Download Total time derivatives of operators in elementary quantum mechanics

Total time derivatives of operators in elementary quantum mechanics
Mark Andrews
Citation: American Journal of Physics 71, 326 (2003); doi: 10.1119/1.1531579
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Total time derivatives of operators in elementary quantum mechanics
Mark Andrews
Department of Physics, The Faculties, Australian National University, ACT 0200, Australia
共Received 25 April 2002; accepted 30 October 2002兲
The use of a total time derivative of operators, that depends on the time evolution of the wave
function as well as on any intrinsic time dependence in the operators, simplifies the formal
development of quantum mechanics and allows its development to more closely follow the
corresponding development of classical mechanics. We illustrate the use of the total time derivative
for a free particle, the linear potential, the harmonic oscillator, and the repulsive inverse square
potential. In these cases, operators whose total time derivative is zero can be found and yield general
properties of wave packets and several useful time-dependent solutions of Schrödinger’s equation,
including the propagator. © 2003 American Association of Physics Teachers.
关DOI: 10.1119/1.1531579兴
I. INTRODUCTION
The expectation value of any operator  changes with
time according to1
冓 冔
d
⳵
i
 ⫹ 具 关 Ĥ, 兴 典 .
具 Â 典 ⫽
dt
⳵t
ប
共1.1兲
The first term on the right-hand side comes from the intrinsic
time dependence in Â, and the second from the way the
states change with time. It has occasionally been suggested2
that the total time derivative of any operator  be defined as
d
⳵
i
Â⬅Â˙ ª Â⫹ 关 Ĥ, 兴 .
dt
⳵t
ប
共1.2兲
冓 冔
共1.3兲
and it can easily be seen that the definition implies that
d
共 ÂB̂ 兲 ⫽Â˙ B̂⫹ÂB̂˙
dt
共1.4兲
and
d
共 f 共 t 兲 Â 兲 ⫽ ḟ Â⫹ f Â˙ ,
dt
II. FORMAL DEVELOPMENT FOR SYSTEMS
DESCRIBED BY A POTENTIAL
We first consider Hamiltonians of the form
共The notation XªY indicates that X is defined to be Y.兲 Then
it is always true that
d
d
 ,
具 Â 典 ⫽
dt
dt
After showing how the formal development of quantum
mechanics and its application to some systems can be simplified, we will find ‘‘invariant operators,’’ that is, operators
whose total time derivative is zero. This property is the appropriate generalization of ‘‘constant of the motion’’ to operators that depend on the time. Invariant operators are used
here to find general properties and specific examples of the
evolution of wave packets and to find the energy eigenvalues
of the harmonic oscillator.
共1.5兲
which implies that the total derivative obeys the same algebraic rules as ordinary derivatives.
Because this total time derivative is not defined to be a
rate of change of anything, some may prefer to use a different symbol for it, for example, D t instead of d/dt, but the
latter notation will be used here. The context determines its
meaning; when applied to an operator, its meaning is as in
Eq. 共1.2兲. Its use eliminates much of the tedious calculation
of commutators in conventional treatments and allows calculations that closely follow those of classical mechanics. For
many systems it also enables simple methods to be used to
obtain the evolution of some wave packets and the propagator. It brings many of the advantages gained by moving to
Heisenberg’s picture of the time-evolution while actually remaining in the Schrödinger picture.
Ĥ⫽
1 2
p̂ ⫹V 共 r̂兲 .
2m
共2.1兲
In this case all the operators are independent of the time, so
calculating the total time derivative requires only the commutator with the Hamiltonian. A calculation of these commutators gives
mr̂˙⫽p̂ and p̂˙⫽⫺ⵜV̂,
共2.2兲
as in classical mechanics.
Ehrenfest’s theorem for the time evolution of expectation
values follows immediately from Eq. 共1.3兲:
m
d
具 r̂典 ⫽ 具 p̂典
dt
and
d
具 p̂典 ⫽⫺ 具 ⵜV̂ 典 .
dt
共2.3兲
The virial theorem also follows easily using Eq. 共1.4兲 from
the expectation value of
1
d
共 r̂"p̂兲 ⫽r̂˙"p̂⫹r̂"p̂˙⫽ p̂2 ⫺r̂"ⵜV̂.
dt
m
共2.4兲
The time derivative of the angular momentum is
d
d
L̂⫽ 共 r̂⫻p̂兲 ⫽r̂˙⫻p̂⫹r̂⫻p̂˙⫽⫺r̂⫻ⵜV̂,
dt
dt
共2.5兲
because p̂⫻p̂⫽0. For a central potential, ⵜV
⫽r ⫺1 (dV/dr)r and therefore L̇⫽0, and L is a constant of
the motion.
326
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© 2003 American Association of Physics Teachers
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These calculations are not only simpler, because the commutators were used only for the equations of motion 共2.2兲,
but easier for students because the calculations closely follow the corresponding classical calculations.
Next consider how the spread of the position and momentum change with time for a free wave packet in one dimension. There is no force, so mx̂˙ ⫽p̂ and p̂˙ ⫽0. To obtain the
spread in position, we use the operator (x̂⫺ 具 x̂ 典 ) 2 . Unlike the
above operators, this operator is time dependent. Up to now
â˙ has just been a convenient shorthand for iប ⫺1 关 Ĥ,â 兴 , but
now the intrinsic time dependence will also play a role. We
use the operator equations of motion to find
m
d
共 x̂⫺ 具 x̂ 典 兲 2 ⫽ 共 x̂⫺ 具 x̂ 典 兲共 p̂⫺ 具 p̂ 典 兲 ⫹ 共 p̂⫺ 具 p̂ 典 兲共 x̂⫺ 具 x̂ 典 兲 ,
dt
共2.6兲
m2
d2
共 x̂⫺ 具 x̂ 典 兲 2 ⫽2 共 p̂⫺ 具 p̂ 典 兲 2 .
dt 2
共2.7兲
If we define (⌬x) 2 ª 具 (x̂⫺ 具 x̂ 典 ) 2 典 and (⌬p) 2 ª 具 (p̂⫺ 具 p̂ 典 ) 2 典 ,
then ⌬p is constant and
m2
d2
共 ⌬x 兲 2 ⫽2 共 ⌬ p 兲 2 .
dt 2
and â˙ ⫽0. Invariant operators will be used in this way to
produce different solutions of Schrödinger’s equation for
several systems.
