Download The evolution of free wave packets

The evolution of free wave packets
Mark Andrews
Citation: American Journal of Physics 76, 1102 (2008); doi: 10.1119/1.2982628
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The evolution of free wave packets
Mark Andrewsa兲
Department of Physics, Australian National University, ACT 0200, Australia
共Received 30 December 2007; revised manuscript received 26 August 2008兲
We discuss four general features of the force-free evolution of wave packets: 共1兲 The spatial spread
of a packet changes with time in a simple way. 共2兲 For sufficiently short durations 共related to the
spread in the momentum of the packet兲 the probability distribution will move with uniform speed
and little change in shape. 共3兲 After a sufficiently long time 共related to the initial spatial spread兲 the
wave function converges to a simple form that is simply related to the momentum distribution of the
packet. In this asymptotic regime the shape of the probability distribution no longer changes, and its
scale increases linearly with the time. 共4兲 There is an infinite denumerable set of simple wave
packets 共the Hermite-Gauss packets兲 that do not change shape at any time during their evolution. ©
2008 American Association of Physics Teachers.
关DOI: 10.1119/1.2982628兴
共1兲 The spread of any wave packet as measured by ⌬x
= 具共x̂ − 具x̂典兲2典1/2 increases in time as
I. INTRODUCTION
The behavior of free wave packets as they evolve in time
is striking to many students of quantum mechanics. A wave
packet usually changes shape and always eventually spreads
out without limit. How different this behavior is from that of
a free classical particle. The reconciliation of the classical
and quantum descriptions is part of the bigger question of the
interpretation of quantum mechanics; here we will consider
only the dynamics of free wave packets as given by
Schrödinger’s equation.
We will consider only one-dimensional systems, but the
results apply to each dimension of a two or threedimensional system because the Hamiltonian Ĥ = p̂2 / 2m in
Cartesian coordinates is a sum of one-dimensional Hamiltonians. 共We will consistently use a hat to indicate an operator.兲
Systems with more than one dimension can display a greater
richness in behavior, because the parameters of the wave
function may be different in each dimension.
Most texts derive the propagator1
K共x,x⬘,t兲 = 冑m/2␲iបt exp关im共x − x⬘兲2/2បt兴,
共1兲
such that the evolution of an arbitrary initial wave packet
␺共x , 0兲 is
␺共x,t兲 =
冕
⬁
K共x,x⬘,t兲␺共x⬘,0兲dx⬘ .
共2兲
−⬁
There are not many wave packets for which this integral can
be easily evaluated, and the only case commonly treated is
the Gaussian packet. Therefore, the evolution of free wave
packets remains obscure to many students.
Ehrenfest’s result that 具p̂典 is constant and 具x̂典 = 具x̂典0
+ 具p̂典t / m is well known. 共Angled brackets will be used to
indicate expectation values; thus, 具â典 represents the expectation value of the operator â.兲 There are four other general
results on the free evolution of wave packets2 that can be
simply derived and greatly improve the general understanding of this evolution.
2
+ 共t − tmin兲2⌬2p/m2 ,
⌬x = 冑⌬min
共3兲
where ⌬min is the minimum value of the spread 共taken at time
tmin兲 and ⌬ p = 具共p̂ − 具p̂典兲2典1/2 does not change with time. If the
wave function is real everywhere 共which can be true only for
an instant兲, the packet has its minimum spread ⌬min at that
instant.
共2兲 For time intervals much less than mប / ⌬2p, the wave
packet moves with speed 具p̂典 / m with little change in the
shape of the probability distribution.
共3兲 The asymptotic evolution for times 兩t兩 Ⰷ m⌬2x / ប converges to a functional form simply related to the momentum
distribution of the packet, given by the Fourier transform of
the initial wave function. There are many forms of initial
wave packet for which the asymptotic evolution can be easily determined, because there are many functions for which
the Fourier transform can be simply evaluated. These two
time regimes cannot overlap because of Heisenberg’s uncertainty relation, ⌬x⌬ p 艌 ប / 2.
共4兲 There is an infinite number of simple wave packets that
do not change shape as they evolve. These wave functions
have the form of a Gaussian multiplied by an Hermite polynomial in x. By shape we mean the functional form of the
probability distribution; the scale 共and consequently the normalization兲 may change with time. Thus, a wave function
␺共x , t兲 such that 兩␺共x , t兲兩2 = ␥共t兲−1 f共x / ␥共t兲兲 does not change
shape. 关The factor ␥共t兲−1 is required to preserve normalization.兴 All wave packets evolve to an asymptotic form whose
shape does not change.
