The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.

A quantum quench is a sudden change in the Hamilton operator of a system. The initial operator $H ̂ i$ is changed to the final one $H ̂ f$. The study of the dynamics after the quench is an active field of research in quantum many-body systems,1 as this presents an interesting example of non-equilibrium physics.

In this paper, we present one simple example of such a quench. Our discussion for a single particle in one dimension complements standard courses as only a few time-dependent problems are discussed in most quantum mechanics textbooks.

As the initial state, we take the ground state $| ψ 0 ⟩$ of the initial Hamiltonian $H ̂ i = T ̂ + V ̂ i$, where $T ̂ = p ̂ 2 / 2 m$ is the operator of the kinetic energy and $V i ( x )$ is the initial attractive potential. At t = 0, the change to $H ̂ f = T ̂ + V ̂ f$, is performed. The time dependence can be described in the Schrödinger or the Heisenberg picture. Which is more appropriate depends on Hf and the initial state, here taken as the ground state of Hi. For the special case of vanishing $V f ( x )$, the dynamics is just the free wave packet evolution of the ground state $| ψ 0 ⟩$ of $H ̂ i$. The description of quenches in the framework of classical statistical mechanics2 is also presented. A comparison with the quantum mechanical results is presented in Secs. III and IV.

A complete analytical description of the dynamics is possible when the initial and final Hamiltonians are of the harmonic oscillator type. We also present results for linear potentials $V f ( x ) = − F x$ in Sec. III.

In Sec. IV, we assume that $V i ( x )$ is a square well potential and $V f ( x )$ vanishes. We present a simple introduction to the concept of Wigner functions.3,4 This allows the description of the broadening of quantum mechanical wave packets similar to the classical case.

In the following, we mainly focus on the probability density:
(1)
to find the particle at position x at time t, where we have introduced the index qm as we later compare quantum quenches with quenches in the framework of classical statistical mechanics.
Instead of directly working with the delta function in Eq. (1), we use its integral representation
(2)
This rarely used expression for $ρ q m ( x , t )$ turns out to be very useful in the following.

The usual approach to obtain $ρ q m ( x , t )$ for times after the quench is to calculate $| ψ ( t ) ⟩ = e − i H ̂ f t / ℏ | ψ ⟩$, using the eigenstates of $H ̂ f$, where $| ψ ⟩$ is the initial state before the quench. This is described for the free time evolution after the quench in Sec. IV.

However, it is sometimes useful to work in the Heisenberg picture in which operators acquire the time dependence $A ̂ ( t ) = e i H ̂ f t / ℏ A ̂ e − i H ̂ f t / ℏ$. Expanding $e i k x ̂$ in Eq. (2) and using $x ̂ n ( t ) = ( x ̂ ( t ) ) n$, one obtains
(3)
In classical statistical mechanics for a single particle, the initial state at time t = 0 is described by the phase space probability density $ρ 0 ( x 0 , p 0 )$ for the initial position x0 and momentum p0 of the particle.2 The average classical particle density at finite times is given by
(4)
where $x x 0 , p 0 ( t )$ is the particle trajectory. For the comparison with the quantum mechanical results in Sec. III, we also make use of the integral representation of the delta function
(5)
If the particle is in thermal equilibrium with a bath at temperature T before the quench the initial probability density $ρ 0 ( x 0 , p 0 )$ is proportional to $e − β ( p 0 2 / 2 m + V i ( x 0 ) )$, with $β = 1 / k B T$. This is an example for a factorized initial distribution $ρ 0 ( x 0 , p 0 ) = ρ 0 ( x 0 ) ρ ̃ 0 ( p 0 )$.
The Hamiltonian of a one-dimensional harmonic oscillator reads
(6)
with λ being the spring constant. The angular frequency of the oscillator is given by $ω 0 = λ / m$.
The quantum mechanical description of the harmonic oscillator is especially simple using the ladder operators. One defines the lowering operator $a ̂$ and its adjoint $a ̂ †$,
(7)
which obey the commutation relation $[ a ̂ , a ̂ † ] = 1 ̂$. The position operator $x ̂$ and the momentum operator $p ̂$ read in terms of $a ̂$ and $a ̂ †$,
(8)
The ground state $| 0 ⟩$ is annihilated by $a ̂$, i.e., $a ̂ | 0 ⟩ = 0$ holds.