Now consider whether eigenstates of an invariant operator
will satisfy Eq. 共3.1兲. The operators 共in Schrödinger’s representation兲 involve differentiation by position only, so if
␾ (r,t) is a solution of â ␾ ⫽ ␣␾ , then f (t) ␾ (r,t) also will be
a solution. Thus it is not true that every eigenfunction will
satisfy Schrödinger’s equation. However, if ␾ (r,t) is an
eigenfunction at a particular time t 0 , then there will be a
unique solution ␺ (r,t) of Schrödinger’s equation 共for all t兲
such that ␺ (r,t 0 )⫽ ␾ (r,t 0 ) and, from Eq. 共3.2兲, (â⫺ ␣ ) ␺
will satisfy Schrödinger’s equation and therefore will remain
zero. That is, ␺ (r,t) will be both an eigenfunction of â 共with
constant eigenvalue ␣兲 and a solution of Schrödinger’s equation. Thus, if there is only one eigenfunction of â, it must
satisfy Eq. 共3.1兲, apart from a time-dependent factor. If there
are several linearly independent eigenfunctions of â, then
linear combinations with time-dependent coefficients may be
required to satisfy Eq. 共3.1兲.
Invariant operators will now be found for some simple
one-dimensional systems.
共2.8兲
关Note that d 2 具 Â 典 /dt 2 ⫽d(d 具 Â 典 /dt)/dt⫽d( 具 dÂ/dt 典 )/dt
⫽ 具 d 2 Â/dt 2 典 , where d 2 Â/dt 2 means the result of the action
in Eq. 共1.2兲 applied twice to Â.] Therefore, ⌬x must have
the form
⌬x⫽⌬x 0 冑1⫹ 共 t⫺t 0 兲 2 / ␶ 2 ,
共2.9兲
where ␶ ⫽m⌬x 0 /⌬ p. Hence, for a free wave packet, the
spread ⌬x goes through a minimum ⌬x 0 and approaches a
linear increase with time at large distances from the minimum.
A similar, but slightly lengthier, calculation can be carried
out for the harmonic oscillator to show the centroid of every
wave packet not only follows the classical oscillation, but its
width oscillates at twice the frequency. For a linear potential
共motion under gravity or a uniform electric field兲 the width
behaves exactly as for the free case.
A comparison of these calculations with the conventional
methods3 will show how much simpler this approach is.
IV. A FREE PARTICLE
The operator equations of motion are mx̂˙ ⫽p̂ and p̂˙ ⫽0.
Classically, mx⫺ pt is constant and similarly mx̂⫺p̂t is invariant. A constant can be added to t, even a complex one;
but a real constant just changes the origin of time, and we
take
âªmx̂⫺ p̂ 共 t⫺i ␶ 兲
Ĥ ␺ ⫽iប
â ␣ ªâ⫺ ␣ ⫽m 共 x̂⫺x̄ 兲 ⫺ 共 p̂⫺ p̄ 兲共 t⫺i ␶ 兲 .
共3.1兲
it will produce another solution, because
冉
Ĥ⫺iប
冊 冉
冊
⳵
⳵ â
â ␺ ⫽ Ĥâ⫺iប ⫺âĤ ␺ ⫽⫺iបâ˙ ␺
⳵t
⳵t
共3.2兲
共4.2兲
Now ␺ ␣ will have â ␣ ␺ ␣ ⫽0 and therefore 具 â ␣ 典 ⫽0, 具 x̂ 典
⫽x̄, and 具 p̂ 典 ⫽ p̄ for an eigenstate of â. Furthermore
â ␣ ⫺â ␣† ⫽2i ␶ 共 p̂⫺ p̄ 兲 ,
1
2
⳵␺
,
⳵t
共4.1兲
To find an eigenstate ␺ ␣ of â with eigenvalue ␣, we first find
the real values x̄ 0 , p̄ 0 such that ␣ ⫽mx̄ 0 ⫹ip̄ 0 ␶ . Then ␣
⫽mx̄⫺ p̄(t⫺i ␶ ) for all time, where x̄, p̄ are the solutions of
the classical equations of motion with initial values x̄ 0 , p̄ 0 .
Next define
III. INVARIANT OPERATORS
An operator is ‘‘invariant’’ if its total time derivative is
zero. This definition includes the usual ‘‘constant of the motion,’’ that is, a time-independent operator that commutes
with the Hamiltonian. It will be seen below that timedependent invariant operators are especially interesting. It is
easy to see that if â and b̂ are invariant then so are â⫹b̂, âb̂,
the adjoints â † and b̂ † , and ␭â if ␭ is a constant.
If an invariant operator â is applied to any solution ␺ of
Schrödinger’s equation
共 ␶ real兲 .
共4.3兲
共 â ␣ ⫹â ␣† 兲 ⫽m 共 x̂⫺x̄ 兲 ⫺ 共 p̂⫺ p̄ 兲 t,
and therefore
2m 共 x̂⫺x̄ 兲 ⫽â ␣ 共 1⫺it/ ␶ 兲 ⫹â ␣† 共 1⫹it/ ␶ 兲 .
Hence, using
关 â ␣ ,â ␣† 兴 ⫽ 关 â,â † 兴 ⫽2បm ␶ ,
we have
4 ␶ 2 具 共 p̂⫺ p̄ 兲 2 典 ⫽ 具 â ␣ â ␣† 典 ⫽ 具 â ␣† â ␣ ⫹2បm ␶ 典 ⫽2បm ␶ ,
4m 具 共 x̂⫺x̄ 兲
2
2
共4.4兲
典 ⫽ 具 â ␣ â ␣† 典 共 1⫹t 2 / ␶ 2 兲 ⫽2បm ␶ 共 1⫹t 2 / ␶ 2 兲 .