The simplest of the wave packets that do not change shape
as they evolve is the real Gaussian, which evolves to become
complex but retains its Gaussian shape. A recent publication3
claims that all wave packets become “approximately Gaussian” after a long enough time. This claim is too broad; there
are many wave packets that will never become anything like
a Gaussian. For example, any antisymmetric wave function
will remain antisymmetric. Furthermore, the Hermite-Gauss
wave packets retain their shape as they evolve and, apart
from the pure Gaussian, none of them could be described as
“approximately Gaussian.”
1102
Am. J. Phys. 76 共12兲, December 2008
http://aapt.org/ajp
© 2008 American Association of Physics Teachers
1102
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II. THE EVOLUTION OVER SHORT PERIODS
We start with the well-known expression for the evolution
of a wave packet in terms of the initial momentum distribution ␾共p兲,4
冕 冋冉
⬁
1
␺共x,t兲 =
冑2␲ប
1 2
i
pt
px −
ប
2m
exp
−⬁
冊册
␾共p兲dp,
共4兲
and express p2 in Eq. 共4兲 in terms of p − p̄ as p2 = p̄2 + 2p̄共p
− p̄兲 + 共p − p̄兲2, where p̄ = 具p̂典.5 We expect ␾共p兲 to be small if
共p − p̄兲2 Ⰷ ⌬2p. Therefore, if ⌬2pt / 2mប Ⰶ 1, the term 共p − p̄兲2
will make a negligible contribution to the phase of the exponential. In other words, the integrand contains a factor
exp关i共p − p̄兲2t / 2mប兴 that is close to unity throughout the
range of p where ␾共p兲 has any significant magnitude. Hence,
␺共x,t兲 ⬇
冉 冊
冑
冕 冋 冉 冊册
冉 冊冉 冊
1
2␲ប
⬁
⫻
exp
ip̄2t
2mប
i
p̄t
exp p x −
ប
m
−⬁
␾共p兲dp
共5a兲
2
ip̄ t
p̄t
␺ x − ,0 .
2mប
m
=exp
共5b兲
The wave function moves with speed p̄ / m and no change in
shape, just a change in phase. A more rigorous approach to
the error in this approximation is given in Sec. VII.
We have taken the initial time to be t = 0, but this analysis
can be applied at any time. Because ⌬ p does not change, the
time scale mប / 2⌬2p for changes in the shape applies throughout the evolution.
If the wave function has any discontinuity, then ⌬ p = ⬁ and
there will be very rapid changes. An example of this behavior will be considered in Sec. VI.
III. THE EVOLUTION FOR LARGE TIMES
For the propagator in Eq. 共1兲 we write the term 共x − x⬘兲2 in
the exponent as x2 − x̄2 − 2共x − x̄兲x⬘ + 共x⬘ − x̄兲2, with x̄ = 具x̂典t=0.
We expect ␺共x⬘ , 0兲 to be small if 共x⬘ − x̄兲2 Ⰷ ⌬2x and therefore
for all times 共future and past兲 with 兩t兩 Ⰷ m⌬2x / ប, the term in
共x⬘ − x̄兲2 will make a negligible change to the phase of the
exponential. Hence
冑
冋
冕 冋
冑 冋
m
im 2 2
exp
共x − x̄ 兲
2␲iបt
2បt
␺共x,t兲 ⬇
⬁
⫻
exp −
−⬁
=
im
共x − x̄兲x⬘ ␺共x⬘,0兲dx⬘
បt
册冉
1
冑2␲ប
冕
␺共x,0兲 =
再
a − 兩x兩 for 兩x兩 艋 a
0
otherwise,
冎
共8兲
show that the asymptotic behavior for large 兩t兩 is such that
兩␺共x , t兲 兩 ⬀t3/2x−2 sin2 共xma / 2បt兲, and the time scale beyond
which this behavior will be approached is ma2 / ប.
IV. THE EVOLUTION OF THE SPREAD ⌬x
To find how 具共x̂ − 具x̂典兲2典 varies with time we use the standard result that for any operator Â共t兲
dt具Â共t兲典 = 具⳵tÂ共t兲典 + 共i/ប兲具关Ĥ,Â共t兲兴典,
共9兲
where Ĥ is the Hamiltonian operator. 共We use dt to denote
the time derivative d / dt and ⳵t for the partial derivative ⳵ / ⳵t.
In this case ⳵tÂ共t兲 will apply only to any explicit dependence
of Â共t兲 on the time; that is, x̂ and p̂ are regarded as being
independent of the time when taking the partial derivative.兲
The calculation is somewhat tedious and is given as a problem in some textbooks.7,8 It becomes much simpler if we
introduce the shorthand notation
DÂ共t兲 ⬅ ⳵tÂ共t兲 + 共i/ប兲关Ĥ,Â共t兲兴,
共10兲
from which it follows that dt具Â共t兲典 = 具DÂ共t兲典, and
D共Â共t兲B̂共t兲兲 = Â共t兲DB̂共t兲 + 共DÂ共t兲兲B̂共t兲.
