then can there be any mathematical method by which we
can make these operators to be self-adjoint in the specified intervals ?
In the present work we deal with a much elementary concept in quantum mechanics related to the reality of the
expectation values of the momentum operator, be it linear or angular. We do not analyze the self-adjointness of
the operators which requires different mathematical techniques. To test the self-adjointness of an operator we have
to see whether the operator is symmetric in a specific spatial interval and the functional domain of the operator
and its adjoint are the same. In the article we always keep in touch with the recent findings from modern research
about self-adjoint extensions but loosely we assume that the operators which we are dealing with are Hermitian. If
something is in contrary we point it out in the main text. A considerable portion of our article deals with the analysis
∗Electronic address: utpalroy@prl.res.in
†Electronic address: sanjana@prl.res.in
‡Electronic address: shreet@prl.res.in
§Electronic address: kaushik@nucleares.unam.mx
http://arxiv.org/abs/0704.0373v1
mailto:utpalroy@prl.res.in
mailto:sanjana@prl.res.in
mailto:shreet@prl.res.in
mailto:kaushik@nucleares.unam.mx
of the fact that the expectation value of the momentum operator in various bound states are zero, a result which
most of the textbooks only quote but never show. In the simpler cases the result can be shown by one or two lines
of calculation, but in nontrivial potentials as the Morse potential, the Coulomb potential the result is established by
using various properties of the special functions such as the associated Legendre and the associated Laguerre.
The presentation of various materials in our article is done in the following way. Next section deals with the
definition of the momentum operator and its properties. Section III deals with the intricacies of the definition of the
momentum operator in spherical polar coordinates and the problems we face when we try to mechanically implement
the quantization condition, which is invariably always written in Cartesian coordinates in most of the textbooks on
quantum mechanics. In section IV we explicitly calculate the momentum expectation values in various potentials and
show that in bound states we always get the expectation value of the linear momentum to be zero. Section V gives
a brief discussion on the Ehrenfest theorem when we are using it to find out the time derivative of the expectation
value of the radial component of momentum in the case of the Hydrogen atom. We end with the concluding section
which summarizes the findings in our article.
Before going into the main discussion we would like to mention about the convention. We have deliberately put a
hat over various symbols to show that they are operators in quantum mechanics. Some times this convention becomes
tricky when we are dealing with angular variables as there the status of these variables is in question. The other
symbols have their conventional meaning. As we are always using the coordinate representation sometimes we may
omit the hat over the position operator as in this representation the position operator and its eigenvalues can be
trivially interchanged.
II. DEFINITION OF THE MOMENTUM OPERATOR AND THE REALITY OF ITS EXPECTATION
VALUE
From the Poisson bracket formalism of classical mechanics we can infer:
[x̂i, p̂j] = ih̄ δi j , (2.1)
where δi j = 1 when i = j and zero for all other cases, and i, j = 1, 2, 3. In the above equation x̂i is the position
operator and p̂j is the linear momentum operator in Cartesian coordinates. From the above equation we can also find
the form of the momentum operator in position representation, which is:
p̂i = −ih̄
. (2.2)
It is interesting to note that the above expression of the momentum operator also gives us the form of the generator
of translations. This is because of the property:
[p̂x, F (x̂)] = −ih̄
dF (x̂)
, (2.3)
where F (x̂) is an arbitrary well defined function of x̂. The above equation ensures that the momentum operator
generates translations along the x direction.
Particularly in one-dimension the expression of the momentum operator becomes p̂x = −ih̄ ∂∂x . We know that the
expectation value of the momentum operator must be real. If we focus on one-dimensional systems to start with,
where the system is specified by the wave-function ψ(x, t), the expectation value of any operator Ô is defined by:
〈Ô〉 ≡
ψ∗(x, t) Ô ψ(x, t) dx , (2.4)
where ψ∗ signifies complex conjugation of ψ and the extent of the system is taken as −∞ < x <∞. From the above
equation we can write,
〈Ô〉∗ =
ψ(x, t) Ô∗ ψ(x, t)∗ dx . (2.5)
If 〈Ô〉 = 〈Ô〉∗ then the condition of the reality of the expectation value becomes:
ψ∗(x, t) Ô ψ(x, t) dx =
ψ(x, t) Ô∗ ψ(x, t)∗ dx . (2.6)
For a three-dimensional system the above condition becomes,
ψ∗(x, t) Ô ψ(x, t) d3x =
ψ(x, t) Ô∗ ψ(x, t)∗ d3x . (2.7)
Now if we take the specific case of the momentum operator in one-dimension we can explicitly show that its expectation
value is real if the extent of the system is infinite and the wave-function vanishes at infinity. The proof is as follows.
The expectation value of the momentum operator is:
ψ∗(x, t) p̂x ψ(x, t) dx = −ih̄
ψ∗(x, t)
∂ψ(x, t)
= −ih̄
ψ∗(x, t)ψ(x, t)|∞−∞ −
ψ(x, t)
∂ψ(x, t)∗
, (2.8)
If the wave functions vanish at infinity then the first term on the second line on the right-hand side of the above
equation drops and we have,
ψ∗(x, t) p̂x ψ(x, t) dx = −ih̄
ψ∗(x, t)
∂ψ(x, t)
= ih̄
ψ(x, t)
∂ψ(x, t)∗
ψ(x, t) p̂∗x ψ(x, t)
∗ dx . (2.9)
A similar proof holds for the three-dimensional case where it is assumed that the wave-function vanishes at the
boundary surface at infinity.