共4.5兲
共4.6兲
Thus, for any eigenstate of â,
⌬ p⫽ 冑បm/2␶
and ⌬x⫽ 冑ប ␶ /2m 冑1⫹t 2 / ␶ 2 ,
共4.7兲
consistent with Eq. 共2.9兲. Note that the minimum value of the
uncertainty product ⌬x⌬p is 21 ប; so at its minimum this
eigenfunction is a minimum uncertainty wave packet.
It is easy to find the form of the wave function ␺ ␣ (x,t).
From â ␣ ␺ ␣ ⫽0, we have
327
Am. J. Phys., Vol. 71, No. 4, April 2003
Mark Andrews
327
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冋
册
m 共 x⫺x̄ 兲
⳵␺ i
␺,
⫽ p̄⫹
⳵x ប
t⫺i ␶
and hence
␺ ␣ 共 x,t 兲 ⫽exp
共4.8兲
冋
册
i
m 共 x⫺x̄ 兲 2
␽ 共 t 兲 ⫹p̄x⫹
,
ប
2 共 t⫺i ␶ 兲
共4.9兲
where ␽ (t) can be determined by substituting ␺ ␣ (x,t) into
Schrödinger’s equation. This substitution gives
␽˙ ⫽ 21 iប/ 共 t⫺i ␶ 兲 ⫺Ē,
共4.10兲
where Ēª 21 p̄ 2 /m, and therefore
␺ ␣ 共 x,t 兲 ⫽
1
¯
冑t⫺i ␶
e i 共 p̄x⫺E t 兲 /ប exp
冉
冊
im 共 x⫺x̄ 兲 2
.
2ប 共 t⫺i ␶ 兲
共4.11兲
Equation 共4.11兲 represents the moving Gaussian wave function found in most texts, but here its expectation values
( 具 x̂ 典 , 具 p̂ 典 ,⌬x,⌬ p) are known without any further integration.
The eigenfunction ␺ ␣ (x,t), with p̄⫽0 and ␶ ⫽0, must
共when appropriately normalized兲 be the propagator for the
free particle. The propagator is just the evolution of the ␦
function, and the eigenfunction 共with eigenvalue x̄) of x̂ is
␦ (x⫺x̄). Therefore, the eigenfunction of x̂⫺p̂t/m must approach ␦ (x⫺x̄) as t→0. Thus the propagator K(x,x ⬘ ,t),
such that
冕
⬁
⫺⬁
K 共 x,x ⬘ ,t 兲 ␺ 共 x ⬘ ,t 0 兲 dx ⬘ ⫽ ␺ 共 x,t 0 ⫹t 兲
V. UNIFORM BUT TIME-VARYING FORCE
冉
冊
im
⌿ 共 x,x ⬘ ,t 兲 ª exp
共 x⫺x ⬘ 兲 .
2បt
冑t
1
共4.13兲
To determine the factor, use the integral
⬁
⫺⬁
Thus C(y)⫹iS(y) satisfies the free Schrödinger equation if
y⫽ 冑m/ ␲ បtx, and ␾ ª 21 ⫹ 12 (1⫺i) 关 C(y)⫹iS(y) 兴 is the
time evolution of the step function and also satisfies âp̂ ␸
⫽0. 关If ␹ ªp̂ ␾ , then â ␹ ⫽0, so that ␹ ⫽ f (t) ␺ , where ␺ is
the
propagator,
and
therefore
␾ (x,t)⫽(i/ប) f (t)
⫻ 兰 x ␺ (x ⬘ ,t)dx ⬘ .] Figure 1 shows 兩 ␾ 2 兩 . Many other invariants can be made from powers, sums, and products of â and
p̂. These can be used to create a vast variety of solutions of
the free Schrödinger equation, but they will not be pursued
further here.
共4.12兲
for any wave function ␺ (x,t), can differ only by a constant
factor from
冕
Fig. 1. The free evolution of the step function. The graph shows 兩 ␾ 2 兩 where
␾ (x) is initially the step function. The shape is valid for all times t⬎0; all
that changes is the scale on the x-axis. The values of x shown are for t
⫽m/ ␲ ប.
⌿ 共 x,x ⬘ ,t 1 兲 ⌿ 共 x ⬘ ,x ⬙ ,t 2 兲 dx ⬘
⫽ 冑2 ␲ iប/m⌿ 共 x,x ⬙ ,t 1 ⫹t 2 兲 ,
共4.14兲
which is easily calculated by direct integration. Because the
propagator must propagate itself, it must be K(x,x ⬘ ,t)
⫽⌿(x,x ⬘ ,t)/ 冑2 ␲ iប/m.
Other invariants: For the free particle, p̂ is also invariant
共but independent of t兲. Thus the spatial derivative of any
solution of the free Schrödinger equation is also a solution,
as are all higher derivatives as follows directly from Schrödinger’s equation. A similar property holds for indefinite spatial integrals and time derivatives and integrals of any solution of the free Schrödinger equation. The spatial derivatives
of the propagator are the evolutions of the derivatives of the
␦ function, and can be related4 to Hermite polynomials using
2
2
共 d/d ␰ 兲 n e ⫺x ⫽ 共 ⫺1 兲 n H n 共 ␰ 兲 e ⫺x .
共4.15兲
The integral of the ␦ function is a step function and the
integral of the propagator can be expressed in terms of the
Fresnel integrals
C 共 y 兲 ⫹iS 共 y 兲 ⫽
冕
y
0
exp共 i ␲ z 兲 dz.
1
2
2
共4.16兲
The Hamiltonian is
Ĥ⫽
1 2
p̂ ⫺E共 t 兲 x̂,
2m
共5.1兲
which could represent a charged particle in a time-varying
electric field. The operator equations of motion are mx̂˙ ⫽p̂,
p̂˙ ⫽E(t) which differ from the free case only in that p̂˙ varies
with time, but is still independent of x̂ and p̂. Thus, all we
need to do is change the invariant â by adding a suitable
time-varying multiple of the unit operator:
âªmx̂⫺p̂ 共 t⫺i ␶ 兲 ⫹ ␴ 共 t 兲 .