共11兲
For a free particle Dx̂ = p̂ / m and Dp̂ = 0. We introduce the
time-dependent operators X̂ = x̂ − 具x̂典 and P̂ = p̂ − 具p̂典. It follows
冊
m
im 2 2
m
exp
共x − x̄ 兲 ␾ 共x − x̄兲 ,
it
2បt
t
where
␾共p兲 =
册
册
is given in Sec. VII. This asymptotic form is important in the
theory of scattering.6
The asymptotic probability distribution is therefore given
by 共m / t兲 兩 ␾共m共x − x̄兲 / t兲兩2. This result has a simple interpretation: m共x − x̄兲 / t is the momentum required for the particle to
be at position x at time t if it was at x̄ at time t = 0. So, the
probability of finding the particle at x is proportional to the
probability that it had the right momentum to get there 共ignoring the fact that the initial distribution was spread over a
distance of order ⌬x about x̄兲.
Let us introduce the times t p = mប / 2⌬2p and tx = 2m⌬2x / ប.
Then t p 艋 tx 共from the uncertainty relation兲 and their equality
implies a Gaussian packet. In general, t p gives the time scale
for changes in shape 共including the changes in spatial scale兲
while tx gives the time scale to reach the asymptotic regime
where there is no further change in shape and 兩␺兩 expands
uniformly with time.
Problem 1. For the initial triangular wave packet with
共6a兲
that DX̂ = P̂ / m and DP̂ = 0. Then, using Eq. 共11兲, we have
DX̂2 = 2R̂ / m, where R̂ = 共P̂X̂ + X̂P̂兲 / 2, and DR̂ = P̂2 / m. Thus,
dt具R̂典 = 具P̂2典 / m and, because 具P̂2典 is constant, it follows that
共6b兲
m具R̂典 = 具P̂2典共t − t0兲,
共12兲
where t0 is a constant. Thus 具R̂典 will become zero at time t
⬁
exp共− ipx⬘/ប兲␺共x⬘,0兲dx⬘ ,
共7兲
−⬁
which is the initial momentum wave function of the packet.
A more rigorous derivation of the error in this approximation
= t0. We can integrate dt具X̂2典 = 2具R̂典 / m to give
具X̂2典 = m−2具P̂2典共t − t0兲2 + 具X̂2典0 ,
共13兲
which is equivalent to Eq. 共3兲. Note that Eq. 共12兲 implies
that, at any time t, we can calculate the time of minimum
1103
Am. J. Phys., Vol. 76, No. 12, December 2008
Mark Andrews
1103
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spread from 具P̂2典 and the value of 具R̂典 at time t. Every wave
packet has its minimum spread when 具R̂典 = 0.
If the wave function ␺ is real at some time, the i in
Schrödinger’s equation ⳵t␺ = −i共ប / 2m兲⳵2x ␺ shows that ␺ will
immediately become complex. Also, 具p̂典 = 0 because 具p̂典
= 兰␺ p̂␺dx = 兰共p̂␺兲ⴱ␺dx = −兰共p̂␺兲␺dx = −具p̂典. Similarly, 具p̂x̂典
= 兰␺ p̂x̂␺dx = 兰共p̂␺兲ⴱx̂␺dx = −兰共p̂␺兲共x̂␺兲dx = −具x̂p̂典. That is,
具R̂典 = 0 and therefore the wave packet has its minimum
spread at that instant. This result also shows that a free wave
packet which is real at some time cannot be real at any other
time. In fact, the probability is symmetric in time about the
instant when the wave function was real, because if ␺共x , t兲
satisfies Schrödinger’s equation, then so does ␺ⴱ共x , 2t0 − t兲.
Because these two functions are equal at t = t0 if ␺共x , t0兲 is
real, they must be equal at all times.
Given a wave packet ␰ that is at rest 共in the sense that
具p̂典␰ = 0 and therefore 具x̂典␰ is constant兲, we can make a moving packet by multiplying the wave function by exp共ipx / ប兲.
With ␺共x , t0兲 = exp共ipx / ប兲␰共x , t0兲, it follows that p̂␺ = p␺
+ exp共ipx / ប兲p̂␰, and therefore 具p̂典␺ = p. Similarly, 具p̂2典␺ = p2
+ 具p̂2典␰, and hence 共⌬ p兲␺ = 共⌬ p兲␰. Also, p̂共x␺兲 = px␺
+ exp共ipx / ប兲p̂共x␰兲, and therefore 具p̂x̂ + x̂p̂典␺ = 2p具x̂典␰ + 具p̂x̂
+ x̂p̂典␰. Thus, 具R̂典␺ = 具R̂典␰, and it follows that the evolution of
the spatial spread is the same as for ␰.
The evolution of ␺ can be obtained from that of ␰ by
applying a Galilean transformation,9
冉冉
t − t0
i
␺共x,t兲 = exp
px − p2
ប
2m
冊冊 冉
冊
t − t0
,t .