III. THE EXPECTATION VALUE OF THE MOMENTUM OPERATOR IN CARTESIAN AND
SPHERICAL POLAR COORDINATES
In non-relativistic version of quantum mechanics we know that if we have a particle of mass m which is present in
a time-independent potential we can separate the Schrödinger equation:
∂ψ(x, t)
∇2 + V (x)
ψ(x, t) , (3.1)
into two equations, one is the time-dependent one which gives the trivial solution e−
h̄ where E is the total energy
of the particle, and the other equation is the time-independent Schrödinger equation:
∇2u(x) + 2m
(E − V (x))u(x) = 0 , (3.2)
where u(x) is the solution of the time-independent Schrödinger equation and the complete solution of the Eq. (3.1)
ψ(x, t) = u(x)e−
h̄ . (3.3)
In the case of the free-particle, where V (x) = 0, we have u(x, t) = eik·x where E = k
and k = |k|. The free-
particle solution is an eigenfunction of the momentum operator with eigen value h̄k. Although if we try to find
out the expectation value of the momentum operator as is done in the last section we will be in trouble as these
wave-functions do not vanish at infinity, a typical property of free-particle solutions. But this problem is not related
to the Hermiticity property of the momentum operator, it is related with the de-localized nature of the free-particle
solution.
In physics many times we require to solve a problem using curvilinear coordinate systems. The choice of our
coordinate system depends upon the specific symmetry which we have at hand. Suppose we are working in spherical
polar coordinates and the solution of Eq. (3.2) can be separated into well behaved functions of r, θ and φ as,
u(x) = u(r, θ, φ) = R(r)Θ(θ)Φ(φ) . (3.4)
If we try to follow the proof of the Hermiticity of the linear momenta components, as done in the last section, in
spherical polar coordinates, then we should write:
〈p̂〉 = −ih̄
u∗(r, θ, φ)∇u(r, θ, φ) dτ ,
= −ih̄
R∗(r)Θ∗(θ)Φ∗(φ)
r sin θ
R(r)Θ(θ)Φ(φ)r2drdΩ ,
(3.5)
where in the above equation er, eθ, eφ respectively are the unit vectors along r, θ and φ, and dΩ = sin θ dθ dφ. τ is
the volume over which we integrate the expression in the above equation. From the last equation we can write:
〈p̂r〉 = −ih̄
|Θ(θ)|2|Φ(φ)|2dΩ
r2R∗(r)
dR(r)
dr , (3.6)
As Θ(θ) and Φ(φ) are normalized, the integration:
|Θ(θ)|2|Φ(φ)|2dΩ = 1 and we can proceed as in Eq. (2.9) as:
〈p̂r〉 = −ih̄
r2R∗(r)
dR(r)
= −ih̄
r2R∗(r)R(r)
2rR∗(r) + r2
dR∗(r)
R(r) dr
. (3.7)
If R(r) vanishes at infinity then the above equation reduces to,
〈p̂r〉 =
r2R(r)
dR∗(r)
+ 2ih̄
r|R(r)|2 dr ,
= 〈p̂r〉∗ + 2ih̄
r|R(r)|2 dr . (3.8)
The above equation implies that 〈p̂r〉 is not real in spherical polar coordinates. The solution of the above problem
lies in redefining p̂r as is evident from Eq. (3.8), and it was given by Dirac [5, 6]. The redefined linear momentum
operator along r can be:
p̂r ≡ −ih̄
= −ih̄
r . (3.9)
This definition of the p̂r is suitable because in this form it satisfies the commutation relation as given in Eq. (2.1)
where now the operator conjugate to r̂ is p̂r. The form of p̂r in Eq. (3.9) shows that for any arbitrary function of r as
F (r) we must still have Eq. (2.3) satisfied. This implies that the modified form of p̂r is still a generator of translations
along the r direction. Up to this point we were following what was said by Dirac regarding the status of the radial
momentum operator. Still everything is not that smooth with the redefined operator as we can see that it turns out
to be singular around r = 0, more over, although the radial momentum acts like a translation generator along r but
near r = 0 it cannot generate a translation towards the left as the interval ends there.
In this regard we can state that the issue of the reality of the radial component of the momentum in spherical polar
coordinates is a topic of modern research in theoretical physics [7, 8]. It has been shown that the operator −ih̄ ∂
not Hermitian and more over it can be shown [4] that such an operator cannot be self-adjoint in the interval [0,∞].
In some recent work [8] the author claims that there can be an unitary operator which connects −ih̄ ∂
to −ih̄1
and as the former operator does not have a self-adjoint extension in the semi-infinite interval so the latter is also not
self-adjoint in the same interval.
If we further try to find out whether 〈p̂θ〉 and 〈p̂φ〉 are real, then we will face difficulties. Working out naively if
we claim that p̂φ =
rsinθ
as suggested by the φ component of Eq. (3.5) we will notice that φ p̂φ does not have the
dimension of action. This means p̂φ or p̂θ is not conjugate to φ or θ. This is a direct representation of the special
coordinate dependence of the quantization condition. Only in Cartesian coordinates the variables conjugate to x, y
and z are px, py and pz. Taking the clue from classical mechanics we know the proper dynamical variables conjugate
to φ̂ and θ̂ are the angular momentum operators, namely L̂θ and L̂φ. In general L̂φ is given by:
L̂φ = −ih̄
, (3.10)
which can be shown to posses real expectation values by following a similar proof as is done in Eq. (2.8) and Eq. (2.9),
if we assume Φ(0) = Φ(2π). In this form it is tempting to say that we can have a relation of the form,
[φ̂, L̂φ] = ih̄ , (3.11)
which looks algebraically correct. But the difficulty in writing such an equation is in the interpretation of φ̂ which
has been elevated from an angular variable to a dynamical operator. In spherical polar coordinates both θ and φ
are compact variables and consequently have their own subtleties. Much work is being done in trying to understand
the status of angular variables and phases [9, 10], in this work we only present one example showing the difficulty of
accepting φ̂ as an operator.