As $H ̂ i$ we use the Hamiltonian in Eq. (6) with $λ → λ i$, and its ground state $| 0 ⟩ i$ as the initial state. Averages of operators $A ̂$ in this ground state we denote by $⟨ A ̂ ⟩$. Then, $⟨ x ̂ ⟩ = 0$ holds as well as $⟨ p ̂ ⟩ = 0$.

As the final potential, we use
(9)
i.e., a harmonic oscillator with spring constant λf shifted to the position $a F = F / λ f$ and angular frequency $ω f = λ f / m$.
The equations of motion and their solutions for the operators $x ̂ ( t )$ and $p ̂ ( t )$ are identical in form to the classical ones for x(t) and p(t). In the quantum mechanical case, the classical initial conditions x0 and p0 are replaced by $x ̂ = x ̂ ( 0 )$ and $p ̂ = p ̂ ( 0 )$. With the definition $Δ A ̂ ≡ A ̂ − ⟨ A ̂ ⟩$, one obtains
(10)
and
(11)
(12)
In order to calculate $ρ q m ( x , t )$, it is useful to express $Δ x ̂ ( t )$ as a linear combination of the ladder operators corresponding to the harmonic oscillator $H ̂ i$,
(13)
where using Eq. (8), $α ( t )$ is given by
(14)
Using $a ̂ | 0 ⟩ = 0$ and $[ a ̂ , a ̂ † ] = 1 ̂$, the expectation value of $( Δ x ̂ ( t ) ) 2$ in the ground state of $H ̂ i$ is given by $| α ( t ) | 2$,
(15)
In order to calculate $ρ q m ( x , t )$, we use the Baker–Hausdorff-identity.5 It states that if $[ A ̂ , [ A ̂ , B ̂ ] ] = 0 = [ B ̂ , [ A ̂ , B ̂ ] ]$,
(16)
For operators $A ̂$ and $B ̂$ linear in the ladder operators, the requirements are fulfilled. Therefore, the expectation value in Eq. (3) can easily be calculated. In the integrand in Eq. (3), we write
(17)
and evaluate the expectation value of $e i k Δ x ̂ ( t )$ using Eqs. (13) and (16),
(18)
Because of $a ̂ | 0 ⟩ i = 0$ which implies $i ⟨ 0 | a ̂ † = 0$, the expectation value in the second equality equals 1. Putting this into the upper part of Eq. (3), the Gaussian integration can be performed and one obtains
(19)
An analogous result is obtained for the momentum probability distribution
(20)
with the width determined by
(21)

Figure 1 shows the time dependence of $⟨ x ̂ ( t ) ⟩$, $Δ x ( t ) = ⟨ ( Δ x ̂ ( t ) ) 2 ⟩$, $Δ p ( t ) = ⟨ ( Δ p ̂ ( t ) ) 2 ⟩$, and the uncertainty product. The initial wave packet makes an oscillatory motion around the center of the harmonic potential in Hf. The width $Δ x ( t )$ and $Δ p ( t )$ oscillate with double frequency and the uncertainty product $Δ x ( t ) Δ p ( t )$ equals $ℏ / 2$ at positions $0 , a F$, and $2 a F$ and is larger at the intermediate positions.

Fig. 1.

Time dependence of $⟨ x ̂ ( t ) ⟩ / a F$ (solid line), $Δ x ( t ) / Δ x ( 0 )$ (dashed line), $Δ p ( t ) / Δ p ( 0 )$ (double-dashed dotted line), and the uncertainty product $2 Δ x ( t ) Δ p ( t ) / ℏ$ (dashed-dotted line) as a function of $ω f t$ for $ω f / ω i = 1 / 2$.