␰ x−p
m
共14兲
It is easy to check that this form of ␺ satisfies Schrödinger’s
equation if ␰ does and that ␺共x , t0兲 = exp共ipx / ប兲␰共x , t0兲.
A Gaussian packet 共that is, a wave function of the form
exp关ax2 + bx + c兴, where a , b, and c may be complex兲 will
remain Gaussian as it evolves, as can be seen by inserting a
Gaussian into the propagation integral in Eq. 共1兲. A real
Gaussian wave packet can be written as ␺共x兲
= exp共−x2 / 2d2兲, apart from normalization and a shift in origin. Then 具x2典 = d2 / 2, 具p2典 = ប2 / 2d2 and the uncertainty product ⌬x⌬ p is ប / 2, which shows that every real Gaussian is a
minimum uncertainty state. A packet can have the minimum
uncertainty property only for an instant, because every
packet evolves with ⌬ p fixed and ⌬x changing. It was shown
above that a Gaussian of the form exp共ipx / ប − x2 / 2d2兲 has
the same ⌬x and ⌬ p as when p = 0 and therefore is also a
minimum uncertainty state. It is shown in many texts10 that
every minimum uncertainty state has this form.
Problem
2. The
initial
wave
packet
␺共x兲
= cos共px / ប兲exp共−x2 / 2d2兲 can be expressed as the superposition of the two packets exp共ipx / ប − x2 / 2d2兲 and
exp共−ipx / ប − x2 / 2d2兲. Apply the results of the previous paragraph and Eq. 共13兲 to each of these two packets to show that
if p Ⰷ ប / d, then ␺共x兲 will evolve into two separate packets,
and if p Ⰶ ប / d, the two packets will overlap indefinitely. 关Use
the values of 具x2典 and 具p2典 for the real Gaussian given in the
previous paragraph. It is not necessary to evaluate any other
integrals or to calculate the explicit time evolution of any
wave functions.兴
Problem 3. 共a兲 Show that the minimum spread ⌬min of any
2
wave packet is given by ⌬min
= 共具X2典具P2典 − 具R典2兲 / 具P2典. 共b兲 For
the general Gaussian packet ␺共x兲 = exp共ax2 + bx + c兲, where a,
b, and c may be complex, it is straightforward to show that
具X2典 = 1 / 4␣, 具P2典 = 共␣2 + ␤2兲 / ␣, and 具R典 = ␤ / 2␣, where a =
−␣ + i␤. Use these results to show that every Gaussian packet
becomes a minimum uncertainty packet at some time.
Problem 4. Consider a wave packet ␺共x兲 that is initially
real and symmetric. Then, its momentum wave function ␾共p兲
is also real and symmetric. From Eq. 共6兲 ␹共x , t兲 = 冑m / t␾共共x
− x̄兲m / t兲 is real and, for large times, has the same magnitude
as the complex evolution ␺共x , t兲 of ␺共x兲. That is, ␹共x , t兲 is an
envelope to the real and imaginary parts of ␺共x , t兲. Although
␹共x , t兲 will have the same spatial spread as ␺共x , t兲, the spread
in momentum will be very different. Show that
具共x̂ − x̄兲2典␹ = 共t/m兲2具共p̂ − p̄兲2典␺共x兲
共15兲
具共p̂ − p̄兲2典␹ = 共m/t兲2具共x̂ − x̄兲2典␺共x兲 .
共16兲
It follows that the uncertainty product ⌬x⌬ p of the envelope
of the completely evolved packet will be the same as that of
the initial packet.
V. THE HERMITE-GAUSS WAVE PACKETS
We will first find the most general solution of the free
Schrödinger equation that is of Gaussian form
共17兲
⌿共x,t兲 = exp关a共t兲x2 + b共t兲x + c共t兲兴.
This form will enable us to generate a sequence of solutions
that are Hermite polynomials multiplied by Gaussians. It is
easy to verify that ⌿共x , t兲 will be a solution if
ȧ =
2iប 2
a ,
m
共18a兲
ḃ =
2iប
ab,
m
共18b兲
ċ =
iប
共2a + b2兲.
2m
共18c兲
Equation 共18a兲 gives a = im / 2ប共t + tc兲, where tc is a constant
that could be complex. The real part of tc would merely
change the origin of t, so we take tc = −i␶ where ␶ is real, and
take ␶ positive to make 兩⌿共x , t兲兩2 normalizable. Then, Eq.