共5.2兲
Then â˙ ⫽⫺E(t)(t⫺i ␶ )⫹ ␴˙ , so that
␴共 t 兲⫽
冕
t
0
共 t ⬘ ⫺i ␶ 兲 E共 t ⬘ 兲 dt ⬘ .
共5.3兲
To generalize this invariant to refer to an arbitrary trajectory, we define
â ␣ ªm 共 x̂⫺x̄ 兲 ⫺ 共 p̂⫺ p̄ 兲共 t⫺i ␶ 兲 ⫽â⫺ ␣ ,
共5.4兲
where x̄, p̄ satisfy the classical equations of motion mx̄˙ ⫽p̄
and p̄˙ ⫽E(t) to ensure that â ␣ is invariant. Here ␣ ⫽mx̄
⫺ p̄(t⫺i ␶ )⫹ ␴ (t) and the real and imaginary parts of the
constant ␣ give the classical motion
p̄⫽ p̄ 0 ⫹
冕
t
0
E共 t ⬘ 兲 dt ⬘ ,
mx̄⫽mx̄ 0 ⫹ p̄ 0 t⫹
冕
t
0
共 t⫺t ⬘ 兲 E共 t ⬘ 兲 dt ⬘ .
共5.5兲
328
Am. J. Phys., Vol. 71, No. 4, April 2003
Mark Andrews
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As before 具 â ␣ 典 ⫽0 gives 具 p̂ 典 ⫽p̄ and 具 x̂ 典 ⫽x̄. Thus 具 x̂ 典 , 具 p̂ 典
exactly follow the classical trajectory. The widths ⌬x, ⌬p
are not affected by ␴ (t) and therefore evolve exactly as they
would if no force were acting. The wave function has the
same form as for the free particle 关see Eq. 共4.11兲兴, except that
x̄, p̄ now follow a trajectory that depends on the field E(t).
冋
␺ ⫽exp
册
i
共 ␽ 共 t 兲 ⫹ p̄x 兲 ⫺ 共 x⫺x̄ 兲 2 /L 2 ,
ប
1
1 2 1
␽˙ ⫽⫺ ប ␻ ⫺
¯p ⫹ m ␻ 2 x̄ 2 ,
2
2m
2
The Hamiltonian is
1 2 1
p̂ ⫹ m ␻ 2 x̂ 2 ,
Ĥ⫽
2m
2
共6.1兲
and the operator equations of motion are mx̂˙ ⫽ p̂, p̂˙
⫽⫺m ␻ 2 x̂. From these equations, we have x̂˙ ⫹ip̂˙ /m ␻
⫽⫺i ␻ (x̂⫹ip̂/m ␻ ), and therefore
âªe i ␻ t 共 x̂⫹ip̂/m ␻ 兲 /L
共6.2兲
is invariant. The divisor Lª 冑2ប/(m ␻ ) is inserted to make
关 â,â † 兴 ⫽1, as usual.
Energy eigenstates: Using this invariant operator allows a
slightly different approach to finding the energy eigenstates.
When multiplied by exp(⫺iEt/ប), an eigenstate with energy
E satisfies Schrödinger’s equation and therefore if â is applied to such an energy eigenstate, it either gives zero or a
solution of Schrödinger’s equation with its only time dependence in the factor exp关⫺i(E⫺ប␻)t/ប兴, that is, an eigenstate
with E lower by ប␻. Similarly â † , which is also invariant,
raises the energy by ប␻. Because the energy eigenvalues
must be positive, if we repeatedly apply â to any energy
eigenstate, we must eventually reach a ground state ␺ 0 such
that â ␺ 0 ⫽0. Then â † â ␺ 0 ⫽0, and
Ĥ⫽ប ␻ 共 â † â⫹ 21 兲 ,
共6.3兲
so the lowest energy eigenvalue is E .0 ⫽ 21 ប ␻ , and every eigenvalue must be of the form (n⫹ 21 )ប ␻ with n⫽0,1,2,... .
The spatial dependence of the wave function ␺ 0 (x,t) is
found from â ␺ 0 ⫽0, which gives
␺ 0 共 x,t 兲 ⫽e ⫺i␻t/2 exp共 ⫺x 2 /L 2 兲 .
共6.4兲
Applying â repeatedly to ␺ 0 (x,t) gives the excited states, in
the usual way.
Coherent states: To deal with an eigenstate of â with eigenvalue ␣, we first find the classical motion x̄, p̄ that has
real initial values x̄ 0 , p̄ 0 such that ␣ ⫽(x̄ 0 ⫹ip̄ 0 /m ␻ )/L.
Then e i ␻ t (x̄⫹ip̄/m ␻ )/L will be constant 共and equal to ␣兲
for all time. Now we define
†
â ␣ ªâ⫺ ␣ ⫽e i ␻ t 关共 x̂⫺x̄ 兲 ⫹i 共 p̂⫺p̄ 兲 /m ␻ 兴 /L.
共6.5兲
Then the solutions of â ␣ ␺ ⫽0 will have 具 â ␣ 典 ⫽0, and therefore 具 x̂ 典 ⫽x̄ and 具 p̂ 典 ⫽p̄; that is, the wave packet evolves
with its centroid at x̄, p̄ following a sinusoidal oscillation. To
find the spread in position, we use
x̂⫺x̄⫽ 12 L 共 â ␣ e ⫺i ␻ t ⫹â ␣† e i ␻ t 兲 ,
共6.6兲
and obtain
具 共 x̂⫺x̄ 兲 2 典 ⫽ 41 L 2 具 â ␣ â ␣† 典 .
共6.7兲
we get ⌬x⫽ L. This reThen using
sult is the same as for the ground state energy eigenstate,
1
2
共6.8兲
and ␽ (t) can be found by inserting Eq. 共6.8兲 into Schrödinger’s equation. The result is
VI. THE SIMPLE HARMONIC OSCILLATOR
关 â ␣ ,â ␣† 兴 ⫽ 关 â,â † 兴 ⫽1,
which corresponds to x̄⫽ p̄⫽0. Similarly, ⌬p⫽ប/L, so these
states always have the minimum uncertainty product.