Fig. 1.

Time dependence of $⟨ x ̂ ( t ) ⟩ / a F$ (solid line), $Δ x ( t ) / Δ x ( 0 )$ (dashed line), $Δ p ( t ) / Δ p ( 0 )$ (double-dashed dotted line), and the uncertainty product $2 Δ x ( t ) Δ p ( t ) / ℏ$ (dashed-dotted line) as a function of $ω f t$ for $ω f / ω i = 1 / 2$.

Close modal

The behavior of $ρ q m ( x , t )$ in Eq. (19) is well known. It is usually obtained by calculating $⟨ x | 0 ( t ) ⟩ i$ introducing the concepts of coherent and squeezed states5,6 as a typical quantum mechanical behavior. The solution presented here without introducing these states is simpler.

Surprisingly, Eq. (19) can also be obtained purely classically. To show this, we consider the classical quench dynamics switching from Hi to Hf when $ρ c l ( x 0 , p 0 ) = ρ 0 ( x 0 ) ρ ̃ 0 ( p 0 )$ and $ρ 0 ( x 0 )$ and $ρ ̃ 0 ( p 0 )$ are both Gaussians. Then, the integration in Eq. (5) can be performed analytically by first calculating
(22)
by Gaussian integrations. The remaining $k −$ integration is also Gaussian and one obtains
(23)
with
(24)
For the case of the canonical ensemble discussed after Eq. (5), one has $⟨ p 0 2 ⟩ / ⟨ x 0 2 ⟩ = m 2 ω i 2$ and the time dependence of $⟨ ( Δ x ( t ) ) 2 ⟩ c l$, apart from the prefactor is the same as in Eq. (15). For the special temperature choice, $k B T = ℏ ω i / 2$, also the prefactor is the same. This corresponds to the initial condition
(25)
where $ψ 0 ( x 0 ) = ⟨ x 0 | ψ 0 ⟩$ is the real wave function for the Gaussian initial state $| ψ 0 ⟩ = | 0 ⟩ i$ considered in this section and $ψ ̃ 0 ( p 0 ) = ⟨ p 0 | ψ 0 ⟩$ is the corresponding real Gaussian momentum amplitude.

It is left as an exercise to repeat the calculation of $ρ q m ( x , t )$ for $| 1 ⟩ i = a ̂ † | 0 ⟩ i$ and $| 2 ⟩ i = ( a ̂ † ) 2 | 0 ⟩ i / 2$ as the initial state by expanding the exponential functions in the expectation value in the second equality in Eq. (18). The results for the special case $H ̂ f = T ̂$ can be found in Ref. 7. There the free time evolution after the quench for the initial state $c p ̂ 2 | 0 ⟩ i$ is also discussed. It is a special linear combination of $| 0 ⟩ i$ and $| 2 ⟩ i$. Results for the general linear combination of these two states are presented in Ref. 8.

We next discuss the case $λ f = 0$ and a finite value of F, i.e., the linear potential $V f ( x ) = − F x$. Performing the limit $ω f → 0$ in Eq. (10) leads to
(26)
independent of the value of F which only appears in the expression for $⟨ x ̂ ( t ) ⟩$ in Eq. (10), given by
(27)
The result for $| ⟨ x | ψ ( t ) ⟩ | 2$ for the linear potential is given by Eq. (19) with the results of Eqs. (26) and (27) inserted. The time dependence of the broadening is identical to the free particle case.
This holds for linear potentials for arbitrary initial states $| ψ ( 0 ) ⟩$. The solution of the Heisenberg equation of motion is given by
(28)
Putting this into the second line of Eq. (3) yields
(29)
This is in accord with the (weak) equivalence principle,9,10 which states that all laws of a freely falling particle are the same as in an unaccelerated reference frame. The presented proof for the broadening of wave packets using the Heisenberg picture by directly addressing the measurable probability density $| ⟨ x | ψ ( t ) ⟩ | 2$ is much simpler than using the Schrödinger picture and calculating $⟨ x | ψ ( t ) ⟩$ first.9,10

In quantum mechanics textbooks, the factor multiplying t2 for the free case F = 0 in Eq. (26) is usually expressed differently as the broadening of the Gaussian wave packet is treated before the harmonic oscillator. Instead of $ω i 2$, the factor is written ($ℏ / ( 2 m ⟨ x ̂ 2 ⟩ ) )$2.