From the solution of the time-independent Schrödinger equation for an isotropic potential we will always have:
Φ(φ) =
eiMφ , (3.12)
where M = 0,±1,±2, ·, ·. Now if φ̂ is an operator we can find its expectation value, and it turns out to be:
〈φ̂〉 = 1
φeiMφe−iMφ dφ ,
= π , (3.13)
and the expectation value of φ̂2 is:
〈φ̂2〉 = 1
φ2eiMφe−iMφ dφ ,
π2 . (3.14)
Consequently ∆φ =
〈φ̂2〉 − 〈φ̂〉2 = π√
. Similarly calculating L̂φ we get:
〈L̂φ〉 =
e−iMφeiMφ dφ ,
= Mh̄ , (3.15)
as expected, and 〈L̂2φ〉 = M2h̄
. This implies ∆Lφ =
〈L̂2φ〉 − 〈L̂φ〉2 = 0. So we can immediately see that the
Heisenberg uncertainty relation between φ̂ and L̂φ, ∆φ∆Lφ ≥ h̄/2 breaks down. This fact makes life difficult and we
have no means to eradicate this problem.
Taking the clue from the φ part we can propose that L̂θ is also of the form −ih̄ ∂∂θ . With this definition of L̂θ let
us try to prove its Hermitian nature as done in Eq. (3.7). Taking R(r) and Φ(φ) in Eq. (3.4) separately normalized,
we can write:
〈L̂θ〉 = −ih̄
Θ∗(θ)
dΘ(θ)
sin θdθ
= −ih̄
sin θΘ∗(θ)Θ(θ)|π0 −
cos θΘ∗(θ) + sin θ
dΘ∗(θ)
Θ(θ) dθ
sin θΘ(θ)
dΘ∗(θ)
+ ih̄
cos θ |Θ(θ)|2 dθ ,
= 〈L̂θ〉∗ + ih̄
cos θ |Θ(θ)|2 dθ . (3.16)
The above equation shows that 〈L̂θ〉 is not real. The rest is similar to the analysis following Eq. (3.8) where now we
have to redefine the angular momentum operator conjugate to θ as [11]:
L̂θ ≡ −ih̄
cot θ
. (3.17)
Unlike the φ case, Θ(θ) are not eigenfunctions of L̂θ. But the difficulties of establishing θ as an operator still persists
and in general θ is not taken to be a dynamical operator in quantum mechanics.
It is known that both θ and φ are compact variables, i.e. they have a finite extent. But there is a difference between
them. In spherical polar coordinates the range of φ and θ are not the same, 0 ≤ φ < 2π and 0 ≤ θ ≤ π. This difference
can have physical effects. As φ runs over the whole angular range so the wave-function corresponding to it Φ(φ) is
periodic in nature while due to the range of θ, Θ(θ) need not be periodic. Consequently there can be a net angular
momentum along the φ direction while there cannot be any net angular momentum along θ direction. And this can
be easily shown to be true. As the time-independent Schrödinger equation for an isotropic potential yields Φ(φ) as
given in Eq. (3.12) similarly it is known that in such a potential the form of Θ(θ) is given by:
Θ(θ) = Nθ P
M (cos θ) , (3.18)
where Nθ is a normalization constant depending on L, M and P
M (cos θ) is the associated Legendre function, which
is real. In the above equation L and M are integers where L = 0, 1, 2, 3, ·, · and M = 0,±1,±2,±3, ·, ·. The
quantum number M appearing in Eq. (3.12) and in Eq. (3.18) are the same. This becomes evident when we solve
the time-independent Schrödinger equation in spherical polar coordinates by the method of separation of variables.
A requirement of the solution is −L ≤ M ≤ L. Now we can calculate the expectation value of L̂θ using the above
wave-function and it is:
〈L̂θ〉 = −ih̄N2θ
PLM (cos θ)
dPLM (cos θ)
cot θPLM (cos θ)
sin θdθ
= −ih̄N2θ
PLM (cos θ)
dPLM (cos θ)
sin θdθ +
PLM (cos θ)P
M (cos θ) cos θ dθ
(3.19)
To evaluate the integrals on the right hand side of the above equation we can take x = cos θ and then the expectation
value becomes:
〈L̂θ〉 = −ih̄N2θ
PLM (x)
dPML (x)
(1− x2) 12 dx
PLM(x)P
1− x2
. (3.20)
The second term in the right hand side of the above equation vanishes as the integrand is an odd function in the
integration range. For the first integral we use the following recurrence relation [12]:
(x2 − 1)
dPLM (x)
=MxPLM (x) − (L+M)PLM−1(x) , (3.21)
the last integral can be written as,
〈L̂θ〉 = ih̄N2θ
x(1− x2)−
2PLM (x)P
M (x) dx
− (L+M)
(1 − x2)−
2PLM (x)P
M−1(x) dx
. (3.22)
PLM (x) = (−1)L+MPLM (−x) , (3.23)
we can see immediately that both the integrands in the right hand side of the above equation is odd and consequently
〈L̂θ〉 = 0 as expected. A similar analysis gives 〈L̂φ〉 = Mh̄. It must be noted that the form of L̂θ still permits it to
be the generator of rotations along the θ direction.
As the motion along φ is closed so there can be a net flow of angular momentum along that direction but because
the motion along θ is not so, a net momentum along θ direction will not conserve probability and consequently for
probability conservation we must have expectation value of angular momentum along such a direction to be zero.
In elementary quantum mechanics text books it is often loosely written that the solution of the time-independent
Schrödinger equation is real when we are solving it for a real potential. But this statement is not correct. The reality
of the solution also depends upon the coordinate system used. Specially for compact periodic coordinates we can
always have complex functions as solutions without breaking any laws of physics.
Before leaving the discussion on angular variables in spherical polar coordinates we want to point out one simple
thing which is interesting. In Cartesian coordinates when we deal with angular momentum we know that:
[L̂i, L̂j] = iǫijk L̂k , (3.24)
where L̂i stands for L̂x, L̂y or L̂z. For this reason there cannot be any state which can be labelled by the quantum
numbers of any two of the above angular momenta. But from the expressions of L̂φ and L̂θ we see that,
[L̂φ, L̂θ] = 0 , (3.25)
and consequently in spherical polar coordinates we can have wave-function solutions of the Schrödinger equation
which are simultaneous eigenfunctions of both L̂φ and L̂θ as P
M (θ).