The differential equation â ␣ ␺ ⫽0 yields
共6.9兲
so that
␽ ⫽⫺ 21 ប ␻ t⫺ 21 x̄p̄.
共6.10兲
These states, which follow the classical oscillation without
change of width, are known as coherent states. The ‘‘raising
operator’’ â ␣† can be applied to these states to obtain displaced excited states that are essentially Hermite–Gaussians
that also follow a classical oscillation.4
Squeezed states: The operator â in Eq. 共6.2兲 is not the
most general invariant that is linear in x̂ and p̂. The form
b̂⫽ ␮ â⫹ v â †
共6.11兲
will be invariant if ␮ and ␯ are constants. To set the scale of
b̂, we also take 关 b̂,b̂ † 兴 ⫽1, which implies that ␮ * ␮ ⫺ ␯ * ␯
⫽1. In terms of x̂ and p̂,
បb̂⫽ ␰ 共 t 兲 p̂⫺ ␲ 共 t 兲 x̂,
共6.12兲
where
␰ ⫽ 12 iL 共 ␮ e i ␻ t ⫺ ␯ e ⫺i ␻ t 兲 , ␲ ⫽⫺ប 共 ␮ e i ␻ t ⫹ ␯ e ⫺i ␻ t 兲 /L.
共6.13兲
Note that ␰ and ␲ have the dimensions of length and momentum, respectively, and satisfy the classical equations of mo˙ ⫽⫺m ␻ 2 ␰ , but can be complex. The phase
tion, m ␰˙ ⫽ ␲ , ␲
assigned to b̂ is not important, so we take ␮ to be real. Then
adding a constant phase ␽ to ␯ is equivalent to shifting the
time by 21 ␽ / ␻ 关because ␮ a(t)⫹ ␯ e i ␽ a † (t)⫽ ␮ e i ␻ t a(0)
⫹ ␯ e ⫺i( ␻ t⫺ ␽ ) a † (0)⫽e 1/2i ␽ b̂(t⫺ 21 ␽ / ␻ )]. Thus, we can also
take ␯ to be real. Instead of the two parameters ␮ and ␯, it is
sometimes convenient to use the ‘‘squeezing factor’’ sª ␮
⫹ ␯ . Then ␮ 2 ⫺ ␯ 2 ⫽1 leads to ␮ ⫺ ␯ ⫽1/s and s⭓1. The
coherent states correspond to ␮ ⫽1, ␯ ⫽0, or s⫽1.
Next we introduce
បb̂ ␣ ª ␰ 共 t 兲共 p̂⫺ p̄ 兲 ⫺ ␲ 共 t 兲共 x̂⫺x̄ 兲 ,
共6.14兲
where x̄, p̄ is a classical oscillation and again the solution of
b̂ ␣ ␺ ⫽0 共and therefore any eigenstate of b̂) will follow the
classical motion in the sense that 具 x̂ 典 ⫽x̄ and 具 p̂ 典 ⫽p̄. From
Eq. 共6.14兲 and using Im(␰*␲)⫽ 21ប, we find
x̂⫺x̄⫽i 共 ␰ * b̂ ␣ ⫺ ␰ b̂ ␣† 兲 ,
p̂⫺p̄⫽i 共 ␲ * b̂ ␣ ⫺ ␲ b̂ ␣† 兲 , 共6.15兲
and therefore, similarly to the derivation of Eq. 共6.7兲, ⌬x
⫽ 兩 ␰ 兩 and ⌬p⫽ 兩 ␲ 兩 . From Eq. 共6.13兲, at t⫽0, ⌬x⫽ 21 L( ␮
⫺ ␯ )⫽ 12 L/s, and ⌬ p⫽បs/L. Then 2⌬x/L and L⌬p/ប oscillate between the two extremes of s and 1/s, each taking its
maximum value when the other takes its minimum. The differential equation corresponding to b̂ ␣ ␺ ⫽0 is easily solved
329
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Mark Andrews
329
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关similarly to Eq. 共6.8兲兴 and the squeezed states of the oscillator are also moving Gaussians.
The propagator requires ␰ ⫽0 to ensure that the eigenfunctions of b̂ will be ␦ (x⫺x̄) and ␰ will be zero at t⫽0 if s
⫽⬁. 共A factor of s must be removed from b̂ ␣ to keep it finite
in this limit.兲
VII. REPULSIVE INVERSE SQUARE POTENTIAL
The Hamiltonian is
Ĥ⫽
1 2
p̂ ⫹c/x 2 .
2m
共7.1兲
Only the repulsive case (c⬎0) will be considered because
the attractive case is too singular and requires special
treatment.5 Such a potential arises, with c⫽ប 2 ᐉ(ᐉ⫹1)/2m,
for the radial part of the wave function 共multiplied by r兲 for
a spherically symmetric system in three dimensions, and the
spherical Bessel functions are familiar as the radial part of
the energy eigenfunctions for the free particle. We will treat
the general case of arbitrary c⬎0 in one dimension. The
repulsive hill is impenetrable so there can be no tunneling
through the origin. Therefore, only x⬎0 need be considered,
and we expect the wave function to vanish as x→0.
The operator equations of motion are mx̂˙ ⫽p̂, ˙p̂ ⫽2c/x 3 .
Therefore
d
1
d
共 x̂p̂ 兲 ⫽ 共 p̂x̂ 兲 ⫽ p̂ 2 ⫹2c/x 2 ⫽2Ĥ
dt
dt
m
共7.2兲
⪠21 共 x̂p̂⫹p̂x̂ 兲 ⫺2tĤ
共7.3兲
and
is invariant. 共We have used the symmetrized form to keep it
Hermitean.兲 Furthermore,
1
1 d 2
m 共 x̂ 兲 ⫽ 共 x̂ p̂⫹p̂x̂ 兲 ⫽â⫹2tĤ,
2 dt
2
共7.4兲
so that
b̂ª 21 mx̂ 2 ⫺tâ⫺t 2 Ĥ⫽ 21 mx̂ 2 ⫺ 21 共 x̂ p̂⫹p̂x̂ 兲 ⫹t 2 Ĥ
共7.5兲
is also invariant. It follows that 具 x 典 is quadratic in t for any
state.