共18b兲 becomes ḃ = −b / 共t − i␶兲 which gives b = b0 / 共t − i␶兲. We
take b0 = 2␭␶ / d, where we have introduced the length d
= 冑ប␶ / m, so that ␭ is a dimensionless constant. Then, it follows that c = − 21 log共t − i␶兲 − 2i␶␭2 / 共t − i␶兲 + c0, where c0 is
constant. Thus,
⌿共x,t兲 =
1
冑t − i␶
冉
exp
冊
ix2/2d2 + 2␭x/d − 2i␭2
+ c0 . 共19兲
共t − i␶兲/␶
关This solution of the free Schrödinger equation is essentially
the same as the propagator K共x , x⬘ , t兲 in Eq. 共1兲, which is a
solution except at t = 0, where it is singular. To obtain ⌿ from
K replace x⬘ by 2i␭d and t by t − i␶, which avoids the singularity.兴
1104
Am. J. Phys., Vol. 76, No. 12, December 2008
Mark Andrews
1104
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Now, we expand the solution in Eq. 共19兲 in powers of ␭:
⬁
␺n共x , t兲␭n / n!. This expansion can be achieved
⌿共x , t兲 = 兺n=0
using the generating function for the Hermite polynomials,
⬁
n
s
exp共2zs − s 兲 = 兺 Hn共z兲 .
n!
n=0
共20兲
2
1
冑t − i␶
冉
exp
冊
2x
i␶x2
+ s − s2 ,
2d 共t − i␶兲 ␥
2
冉 冊 冉
t + i␶
␺n共x,t兲 =
冑t − i␶ t − i␶
n/2
共21兲
冊
itx2
x2
exp
Hn共x/␥兲
2 −
2␶␥
2␥2
共22兲
is a solution of the free Schrödinger equation. These
Hermite-Gauss wave packets do not change their shape as
they evolve, but the scale ␥ does change.
The normalized probability distributions of the first few
are
兩␺0共x,t兲兩2 =
兩␺1共x,t兲兩2 =
兩␺2共x,t兲兩2 =
1
exp共− x2/␥2兲,
␥冑␲
共23兲
2 x2
2 2
2 exp共− x /␥ 兲,
冑
␥
␥ ␲
1
2␥冑␲
共24兲
共1 − 2x2/␥2兲2 exp共− x2/␥2兲.
共25兲
These probability distributions have the same spatial form as
the energy eigenfunctions of the harmonic oscillator.
It follows from the completeness of the harmonic oscillator eigenfunctions that this set of solutions forms a complete
basis for the expansion of an arbitrary wave function at any
particular time. This result provides an alternative method of
calculating the free evolution of any wave function
Each of the solutions ␺n共x , t兲 remains centered on the origin. To obtain the solutions that move uniformly with speed
p / m, we can either multiply by exp共ipx / ប兲 and carry out the
Galilean transformation in Eq. 共14兲 or we can modify the
generating solution by replacing ␭ by ␭ + pd / 2ប, so that
⌿共x,t兲 =
1
冑t − i␶
冋
⫻ exp i
− ␭2
冋冉
exp
i
p 2t
px −
ប
2m
冊册
册
0.6
0.4
0.8
0.3
0.1
0.0
x
0
1
2
3
4
Fig. 1. The probability distribution 兩␺共x , t兲兩2 of the wave packet in Eq. 共27兲,
whose initial probability distribution is shown by the solid curve. Only
half of the packet is shown because it remains symmetric. The probability
is
shown
before
the
asymptotic
regime
for
times
t
= 0 , 0.1, 0.2, 0.4, 0.6, 0.8, 1.0 in units of ␶. The distance x is in units of
共ប␶ / m兲1/2.
VI. EXAMPLES
Our first example is the second spatial derivative of the
simple Gaussian packet ␹共x , t兲 = ␬ exp共−␬2x2兲, where ␬
= 关m / 2iប共t − i␶兲兴1/2. The packet ␹共x , t兲 is ⌿共x , t兲, with ␭ = 0 in
Eq. 共19兲 and is the same as ␺0共x , t兲 in Eq. 共22兲. Thus, we
consider ␹2共x , t兲 ⬀ ⳵2x ␹共x , t兲 which has the form
␹2共x,t兲 = N␬3共2␬2x2 − 1兲e−␬
2x2
,
共27兲
where N is the normalization constant 共3冑␲ / 32兲−1/2. The
function ␹2共x , t兲 must satisfy the free Schrödinger’s equation
because the spatial derivative of any solution will be a solution. Although the packet ␺2共x , t兲 in Eq. 共22兲 does not change
shape as it evolves, we will show that the similar packet
␹2共x , t兲 does change shape even though it also is smooth
共infinitely differentiable兲 everywhere and goes to zero exponentially at large distances.
Initially, the wave function ␹2共x , t兲 is real. It is also symmetric about x = 0 and will maintain this symmetry. As shown
in Fig. 1, there is a peak centered on the origin with a lower
peak on either side. We will see that as it evolves, the central
peak diminishes and eventually disappears while the outer
peaks move further out and spread. The initial packet has
⌬2x = 7ប␶ / 6m and ⌬2p = 5បm / 2␶, and hence t p = ␶ / 5 and tx
= 7␶ / 3. We expect that the shape of the probability distribution will not change significantly for times much less than t p.