For real V (x), we expect the solution of the time-independent Schrödinger equation u(x) to be real, when we
are solving the problem in Cartesian coordinates. In all these cases the expectation value of the linear momentum
operators must vanish. The reason is simple and can be understood in one-dimensional cases where with real u(x) we
directly see that the integral
∂u(x)
dx is real and so
∗(x) p̂x u(x) dx becomes imaginary as p̂x contains
i, as is evident from the first line in Eq. (2.9). So if the expectation value of the momentum operator has to be
real then the only outcome can be that for all those cases where we have a time-independent solution in a bounded
region of space, with a real potential and working in Cartesian coordinates, the expectation value of the momentum
operator must vanish. The above statement is true in curvilinear coordinates also, but in those cases the definition of
the momentum operators have to be modified. This fact becomes clear when we write the relationship between the
probability flux and the expectation value of the momentum operator. The probability flux for a particle of mass m
j(x, t) = −
[ψ∗(x, t)∇ψ(x, t) − (∇ψ∗(x, t))ψ(x, t)] ,
Im (ψ∗(x, t)∇ψ(x, t)) , (3.26)
where ‘Im’ implies the imaginary part of some quantity. Most of the elementary quantum mechanics books then
proceeds to show that:
d3x j(x, t) =
, (3.27)
which is obtained from Eq. (3.26) by integrating both sides of it over the whole volume. From Eq. (3.26) we immediately
see that if the solution of the time-independent Schrödinger equation is real we will have j(x, t) = 0 and consequently
from Eq. (3.27), 〈p̂〉 = 0. But this statement is also coordinate dependent, which is rarely said in elementary textbooks
of quantum mechanics. Eq. (3.26) evidently does not hold in spherical polar coordinates. If we take Eq. (3.4) as the
solution in a general isotropic central potential and use the general form of ∇ in spherical polar coordinates then it
can be seen that jr(r, θ, φ, t) = 0 for a real potential. But then Eq. (3.27) does not hold as here p̂r is simply the radial
component of ∇ and not as given in Eq. (3.9), and we know 〈 d
〉 is not zero. The reason why Eq. (3.26) is not suitable
in spherical polar coordinates is related to the fact that in deriving Eq. (3.26) one assumes that the probability density
of finding the quantum state within position x and x + dx at time t is |ψ(x, t)|2. But this statement is only true in
Cartesian coordinates, in spherical polar coordinates the probability density of the system to be within a region r and
r + dr, θ and θ + dθ, φ and φ+ dφ is not |ψ(r, θ, φ)|2 but |ψ(r, θ, φ)|2r2 sin θ and consequently the steps which follow
leading to Eq. (3.26) in Cartesian coordinates are not valid in spherical polar coordinates. In general Eq. (3.26) will
not be valid in any curvilinear coordinate system.
The next section contains the actual calculations of the expectation values of the momentum operator in various
cases where we have bound state solutions. In all the relevant cases discussed in this article it is seen that although
〈p̂x〉 = 0 but 〈p̂2x〉 is not zero as it is related to the Hamiltonian operator. In all the cases we must have,
〈(p̂x)s〉 = 0 , s = odd integer . (3.28)
The above equation can be guessed from the reality of the expectation value of the momentum operator.
IV. MOMENTUM EXPECTATION VALUES IN VARIOUS BOUND STATES
In this section we will calculate the momentum expectation values in various bound states with stiff or slowly
varying potentials.
A. Particle in one-dimensional stiff potential wells
1. Infinite square well potential
In this case we consider a particle to be confined in region −L
along the x-axis where the potential is specified
V (x̂) = ∞ , |x| ≥
= 0 , |x| <
. (4.1)
In this case the solution of the time-independent Schrödinger equation, Eq. (3.2), satisfies the boundary condition,
= 0 , (4.2)
and as the potential has parity symmetry about x = 0 we have two sets of solutions, the odd solutions:
u(o)n (x) =
, (4.3)
and the even solutions:
u(e)n (x) =
(2n− 1)πx
. (4.4)
In the above equations n is a positive integer. Both of these functions, u
n (x) for the odd case and u
n (x) for the
even case, are real and are not momentum eigenstates. But the momentum expectation values can be found out from
the above solutions. For the odd solutions we have:
〈p̂x〉 = −ih̄
u(o)n (x)
n (x)
= −4inπh̄
= 0 , (4.5)
as expected. Similarly for the even solutions it is also easy to show that the expectation value of the momentum
operator vanishes.
2. Finite square well potential
In this case,
V (x̂) = 0 , |x| ≥ a ,
= −V0 , |x| < a , (V0 > 0) . (4.6)
If we are not interested in the normalization constant of the bound state solution then the solution of the time-
independent Schrödinger equation in this case is:
u(x) ∼ e−κ|x| , |x| > a ,
∼ cos(kx) , |x| < a , (even parity)
∼ sin(kx) , |x| < a , (odd parity) , (4.7)
where,
2m(−|E|+ V0)
, (4.8)
2m|E|
. (4.9)
In this case the expectation value of the momentum operator is:
〈p̂x〉 ∼ −ih̄
du(x)
∼ −ih̄
e2κxdx−
e−2κxdx
sin(kx) cos(kx) dx
= 0 , (4.10)
where the first two lines of the above equation holds up to a constant arising from the normalization of the wave-
function. In deriving the last equation we have taken the odd parity solution, but the result remains unaffected if we
take the even parity solution as well.