Also the eigenfunction of b̂, with arbitrary eigenvalue ␤,
will give the propagator for the system because as t→0, it
must approach an eigenfunction of x̂ 2 , that is, ␦ (x 2
⫺2 ␤ /m) apart from a factor. 共The variable x 2 is just as good
as x for this system because we need only consider positive
values of x.兲
The solution of the differential equation b̂ ␸ ⫽ ␤ ␸ is
2
␸ ⫽exp共 i 21 mx 2 /បt 兲 冑xJ ␯ 共 冑2m ␤ /ប 2 x/t 兲 ,
共7.6兲
where ␯ ª ⫹2cm/ប . A discussion of how to find this solution is given in the Appendix. The positive value of ␯ gives
the regular solution; the negative value also gives a solution,
but it is singular at x⫽0. As before, this eigenfunction requires a time-dependent factor to also satisfy Schrödinger’s
equation. The factor can be found to be exp(i␤/បt)/t by inserting ␸ in Eq. 共7.6兲 into Schrödinger’s equation. Thus
2
1
4
2
␺ ⫽exp关 i 共 ␤ ⫹ 12 mx 2 兲 /បt 兴共 冑x/t 兲 J ␯ 共 冑2m ␤ /ប 2 x/t 兲
共7.7兲
is the eigenfunction of b̂ that also evolves as a wave function.
As before, a more general wave function will be obtained
by replacing t by t⫺i ␶ . The propagator results from the limit
␶ →0, while for sufficiently large ␶, the wave function corresponds to a localized ‘‘particle’’ coming toward the origin,
being slowed to a standstill by the repulsive force and then
retreating, as in the classical case. The details of this calculation, and the exact normalization and calculation of 具 x 2 典 ,
具 Ĥ 典 , and 具 â 典 , will be pursued in the Appendix.
VIII. RELATION TO THE HEISENBERG PICTURE
For the operator Â, the equivalent in the Heisenberg
picture6 is  H ªT̂ † ÂT̂, where T̂ is the time-evolution operator such that ␺ (t)⫽T̂ ␺ (0) and iបdT̂/dt⫽ĤT̂. It follows
that
ih
⳵ Â
d H
⫽ 关 Â H ,Ĥ H 兴 ⫹ihT̂ †
T̂.
dt
⳵t
共8.1兲
That is,
d H †
dÂ
⫽T̂
T̂ .
dt
dt
共8.2兲
共Note that where d/dt is applied to an operator in Heisenberg’s picture, it does not have the special meaning it has
when applied to an operator in Schrödinger’s picture兲.
Thus the total time derivative of any operator is the Schrödinger picture version of the time derivative of its Heisenberg picture version, which implies that the operator equations of motion will have the same form in the Heisenberg
picture as they have in Schrödinger picture using the total
time derivative.
The formal development in Sec. II and the determination
and application of the invariant operators could therefore be
presented in the Heisenberg picture, but the advantage of
using the total time derivative in an elementary course is that
the Heisenberg picture requires much more mathematical
machinery. One must deal with exponentials of operators
and, for Hamiltonians that depend on the time, the determination of the explicit form of the transformation is often
quite difficult.
IX. DISCUSSION
Time-dependent invariant operators that are linear in x̂ and
p̂ can be found4 for any Hamiltonian that is quadratic 共or
linear兲 in x̂ and p̂, even with arbitrary time dependence in
the Hamiltonian. These systems include, for example, harmonic oscillators that are driven 共a time-varying linear term
in H兲 or have varying frequency, as well as the cases treated
previously. The case of V⫽c/x 2 shows that the method is not
restricted to quadratic Hamiltonians. However, I have not
found a useful invariant for the important case of V⫽c/x.
An indication that this potential might be more difficult can
be seen from the corresponding classical system. For all classical one-dimensional potentials, an integral gives the time as
a function of x and the energy. In the previous examples, the
time 共or some function of the time兲 can be expressed in a
form that is linear in the velocity and the energy and then the
eigenfunctions of the corresponding quantum operator can be
found easily. But for V⫽c/x, the classical result is highly
non-linear in the energy.
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Am. J. Phys., Vol. 71, No. 4, April 2003
Mark Andrews
330
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In summary, the use of the total derivative of operators 共in
Schrödinger’s picture of the time evolution兲 simplifies the
formal development of quantum mechanics and its application to particular systems. This approach eliminates many of
the tedious calculations of commutators that are required in
the usual treatment and allows a development that more
closely follows the familiar classical treatment. Furthermore,
it makes it easy to use the simple invariant operators that can
be found for many of the systems often studied in elementary
quantum mechanics to find simple wave packets that illustrate the dynamics of the system and to find the propagator.
APPENDIX: ANALYSIS OF THE WAVE FUNCTION
FOR THE POTENTIAL VÄcÕx 2
The equation b̂ ␸ ⫽ ␤ ␸ is, in Schrödinger’s representation,
an ordinary differential equation 共ODE兲 in x with t as a parameter. Its solution, apart from a possible time-dependent
factor, will satisfy Schrödinger’s equation, which is a partial
differential equation 共PDE兲. But b̂ ␸ ⫽ ␤ ␸ is not easy to solve
even though it is only an ODE. 关The solution was given in
Eq. 共7.7兲.兴 Equation-solving programs such as Maple or
Mathematica will probably not solve it because they will not
attempt to factor out the required factor of 冑x exp( 12imx2/បt)
to reduce the equation to Bessel’s form.