No change can be seen in Fig. 1 until t ⬇ t p / 3 ⬇ 0.06␶. The
maximum rate of change is at t ⬇ 0.6 and x = 0 when visible
change starts for ␦t ⬇ t p / 30.
The Fourier transform of ␹2 gives the asymptotic form as
␹2共x,t兲 ⬇ Ñ
冉
冊
x2
m␶x2
,
5/2 exp −
t
2ht2
共28兲
with Ñ = 2共m5␶5 / 9␲ប5兲1/4; this form is expected to be valid
for t Ⰷ tx. No error in this approximation can be seen in Fig.
2 after t ⬇ 7tx.
As a second example consider the square wave packet,
with
␶共x − pt/m兲2
␶共x − pt/m兲
+ 2␭
2
2d 共t − i␶兲
d共t − i␶兲
共t + i␶兲
.
共t − i␶兲
0.5
Ψx, t2
1.0
where s = 冑共t + i␶兲 / 共t − i␶兲␭ and ␥ = d冑1 + t2 / ␶2. Because
⌿共x , t兲 must be a solution for all values of ␭, it follows that
for each n
1
0.6
0.2
If we apply Eq. 共20兲 directly to Eq. 共19兲 with s
= ␭冑2i␶ / 共t − i␶兲, we would find a complex value for the argument z of Hn共z兲. However, if we take c = −␭2, then z will
be real and
⌿共x,t兲 =
0
0.1
0.2
0.4
0.7
共26兲
Equation 共26兲 can be converted, using Eq. 共20兲 as before,
into a sum over Hermite-Gaussians centered on x − pt / m.
␺共x,0兲 =
再
1/冑a, 兩x兩 ⬍ a/2
0,
兩x兩 艌 a/2.
冎
共29兲
From Eq. 共7兲 the momentum wave function is
1105
Am. J. Phys., Vol. 76, No. 12, December 2008
Mark Andrews
1105
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0.4
16
6
t Ψx, t
2
4
共1兲 For a more rigorous approach to the short-time behavior, we write the exact form of Eq. 共5兲 as ␺共x , t兲
= exp共−ip̄2t / 2mប兲␺共x − p̄t / m , 0兲 + ␦␺, where
3
0.3
␦␺共x,t兲 =
0.2
0.1
0.0
1
2
3
4
Fig. 2. The probability distribution of the same wave packet as in Fig. 1 for
times t = 3 , 4 , 6 , 16 when the packet is approaching its asymptotic form. In
the asymptotic region t 兩 ␺共x , t兲兩2 is a function of x / t only, as in Eq. 共28兲, so
we use x / t as the horizontal variable, which hides the substantial spreading
of these packets. For t ⬎ 7tx ⬇ 16␶ the exact distribution is indistinguishable
from the asymptotic form.
␾共p兲 =
冑
a sin共ap/2ប兲
,
2␲ប ap/2ប
共30兲
and therefore for t Ⰷ ma2 / ប,
␺共x,t兲 ⬇
冑
冋 册
am
imx2 sin共amx/2បt兲
exp
.
2␲iបt
2បt
amx/2បt
共31兲
The exact evolution can be carried out using Eq. 共2兲 in terms
of error functions with a complex argument or Fresnel functions with a real argument. Figure 3 shows the exact probability distribution for times well before the asymptotic regime. Figure 4 shows the exact probability distribution for
times when the exact distribution is approaching its
asymptotic form.
In this case the initial value of ⌬2x is a2 / 12 so that tx
= ma2 / 6ប. Figure 4 shows that the asymptotic form is a close
approximation for t ⬎ 3tx. The short-time theory in Sec. II
does not apply because ⌬ p = ⬁, and there are rapid changes
for early times. Discontinuous wave functions are unphysical; an infinite energy would be required to produce them.
Nevertheless, they can be useful as mathematically simple
states which can be closely approximated by physical ones.
The discussion above is sufficiently simple and complete
to be presented in an elementary course in quantum mechanics, but the following related topics might be helpful and
serve as extensions for interested students.
0.01
1
0.5
− ip̄2t
⫻
2mប
− i共p − p̄兲
2mប
0.001
2
⬁
exp
−⬁
i
p̄t
p x−
ប
m
− 1 ␾共p兲dp.
共32兲
We next use the Cauchy-Schwarz inequality
with
g共p兲 = 共p
兩兰f共p兲g共p兲dp兩2 艋 兰 兩 f共p兲兩2dp兰 兩 共p兲兩2dp,
− p̄兲␾共p兲 and 兩f共p兲 兩 = 兩sin关共p − p̄兲2t / 4mប兴 / 共p − p̄兲兩, and the in⬁
sin2共z2兲 / z2dz = 冑␲, and obtain
tegral 兰−⬁
兩␦␺兩2 艋 冑t/␲mប3⌬2p .