3. Dirac-delta potential
In this case the potential is:
V (x̂) = −V0 δ(x̂) , (V0 > 0) . (4.11)
In this case there can be one bound state solution which is obtained after solving the Eq. (3.2). Demanding that the
solution u(x) satisfies the boundary conditions:
u(x = −ǫ) = u(x = +ǫ) , (4.12)
= −2mV0
u(x = 0) , (4.13)
where ǫ is an infinitesimal quantity tending to zero, we get the form of the solution which is:
u(x) =
κ eκx , x ≤ 0 , (4.14)
κ eκx , x ≥ 0 , (4.15)
where κ = mV0
and the energy of the bound state is E = −mV
The expectation value of the momentum operator in this case is:
〈p̂x〉 = −ih̄
du(x)
= −ih̄κ
e2κx dx −
e−2κx dx
= 0 . (4.16)
In this case, also from Hermiticity of the momentum operator we see that Eq. (3.28) holds true.
B. Particle in one-dimensional slowly varying potentials
1. Linear harmonic oscillator potential
In the case of the linear harmonic oscillator we have:
V (x̂) =
mω2x̂2 , (4.17)
where ω is the angular frequency of the oscillator. The solution of Eq. (3.2) in this case, using the series solution
method, yields:
un(q) = Nn e
2 Hn(q) , (4.18)
where n = 0, 1, 2, ·, · and q =
αx where α = mω
. Hn(q) are Hermite polynomials of order n and Nn is the
normalization constant given by,
π n! 2n
. (4.19)
The momentum expectation value in this case turns out to be,
〈p̂x〉 = −ih̄
un(q)
dun(q)
= −ih̄
Hn(q)
dHn(q)
q e−q
H2n(q) dq
= 0 . (4.20)
The first integral on the right side of the second line of the last equation vanishes because,
dHn(q)
= 2nHn−1(q) and
consequently the integral transforms into the orthogonality condition of the Hermite polynomials. The second integral
on the second line of the right side of the above equation vanishes because the integrand is an odd function of q.
The linear harmonic oscillator (LHO) has some very interesting properties. To unravel them we have to digress a
bit from the wave-mechanics approach which we have been following and follow the Dirac notation of bra and kets.
The Hamiltonian of the LHO in one-dimension is:
mω2x̂2 , (4.21)
which can also be written as:
Ĥ = h̄ω
â†â+
, (4.22)
where â and â† are the annihilation and the creation operators given by:
, â† ≡
x̂− ip̂x
. (4.23)
It can be seen clearly from the above definitions that â is not an Hermitian operator. More over from the definition
of the operators we see that,
[â, â†] = 1 . (4.24)
Conventionally the number operator is defined as:
N̂ ≡ â†â , (4.25)
and its eigen-basis are the number states |n〉 such that,
N̂ |n〉 = n|n〉 . (4.26)
The Hamiltonian of the LHO can be written in terms of the number operator and consequently the number states
are energy eigenstates. In this basis the action of the annihilation and creation operators are as:
â|n〉 =
n |n− 1〉 , (4.27)
â†|n〉 =
n+ 1 |n+ 1〉 . (4.28)
From the definitions of the annihilation and creation operators we can write the momentum operator as:
p̂x = −i
(â− â†) . (4.29)
From Eq. (4.27), Eq. (4.28) and the above equation we can write the matrix elements of the momentum operator as:
〈n′|p̂x|n〉 = i
n δn′, n−1 +
n+ 1 δn′, n+1
. (4.30)
The above equation shows that the momentum operator can connect two different energy eigenstates.
In the case of LHO, except the number operator states, we can have another state which is an eigenstate of the
annihilation operator â. This state is conventionally called the coherent state and it is given as:
|α〉 = e−
|n〉 , (4.31)
where α is an arbitrary complex number. Now from Eq. (4.29) we can find the momentum expectation value of the
coherent state and it is,
〈p̂x〉α ≡ 〈α|p̂x|α〉 =
Im(α) . (4.32)
From the above equation we can see that although the expectation value of the momentum operator is zero in the
energy eigen-basis but it is not so when we compute the momentum expectation value in the coherent state basis,
which is essentially a superposition of energy eigenstates. It must be noted that the momentum expectation value is
non zero only when the parameter α has an imaginary part.
2. Pöschl-Teller potential
Among the potentials belonging to the hypergeometric class the Pöschl-Teller potentials have been the most ex-
tensively studied and used. This class of potentials consist of trigonometric as well as the hyperbolic type. The
trigonometric versions have found applications in molecular and solid state physics and the hyperbolic variants have
been used in various studies related to black hole perturbations.
In the present work we use the trigonometric, symmetric Pöschl-Teller potential given by:
V (x̂) = V0 tan
2(ax̂) , (4.33)
where V0 can be parameterized as:
λ(λ− 1) , (4.34)
with for some positive number λ > 1 and a is some scaling factor. The energy eigenvalues of the bound state solutions
En = −
h̄2a2
(n2 + 2nλ+ λ) , (4.35)
and the solution of the time-independent Schrödinger equation is,
un(x) = Nn
cos(ax)P
1/2−λ
n+λ−1/2 (sin(ax)) , (4.36)
where,
a(n+ λ)Γ(n + 2λ)
Γ(n+ 1)
, (4.37)
is the normalization constant and Pµν (x) is the associated Legendre function. At this point it is fair to point out that
Pµν (x) is not the Legendre polynomial P
M (x) appearing in Eq. (3.18), as µ and ν need not be integers as L and M .
Pµν (x) is not a polynomial but the function appearing in the right hand side of Eq. (4.36) is a polynomial.