Surprisingly, one way to obtain the solution is to convert
the equation to a PDE using the requirement that the solution
also satisfy Schrödinger’s equation. First write b̂ª 21 mx̂ 2
⫺t(x̂p̂⫺ 21 iប)⫹t 2 Ĥ, and then insert p̂⫽⫺iប ⳵ / ⳵ x and Ĥ
⫽iប ⳵ / ⳵ t to give the PDE,
tx
冋
冉
⳵␸ 2 ⳵␸
1
1
i
␤ ⫺ mx 2
⫹t
⫹ t⫹
⳵x
⳵t
2
ប
2
冊册
␸ ⫽0.
共A1兲
Such first-order, linear PDEs can be solved easily by standard methods 共or using equation-solving programs兲, and it is
easy to verify directly that ␸ ⫽exp关i(␤⫹ 21mx2)/(បt)兴 f(x/t)/冑x
satisfies Eq. 共A1兲, if f is an arbitrary function of x/t. If we
substitute this form of ␸ into Schrödinger’s equation, we see
that f (u), with u⫽x/t, must satisfy the ODE
u 2 f ⬙ ⫺u f ⬘ ⫹ 共 c 1 ⫹c 2 u 2 兲 f ⫽0,
共A2兲
where c 1 ⫽ 34 ⫺2cm/ប 2 and c 2 ⫽2m ␤ /ប 2 . It is not difficult to
recognize that Eq. 共A2兲 can be converted to Bessel’s equation by dividing f (u) by u. Thus
␸ ⫽exp关 i 共 ␤ ⫹ mx 兲 / 共 បt 兲兴 uJ ␯ 共 冑2m ␤ /ប u 兲 / 冑x,
2
1
2
2
共A3兲
where ␯ ⫽1⫺c 1 ⫽ ⫹2cm/ប . The form of Eq. 共A3兲 agrees
with Eq. 共7.7兲.
Expectation values: As before, t can be complex and the
function
2
1
4
冋
册
1
i ␤ ⫹ 2 mx 2 冑x
␺ ⫽exp
J
ប t⫺i ␶ t⫺i ␶ ␯
冉冑
2m ␤ x
ប 2 t⫺i ␶
冊
B̂ª mx̂ ⫺ 共 t⫺i ␶ 兲共 x̂ p̂⫹ p̂x̂ 兲 ⫹ 共 t⫺i ␶ 兲 Ĥ,
2
1
2
⫻
共A4兲
2
共A5兲
and ␺ satisfies the time-dependent Schrödinger equation for
the potential V⫽c/x 2 .
To calculate expectation values, we first normalize the
wave function using
1
2 ␶ 共 ␤ r ⫹ 2 mx 2 兲 ⫹ ␤ i t
ប
t 2⫹ ␶ 2
册
x
J 共 a * x 兲 J ␯ 共 ax 兲 ,
t ⫹␶2 ␯
共A6兲
2
where aª 冑2m ␤ /ប 2 /(t⫺i ␶ ) and ␤ ⫽ ␤ r ⫹i ␤ i . Now apply
the integral
冕
⬁
0
exp共 ⫺ 21 ␭x 2 兲 J ␯ 共 ax 兲 J ␯ 共 bx 兲 x dx
⫽␭ ⫺1 exp关 ⫺ 21 ␭ ⫺1 共 a 2 ⫹b 2 兲兴 I ␯ 共 ab/␭ 兲
共A7兲
关for Re(␯)⬎⫺1] given by Weber in 1868. We substitute b
⫽a * and ␭⫽2 ␶ m/ប(t 2 ⫹ ␶ 2 ), and obtain
7
冕 ␺ *␺
⬁
dx⫽
0
冉 冊冉 冊
␤r
兩␤兩
ប
exp ⫺
I
.
2␶m
ប␶ ␯ ប␶
共A8兲
To determine 具 x̂ 2 典 we differentiate Weber’s integral in Eq.
共A7兲 with respect to ␭ to obtain8
冕
⬁
0
exp共 ⫺ 21 ␭x 2 兲 J ␯ 共 ax 兲 J ␯ 共 bx 兲 x 3 dx
⫽2␭ ⫺2 e ⫺w 兵 共 ␯ ⫹1⫺w 兲 I ␯ 共 z 兲 ⫹zI ␯ ⫹1 共 z 兲 其 ,
共A9兲
where wª (a ⫹b )/␭ and zªab/␭. Equation 共A9兲 implies that
2
1
2
2
冋
册
ប
t
I ␯ ⫹1 共 z 兲 ␤ r
1
␯ ⫹1⫹z
⫺
,
m 具 x̂ 2 典 ⫽ ␤ r ⫹ ␤ i ⫹ 共 t 2 ⫹ ␶ 2 兲
2
␶
2␶
I ␯共 z 兲
ប␶
共A10兲
and z⫽ 兩 ␤ 兩 /ប ␶ . But Eq. 共A5兲 can be written as
B̂ª 21 mx̂ 2 ⫺ 共 t⫺i ␶ 兲 â⫺ 共 t 2 ⫹ ␶ 2 兲 Ĥ,
共A11兲
and therefore, taking the real and imaginary parts of the expectation value of B̂, ␤ i ⫽ ␶ 具 â 典 , and
1
2
m 具 x̂ 2 典 ⫽ ␤ r ⫹ 共 t/ ␶ 兲 ␤ i ⫹ 共 t 2 ⫹ ␶ 2 兲 具 Ĥ 典 .
共A12兲
If we compare this form with Eq. 共A10兲, we see that
具 Ĥ 典 ⫽
冋
册
ប
I ␯ ⫹1 共 z 兲 ␤ r
␯ ⫹1⫹z
⫺
.