共33兲
Equation 共33兲 is a rigorous bound on the change in the
wave function apart from the shift p̄t / m. For 兩␦␺兩 to be small
compared to ␺ with 兩␺兩2 ⬃ 1 / 2⌬x 共from the normalization of
␺兲, we require t⌬4p / ␲mប3 Ⰶ 1 / 4⌬2x and hence t⌬2p / ␲mប3
Ⰶ 1 / 4⌬2x ⌬2p 艋 1 / ប2 共from the uncertainty relation兲. That is,
we require t Ⰶ ␲mប / ⌬2p, in agreement with the result in
Sec. II.
共2兲 For a more rigorous approach to the error in the
asymptotic approximation, we write the exact form of
Eq. 共6兲 as
冑 冋
册冉
冊
m
im 2 2
m
exp
共x − x̄ 兲 ␾ 共x − x̄兲 + ␦␺ ,
it
2បt
t
␺共x,t兲 =
共34兲
where
␦␺ =
冑
冋
m
im 2 2
exp
共x − x̄ 兲
2␲iបt
2បt
册冕
⬁
f共x⬘兲g共x⬘兲dx⬘ ,
−⬁
共35兲
with
冋
im
共x̄ − x兲x⬘
f共x⬘兲 = exp
បt
VII. FURTHER CONSIDERATIONS
0.1
2␲ប
exp
⫻ exp
xt
0
冉 冊 冕 冋 冉 冊册
冑
冉 冋
册 冊
1
冋
册
exp −
册
im
共x⬘ − x̄兲2 − 1
បt
x⬘ − x̄
g共x⬘兲 = 共x⬘ − x̄兲␺共x⬘,0兲.
, 共36兲
共37兲
The Cauchy-Schwarz inequality gives
兩␦␺兩2 艋 冑m3/␲ប3t3⌬2x .
共38兲
0.5
0.2
0.1
0.15
0
0.10
Ψx, t2
t Ψx, t2
0.05
x
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 3. The probability distribution 兩␺共x , t兲兩2 of the initially square wave
packet in Eq. 共29兲 for several values of the time t = 0 , 0.001, 0.01, 0.1 for
short times when the packet shows complicated behavior, but has not spread
appreciably. The distance x is in units of a, the width of the initial packet.
The times are in units of ma2 / ប.
0.00
xt
0
5
10
15
Fig. 4. The probability distribution of the initially square packet for times
t = 0.1, 0.2, 0.5 when the packet is approaching its asymptotic form. We show
t 兩 ␺共x , t兲兩2 as a function of x / t. For t 艌 0.5 the exact distribution is indistinguishable from the asymptotic form.
1106
Am. J. Phys., Vol. 76, No. 12, December 2008
Mark Andrews
1106
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Equation 共38兲 is a rigorous bound to the error in the
asymptotic approximation. For 兩␦␺兩 to be small compared to
␺ we require 兩␦␺兩2 Ⰶ 1 / 2⌬x, which leads to t Ⰷ 共4 / ␲兲1/3m⌬2x ,
in agreement with Sec. II.
If the wave packet has discontinuities, as in the square
wave in Eq. 共29兲, this derivation of the error is applicable
only when the initial time is taken to be the instant when the
discontinuities exist. At any other time ⌬x does not exist
共because m2d2t 具X̂2典 = 2具P̂2典, as in Sec. IV, and 具P̂2典 = ⬁.兲 The
divergence of 具x̂2典 can easily be seen for the asymptotic wave
function of the initially square packet from Eq. 共31兲.
共3兲 The fact that the Hermite-Gauss packets in Sec. V do
not change shape is consistent with the asymptotic results in
Sec. III because the Fourier transform of a Hermite-Gaussian
is also a Hermite-Gaussian,
1
冑2␲
冕
⬁
2
2
e−z /2Hn共z兲e−ipzdz = i−ne−p /2Hn共p兲.
共39兲
−⬁
As observed by Lipson,11 the Fourier transform of a
Hermite-Gaussian, in its role as an energy eigenfunction of
the harmonic oscillator, must take the same form as a function of p because the Hamiltonian is symmetric under the
interchange of p and x 共apart from constant factors兲.
Another way to obtain the Hermite-Gauss family of solutions is to first create ␺0共x , t兲 as in Eq. 共22兲 as an eigenfunction of the invariant operator12,13 â = x̂ − 共t − i␶兲p̂ / m and then
to repeatedly apply ↠to obtain the solutions with n
= 1 , 2 , 3 , . . . . The function ␺0共x , t兲 satisfies Schrödinger’s
equation because all eigenfunctions of invariant operators
must do so 共apart from a possible time-dependent factor兲 and
the others do because the result of applying an invariant operator to any solution of Schrödinger’s equation will also be
a solution.