Now as claimed in the text let us show that the momentum expectation value is indeed zero. Before we proceed
let us simplify the notation a bit by calling µ = 1/2− λ and ν = n+ λ − 1/2. Substituting z = ax we can write the
momentum expectation value as:
〈p̂x〉 = −ih̄N2n
∫ π/2
cos(z)Pµν (sin(z))
cos(z)Pµν (sin(z))
. (4.38)
Note the limits of the integration range from π/2 to −π/2 since at this value the potential becomes infinity hence we
need not consider the integration range to be the whole real line. For the sake of convenience let us make a change
of variable; letting y = sin(z) the above integral becomes:
〈p̂x〉 = −ih̄N2n
dy (1− y2)1/4Pµν (y)
(1 − y2)1/4Pµν (y)
. (4.39)
Taking the derivative inside the integral we get:
〈p̂x〉 = −ih̄N2n
(1− y2)1/2Pµν (y)
dPµν (y)
− y(1− y
2)−1/2
Pµν (y)P
ν (y)
. (4.40)
It is known that for associated Legendre functions [13],
Pµν (−x) = cos[(µ+ ν)π]Pµν (x)−
sin[(µ+ ν)π]Qµν (x) , (4.41)
where Qµν (x) is the other linearly independent solution of the associated Legendre differential equation. As in our
case µ+ ν = n so Pµν (x) will have definite parity. As P
ν (x) has definite parity so the contribution of the second term
in the above integral vanishes since the total integrand is an odd function. The first integral is similar to the one in
Eq. (3.20) and, due to the typical parity property of Pµν (x) as shown in Eq. (4.41), it also vanishes. Consequently we
have 〈p̂x〉 = 0 as expected.
3. Morse potential
Diatomic molecule is an exactly solvable system, if one neglects the molecular rotation. The most convenient model
to describe the system, is the Morse potential [14]:
V (x̂) = D(e−2βx̂ − 2e−βx̂) , (4.42)
where x = r/r0 − 1, which is the distance from the equilibrium position scaled by the equilibrium value of the
inter-nuclear distance r0. D is the depth of the potential, called dissociation energy of the molecule and β being a
parameter which controls the width of the potential.
In terms of the above scaled variable x, the time-independent Schrödinger equation becomes:
d2u(x)
+D(e−2βx − 2e−βx)u(x) = Eu(x) . (4.43)
Here µ is the reduced mass of the molecule and the corresponding bound state eigen function comes out to be:
uλn(ξ) = Ne
−ξ/2ξs/2Lsn(ξ) , (4.44)
where the variables are described as,
ξ = 2λe−y; y = βx; 0 < ξ <∞ , (4.45)
n = 0, 1, ..., [λ− 1/2] , (4.46)
which is nothing but the quantum number of the vibrational bound states. Here [ρ] denotes the largest integer smaller
than ρ, thus total number of bound states is [λ− 1/2] + 1. The parameters,
2µDr20
and s =
− 8µr
E , (4.47)
satisfy the constraint condition s+2n = 2λ− 1. We note that the parameter λ is potential dependent and s is related
to energy E. In Eq. (4.44), Lsn(y) is the associated Laguerre polynomial and N is the normalization constant [15]:
β(2λ− 2n− 1)Γ(n+ 1)
Γ(2λ− n)r0
. (4.48)
We are looking for the expectation value of linear momentum for a vibrating diatomic molecule, and its expression
〈p̂x〉 = −ih̄
u∗n(ξ)
un(ξ)dx . (4.49)
In terms of the changed variable ξ = 2λe−βx the integration limit changes to ∞ to 0 and the expectation value
becomes:
〈p̂x〉 = −ih̄
u∗n(ξ)
un(ξ)dξ
= ih̄N2
e−ξξs(Lsn(ξ))
2dξ +
e−ξξs−1(Lsn(ξ))
e−ξξsLsn(ξ)
Lsn(ξ)dξ
= ih̄N2
I2 + I3
. (4.50)
Integral I1 is the orthogonality relation of the associated Laguerre polynomials, which is:
e−ξξsLsn(ξ)L
m(ξ)dξ =
Γ(s+ n+ 1)
Γ(n+ 1)
δm,n . (4.51)
To evaluate the second integral one uses the normalization integral of Morse eigenstates. The normalization relation
u∗(ξ)u(ξ)dr =
|N |2r0
e−ξξs−1(Lsn(ξ))
2dξ = 1 . (4.52)
The above integral involving ξ, is explicitly I2. N , being the normalization constant as given in Eq. 4.48. Thus it is
very straight forward to evaluate I2 from the above relation as,
Γ(n+ s+ 1)
s Γ(n+ 1)
. (4.53)
The last integrand I3 includes a differentiation which can be written as [16]:
Lsn(ξ) = −Ls+1n−1(ξ) . (4.54)
Writing the right hand side of the above equation as a summation [17]:
Ls+1n =
Lsm , (4.55)
and substituting the derivative term in integral I3 we obtain:
I3 = −
e−ξξsLsn(ξ)L
m(ξ)dξ . (4.56)
In the above integral m 6= n because m can go only upto (n− 1). Thus the integral vanishes. Now let us see what is
the expectation value of momentum observable, after evaluating the three integrals above. Substituting the non-zero
values I1 and I2 in Eq. 4.50, it is clear that the expectation value of momentum is zero as has been expected.
C. Position expectation values for various potentials
After a thorough discussion about the momentum expectation values for various solvable one-dimensional potentials,
it is worth spending some time discussing about the average position of the particle inside the bound states. Among
all the above examples, in each case we had V (x) = V (−x) except the Morse potential as Morse potential is not an
example of a symmetric potential: V (x) 6= V (−x).
In deriving the expectation values of momentum for above symmetric cases, we often considered that the integrals of
odd functions over the symmetric limits vanishes. This result does not hold true for the asymmetric Morse potential.
Already we have shown that the momentum expectation value: < p >= 0 for all the above potentials. When it
comes to the expectation values of position, one can easily see that < x >= 0 for symmetric potentials whose centers
are at the origin. On the other hand if this is not the case, suppose the infinite square well is defined in the range
0 ≤ x ≤ L also then the expectation value of position does not vanish. It becomes L/2. Thus, more accurately the
average position of the particle is dependent on the symmetry of the potential where as the average momentum is
solely guided by the reality of it’s eigenvalues and consequently it is zero always.