2␶
I ␯共 z 兲
ប␶
共A13兲
The propagator: A more symmetrical form for the eigenfunction of b̂, with eigenvalue 21 mx ⬘ 2 , is
2
is the eigenfunction, with eigenvalue ␤, of the invariant operator
1
2
冋
␺ * ␺ ⫽exp ⫺
⌿ 共 x,x ⬘ ,t 兲 ⫽
冑xx ⬘
t
冋
exp
册冉 冊
im 2
mxx ⬘
. 共A14兲
共 x ⫹x ⬘ 2 兲 J ␯
2បt
បt
As shown in Sec. V, this eigenfunction must be the propagator apart from a factor independent of x and t that will now
be determined. The eigenfunction satisfies Schrödinger’s
equation and therefore should propagate itself. If we use the
integral in Eq. 共A7兲, we can evaluate the relevant integral to
give
冕
⬁
0
⌿ 共 x,x ⬘ ,t 1 兲 ⌿ 共 x ⬘ ,x ⬙ ,t 2 兲 dx ⬘
⫽i ␯ ⫹1 共 ប/m 兲 ⌿ 共 x,x ⬙ ,t 1 ⫹t 2 兲 ,
共A15兲
where the relation I ␯ (iz)⫽i ␯ J ␯ (z) 关for Im(z)⬎0] has been
used. Hence the propagator K, such that
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Am. J. Phys., Vol. 71, No. 4, April 2003
Mark Andrews
331
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冕
⬁
0
K 共 x,x ⬘ ,t 兲 ␺ 共 x ⬘ ,t 0 兲 dx ⬘ ⫽ ␺ 共 x,t 0 ⫹t 兲
共A16兲
for any wave function ␺ (x,t), is
K 共 x,x ⬘ ,t 兲 ⫽i ⫺ 共 ␯ ⫹1 兲 共 m/ប 兲 ⌿ 共 x,x ⬘ ,t 兲 .
共A17兲
The single-hump regime: The argument uªax of the
Bessel function in ␺ has 兩 u 2 兩 ⫽z␭x 2 . But ␺ * ␺ contains a
factor exp(⫺ 12␭x2). Hence if 兩 ␤ 兩 Ⰶប ␶ , that is zⰆ1, then 兩␺兩
will become small 共because of the exponential factor兲 before
the argument of the Bessel function leaves the region where
兩 u 兩 Ⰶ1, and therefore8 J ␯ (u)⬇( 21 u) ␯ /⌫( ␯ ⫹1). In this regime
the spatial form of ␺ is x ␯ ⫹1/2 exp关 21imប⫺1x2/(t⫺i␶)兴, and 兩␺兩
has just a single hump. If we substitute I ␯ (z)⬇( 21 z) ␯ /⌫( ␯
⫹1) into Eq. 共A13兲, we obtain 具 Ĥ 典 ⬇ 21 ប ␶ ⫺1 ( ␯ ⫹1), and
1
2
m 具 x̂ 2 典 ⬇ 具 Ĥ 典 共 t 2 ⫹ ␶ 2 兲 .
共A18兲
Equation 共A18兲 can be compared with the classical motion
1
2
mx 2 ⫽Et 2 ⫹ 21 mc/E.
共A19兲
The first term Et 2 corresponds as closely as possible to the
quantum term 具 Ĥ 典 t 2 , while the second term, specifying the
closest approach to the origin, is 具 Ĥ 典 ␶ 2 ⬇ 41 ប 2 ( ␯ ⫹1) 2 / 具 Ĥ 典 in
the quantum case. This quantity is to be compared with
1
1
1 2
1
2
2 mc/E noting that 2 mc⫽ 4 ប ( ␯ ⫺ 4 ). The closest approach
in the quantum case has a lower limit for 21 m 具 x̂ 2 典 of
9 2
16 ប / 具 Ĥ 典 for a very weak potential (c⬇0) and is always
greater than in the classical case with the same energy. This
behavior is due to the existence of a completely repulsive
barrier at x⫽0 共because ␺ is always zero there兲, and therefore ␺ is squeezed against this barrier, whereas the classical
particle will approach the origin arbitrarily closely for sufficiently small c. In this regime, the wave packet motion is
similar to that of the classical particle coming toward the
origin, slowing to a standstill, and then retracing its inward
trajectory; however, there are quantum effects that are greatest near the closest approach to the origin.
Figure 2 shows two examples of the evolution of this
wave function; one has zⰆ1 and therefore has a single hump
while the other has zⰇ1 and is shown at a time when it has
two maxima.
Fig. 2. Two wave packets for the potential V⫽c/x 2 . In each case the wave
function is as in Eq. 共A4兲; 兩 ␺ 2 兩 is plotted with m⫽ប⫽c⫽1 and eigenvalue
1
␤ ⫽ 2 . In case 共a兲, ␶ ⫽10 and the wave function is shown for t⫽0 共solid兲
and for t⫽20 共dashed兲. This case is in the ‘‘single-hump’’ regime. In case
共b兲, ␶ ⫽0.05 and the wave function is shown for t⫽0 共solid line兲 and for t
⫽2 共dashed line兲. For t⫽0, the maximum value is about 2.5. This case is in
the regime where the Bessel function gives rise to multiple maxima.
Eugen Merzbacher, Quantum Mechanics, 2nd ed. 共Wiley, New York,
1970兲, p. 165.
2
L. D. Landau and E. M. Lifshitz, Quantum Mechanics 共Pergamon, New
York, 1958兲, Chap. 9. Also see Ref. 1, p. 337, Eq. 共15.13兲.
3
Eugen Merzbacher, Quantum Mechanics, 3rd ed. 共Wiley, New York,
1998兲, p. 49, Problems 2 and 3.
4
Mark Andrews, ‘‘Invariant operators for quadratic Hamiltonians,’’ Am. J.
Phys. 67, 336 –343 共1999兲.
5
For a recent discussion of the attractive case 共in three dimensions兲, see
Sydney A. Coon and Barry R. Holstein, ‘‘Anomalies in quantum mechanics: The 1/r 2 potential,’’ Am. J. Phys. 70, 513–519 共2002兲.
6
See Ref. 3, Chap. 14, Sec. 2.
7
G. N. Watson, Theory of Bessel Functions, 2nd ed. 共Cambridge University
Press, Cambridge, 1944兲, Chap XIII, Sec. 13.31, p. 395.
8
For relations involving Bessel’s functions, see Milton Abramowitz and
Irene A. Stegun, Handbook of Mathematical Functions 共NBS, Washington,
DC, 1964兲, Chap. 9.
1
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