共4兲 The spatial derivative of any solution of the free
Schrödinger equation is also a solution. We start with the
simple Gaussian packet ␹ = ␬ exp共−␬2x2兲, where ␬
= 关m / 2iប共t − i␶兲兴1/2, as in the first example in Sec. VI. Taking
successive derivatives of ␹ gives a sequence of packets that
also have the form of a Hermite polynomial multiplied by
the original Gaussian and by a phase factor, but in this case
the probability distribution changes shape as it evolves.
If we define ␹n = ⳵nx ␹ and use Rodrigues’ equation
共d/d␰兲n exp共− ␰2兲 = 共− 1兲nHn共␰兲exp共− ␰2兲,
共40兲
which is equivalent to the generating equation Eq. 共20兲, we
obtain
2 2
␹n ⬀ ␬n+1e−␬ x Hn共␬x兲.
共41兲
The Fourier transform of this combination of a Gaussian
and an exponential is different from that in Eq. 共39兲,
1
冑2␲
冕
⬁
−⬁
2
e−z Hn共z兲e−ipzdz =
共− i兲n
冑2
2
e−p /4 pn .
共42兲
The momentum distribution of ␹n is not a HermiteGaussian, and therefore these packets change shape as they
evolve. The first example in Sec. VII illustrates the case
n = 2.
VIII. CONCLUSION
To gain insight into the nature of the free evolution of a
wave packet, it helps to extend the evolution to the distant
past as well as forward. The centroid 具x̂典 of the packet moves
with constant speed 具p̂典 / m, and we know from Sec. IV that
every packet will contract and then expand, so that ⌬x will
take its minimum value ⌬min at some time tmin. From Eq. 共3兲
th = m⌬min / ⌬ p gives the time scale for the onset of linear
dependence of ⌬x on the time; that is, for 兩t − tmin兩 Ⰷ th, ⌬x
⬀ t. We also know from Sec. III that for 兩t − tmin兩 Ⰷ tx
2
= 2m⌬min
/ ប the shape of the wave packet will not change,
but the scale will change linearly with time 共consistent with
the linear change in ⌬x兲. We also found the time scale t p
= mប / 2⌬2p for short-term changes in the wave function 共and
hence th is the harmonic mean of tx and t p兲. Therefore,
changes in shape 共other than in scale兲 can occur only in the
waist region around tmin and such changes cannot be rapid
over times of the order of t p. Even in this waist region, there
may be no change in shape, as exemplified by the HermiteGauss packets.
In more than one dimension the time of minimum spread
and the rate of spreading may be different in each dimension,
but our results apply to each dimension.
a兲
Electronic mail: [email protected]
Eugen Merzbacher, Quantum Mechanics, 2nd ed. 共Wiley & Sons, New
York, 1970兲, p. 163, Eq. 8.91.
2
We use the term wave packet to indicate that the wave function is sufficiently confined in space for 具x̂2典 to exist.
3
Katsunori Mita, “Dispersion of non-Gaussian free particle wave packets,”
Am. J. Phys. 75, 950–953 共2007兲.
4
Richard W. Robinett, Quantum Mechanics 共Oxford University Press,
New York, 1997兲, Eq. 共3.17兲.
5
Hans C. Ohanian, Principles of Quantum Mechanics 共Prentice Hall,
Englewood Cliffs, NJ, 1990兲, Sec. 2.4.
6
See, for example, M. Daumer, D. Durr, and S. Goldstein, “On the quantum probability flux through surfaces,” J. Stat. Phys. 88, 967–977 共1997兲,
Eq. 共6兲, and papers referenced therein.
7
Reference 1, p. 170, Ch. 8, Prob. 8.
8
Claude Cohen-Tannoudji, Bernard Diu, and Franck Lalöe, Quantum Mechanics 共Wiley & Sons, Paris, 1977兲, Vol. 1, Complement L111, Exer. 4.
9
Eugen Merzbacher, Quantum Mechanics, 3rd ed. 共Wiley & Sons, New
York, 1998兲, p. 75.
10
Reference 1, p. 161, Eq. 共8.78兲.
11
S. G. Lipson, “Self-Fourier objects and other self-transform objects:
Comment,” J. Opt. Soc. Am. A 10, 2088–2089 共1993兲.
12
Mark Andrews, “Total time derivatives of operators in elementary quantum mechanics,” Am. J. Phys. 71, 326–332 共2003兲.
13
Mark Andrews, “Invariant operators for quadratic Hamiltonians,” Am. J.
Phys. 67, 336–343 共1999兲.
1
1107
Am. J. Phys., Vol. 76, No. 12, December 2008
Mark Andrews
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