Below we will briefly discuss how the asymmetry of the potential affects the expectation value of x in the case of
the Morse potential. The expectation value of the position operator is:
〈x̂〉 =
uλ∗n (ξ)xu
n(ξ)dx. (4.57)
The eigen function and the variables are respectively substituted from Eq. (4.44) and Eq. (4.45). We obtain
〈x̂〉 = N
ln(2λ)
e−ξξs−1(Lsn(ξ))
2dξ +
e−ξξs−1(Lsn(ξ))
2 ln(ξ)dξ
. (4.58)
The first integral is already been obtained in Eq. (4.53). This result is independent of the quantum number n. The
second integral (say I) is not that straight forward, because it contains associated Laguerre polynomial, logarithm,
exponential and monomial functions. Here at best we can evaluate the integral atleast for some specific n as, n = 0
or n = 1, when the Laguerre polynomial is respectively replaced by 1 and (−ξ + s + 1). For the ground state wave
function (n = 0), I would be
In=0 =
e−ξξs−1 ln(ξ)dξ, (4.59)
which can be written in terms of Ψ(s) and Γ function [18]:
In=0 = Γ(s)Ψ(s), (4.60)
where, Ψ(s) is the logarithmic factorial function, defined as
d(ln(s)!
= Ψ(s). For n = 0, first integral reduces
to Γ(s) from Eq. (4.53). Above two evaluations gives the ground state expectation value:
〈x̂〉n=0 =
[ln(s+ 1)−Ψ(s)] . (4.61)
For n = 1, one can proceed in the same way
〈x̂〉n=1 =
e−ξξs+1 ln(ξ)dξ + (s+ 1)2
e−ξξs−1 ln(ξ)dξ2(s+ 1)
e−ξξs ln(ξ)dξ
(4.62)
Γ(s+ 2)Ψ(s+ 2) + (s+ 1)2Γ(s)Ψ(s)− 2(s+ 1)Γ(s+ 1)Ψ(s+ 1)
which simplifies to give the expectation value corresponding to the second eigen state:
〈x̂〉n=1 =
ln(s+ 3)−Ψ(s+ 2) + 3
(s+ 2)
. (4.63)
Other expectation values for n > 1 can also be obtained in a similar fashion.
The important point which is to be noted here is, though the average momentum vanishes, the average position is
non-zero for Morse potential and remain so, irrespective of the choice of coordinate origin. This result is also true for
all eigen states of the same Hamiltonian.
D. Momentum expectation value for a three-dimensional slowly varying spherically symmetric potential
In three dimensions, for a spherically symmetric potential the solution of the Schrödinger equation is given in
Eq. (3.4). Here we have assumed that the variables can be separated. The expectation values of L̂θ and L̂φ have been
evaluated in section III. In this section we take the case of the Hydrogen atom and calculate the expectation value of
the radial component of the linear momentum.
1. The Hydrogen atom
In this case,
V (r̂) = −e
. (4.64)
where e is the electronic charge and r =
x2 + y2 + z2. Now we have to write Eq. (3.2) in spherical polar coordinates
and the solution of the time-independent Schrödinger equation is:
unLM (r, θ, φ) = Nr RnL(r)YLM (θ, φ) ,
= Nr e
−r/na0
L2L+1n−L−1
YLM (θ, φ) , (4.65)
where a0 =
is the Bohr radius and m is the reduced mass of the system comprising of the proton and the electron.
n is the principal quantum number which is a positive integer, L2L+1n−L−1(x) are the associated Laguerre polynomials,
YLM (θ, φ) are the spherical-harmonics, and Nr is the normalization arising from the radial part of the eigenfunction.
The values which L and M can take is discussed in section III. The radial normalization constant is given by:
(n− L− 1)!
(n+ L)!2n
. (4.66)
The spherical-harmonics are given by,
YLM (θ, φ) =
2L+ 1
(L −M)!
(L +M)!
PLM (cos θ)e
iMφ , (4.67)
where PML (cos θ) are the associated Legendre functions. It is noted that although the Coulomb potential is a real
potential but the solution in spherical polar coordinates is not real, eiMφ, is complex. The spherical-harmonics are
ortho-normalized according to the relation,
dθ dφ sin θ YLM (θ, φ)YL̃ M̃ (θ, φ) = δLL̃ δMM̃ . (4.68)
Let us write the eigenfunctions in terms of dimensionless quantity: ρ = 2r/na0 ≡ αr. Also we define k ≡ (2L + 1)
and nr ≡ (n − L − 1) for the sake of convenience. With this amount of notational machinery the eigenfunctions can
be written as:
unLM (r, θ, φ) = Nr RnL(ρ)YLM (θ, φ) . (4.69)
The radial momentum expectation value in this case is not given by −ih̄〈 ∂
〉, its form is (already discussed in section
III):
〈p̂ρ〉 = −ih̄Ñ2
dρ ρ2R∗nL(ρ)
RnL(ρ)
dΩ [YLM (θ, φ)]
2 . (4.70)
Where Ñ2 = N2r /α
2. The integral for the spherical harmonics yields identity. The radial expectation value then
becomes,
〈p̂ρ〉 = −ih̄Ñ2
e−ρρk+1[Lknr (ρ)]
2 + (L+ 1) e−ρρk[Lknr (ρ)]
2 + e−ρρk+1Lknr (ρ)
[Lknr (ρ)]
. (4.71)
Using the recurrence relation [16]:
Lknr (ρ) = ρ
−1 [nr Lknr (ρ)− (nr + k)L
nr−1(ρ)
, (4.72)
the expectation value integral acquires the form:
〈p̂ρ〉 = −ih̄Ñ2
e−ρρk+1[Lknr (ρ)]
2 + (nr + L+ 1) e
−ρρk[Lknr(ρ)]
2 + e−ρρkLknr (ρ)L
nr−1(ρ)
. (4.73)
The third contribution of the becomes zero from the orthogonality property of the associated Laguerre polynomials
as given in Eq. (4.51). The contribution from the second term can also be found similarly. To find the share of the
first term we make use of [19]:
dρ e−ρ ρk+1 [Lknr (ρ)]
(nr + k)!
(2nr + k + 1). (4.74)
Collecting all the contributions we get the radial expectation value to be zero as expected.
V. A DISCUSSION ON HEISENBERG’S EQUATION OF MOTION AND EHRENFEST THEOREM
The time evolution of any operator Ô in the Heisenberg picture is given by:
[Ô, Ĥ ] , (5.1)
where Ĥ is the Hamiltonian of the system. The Hamiltonian of a quantum system comprising of a particle of mass
m is given by:
+ V (x̂) . (5.2)
From the above two equations we can write the time evolution of the momentum operator in one dimension, in
Cartesian coordinates as:
[p̂x, Ĥ] = −
V (x̂) , (5.3)
which is the operator version of Newton’s second law in a time independent potential. Now if we take the expectation
values of both sides of Eq. (5.3) in any basis we get:
d〈p̂x〉
V (x̂)
, (5.4)
and historically the above equation is called the Ehrenfest theorem, which was deduced in a different way by P.
Ehrenfest. Using the Ehrenfest theorem we can deduce that the rate of change of the expectation value of the
momentum operator is zero in the case of the linear harmonic oscillator. In the case of the linear harmonic oscillator
we have:
V (x̂) = mω2x̂ , (5.5)
and it can be trivially shown that 〈x̂〉 = 0. This directly implies that,
d〈p̂x〉
= 0 , (5.6)
for the linear harmonic oscillator. The above equation shows that the expectation value of the momentum along x
direction is constant, and this constant is zero is known from other sources.
Next we focus on the Hydrogen atom. The Hamiltonian of the Hydrogen atom is:
Ĥ = − h̄
2 − e
, (5.7)
where,
2 = −h̄2
sin θ
sin θ
sin2 θ
, (5.8)
whose eigenvalues are of the form h̄
L(L+ 1) in the basis YLM (θ, φ). In the expression of the Hamiltonian m is the
reduced mass of the system comprising of the proton and electron. Next we try to apply Heisenberg’s equation to the
radial momentum operator. Noting that the first term of the Hamiltonian is nothing but p̂2r the Heisenberg equation
= − L̂
. (5.9)
The above equation is the operator form of Newton’s second law in spherical polar coordinates. Next we evaluate the
expectation value of both the sides of the above equation using the wave-functions given in Eq. (4.65). We know,
n3 a20(L +
, (5.10)
a30 n
3L(L+ 1
)(L + 1)
. (5.11)
Using the above expectation values in Eq. (5.9) and noting that 〈L̂2〉 = h̄2L(L+1) we see that the time derivative of
the expectation value of the radial momentum operator of the Hydrogen atom vanishes.
The above analysis shows that the form of the Ehrenfest theorem as given in Eq. (5.4) is only valid in Cartesian
coordinates. In the case of the Hydrogen atom if we used Eq. (5.4) we should have never got the correct result.
VI. CONCLUSION
In the present work we have emphasized on the reality of the momentum expectation value and using the reality
of the expectation value as a bench mark we did find out the form of the momentum operator in spherical polar
coordinate system. We found that most of the concepts which define the momentum operator in Cartesian coordinates
do not hold good in spherical polar coordinates and in general in any other coordinate system. The reason being
that whenever we do an integration in curvilinear coordinates the Jacobian of the coordinate transformation matrix
comes inside the picture and the Cartesian results start to falter if we do not change the rules appropriately. The
forms of the momentum along the radial direction and the form of the angular momentum operators are derived in
section III. The status of the angular variables was briefly discussed in the same section. We explicitly calculated the
expectation values of the momentum operator in various important cases and showed that the expectation value of
the momentum operator do really come out to be zero as expected. Although the expectation value of the momentum
operator vanishes in most of the bound states, with a real potential, the expectation value of the position is not
required to vanish. The expectation value of the position operator is directly related with the parity property of the
potential which was briefly discussed in subsection IVC. At the end we calculated the Heisenberg equation of motion
for the radial momentum operator for the Hydrogen atom and showed its formal semblance with Newton’s second
law. It was also shown that if we properly write the Heisenberg equation of motion in spherical polar coordinates
then Ehrenfest’s theorem follows naturally.
In short we conclude by saying:
1. the forms of the various momentum operators, in most of the coordinate systems, in quantum mechanics can be
obtained by imposing the condition of the reality of their eigenvalues. The form of the probability conservation
equation and Ehrenfest theorem must be modified in curvilinear coordinates to yield meaningful results.
2. There are obvious problems in elevating the status of angular variables to dynamical variables in quantum
mechanics.
3. For compact variables, if the variable is periodic the expectation value of the angular momentum conjugate to
it is non-zero. If the compact variable is not periodic then the angular momentum conjugate to it must vanish.
4. The momentum expectation values in cases of bound state motions vanish, whereas the position expectation
values in those cases depends on the symmetry of the potential.
Acknowledgements
The authors thank Professors D. P. Dewangan, S. Rindani, J. Banerji, P. Panigrahi and Ms. Suratna Das for
stimulating discussions and constant encouragements.
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http://mathworld.wolfram.com/LaguerrePolynomial.html, Equation 24.
http://mathworld.wolfram.com/LaguerrePolynomial.html
	Introduction
	Definition of the momentum operator and the reality of its expectation value
	The expectation value of the momentum operator in Cartesian and spherical polar coordinates
	Momentum expectation values in various bound states
	Particle in one-dimensional stiff potential wells
	Infinite square well potential
	Finite square well potential
	Dirac-delta potential
	Particle in one-dimensional slowly varying potentials
	Linear harmonic oscillator potential
	Pöschl-Teller potential
	Morse potential
	Position expectation values for various potentials
	Momentum expectation value for a three-dimensional slowly varying spherically symmetric potential
	The Hydrogen atom
	A discussion on Heisenberg's equation of motion and Ehrenfest theorem
	Conclusion
	References