For the case $λ f = 0$ and F = 0, the free broadening of a Gaussian wave packet using operator manipulations was presented in this journal recently.11

In Sec. III, we used the Heisenberg picture to calculate the time dependence of the probability density ρqm after the quench. Here, we use the Schrödinger picture to obtain $ρ q m ( x , t )$ for the quench in which $V f ( x ) ≡ 0$.

We begin with the usual approach to obtain $ρ q m ( x , t )$ by calculating $⟨ x | ψ ( t ) ⟩$ and taking its absolute square. In the second part of this section, we discuss the additional insight one can obtain by using Eq. (2) instead.

In the first approach, one uses the eigenstates of $H ̂ f$. For the case $H ̂ f = T ̂$, those are given by the momentum states $| p ⟩$. Inserting the unit operator expressed in terms of the momentum states and using $⟨ x | p ⟩ = e ipx / ℏ / 2 π ℏ$ yields
(30)
with $ϵ p = p 2 / 2 m$ and the momentum representation
(31)
of the arbitrary initial state $| ψ ⟩$.

If the wave function $ψ ( x ) = ⟨ x | ψ ⟩$ is Gaussian, the same holds for $ψ ̃ ( p )$ and the integration in Eq. (30) can be performed analytically. This is presented in almost all quantum mechanics textbooks. For generic $ψ ( x )$, the integration in Eq. (30) has to be performed numerically.

In this section, we take as the initial state $| ψ 0 ⟩$ the ground state of an attractive square well potential $V i ( x ) = V i Θ ( a − | x | )$, with Θ being the step function and $V i < 0$. Its wave function is given by
(32)
with $k 0 = 2 m ( E 0 − V i ) / ℏ$ and the ground state energy E0.
In order to have continuous first derivatives at $x = ± a$
(33)
has to hold with $tan ( k 0 a ) > 0$. The normalization constant is given by $c 0 = [ 1 / κ + a ( 1 + κ 2 / k 0 2 ) + κ / k 0 2 ] − 1 / 2$. The dependence of E0 on Vi is irrelevant in the following.

There are two interesting limiting cases:

• The limit $a → 0 , V i → − ∞$ with arbitrary $κ > 0$, i.e., an attractive delta potential with the ground state wave function $ψ 0 ( x ) = κ e − κ | x |$.

• The infinitely deep potential: For $a k 0 → π / 2$, the ratio $κ / k 0$ tends to infinity and one obtains $ψ 0 ( x ) = cos ( k 0 x ) Θ ( a − | x | ) / a$.

The momentum representation of $| ψ 0 ⟩$ can be calculated analytically. Using Eqs. (31) and (32), one obtains with $p = ℏ k$
(34)
Inserting this result into the integral in Eq. (30), it can be calculated numerically to obtain $⟨ x | ψ ( t ) ⟩$. Its absolute value squared is shown in Fig. 2 for $k 0 a = π / 2$, i.e., the ground state of the infinitely deep well, for four different times. It shows that the probability density to find the particle at the origin is larger for $t = 0.14 t 0$ than at the initial time t = 0, where $t 0 = m a 2 / ℏ$. For $t = 0.07 t 0$, it has a minimum at the origin.
Fig. 2.

Probability density ρqm times a as a function of x/a for four different times: T = 0 (solid line), $t = 0.07 t 0$ (dashed-dotted line), $t = 0.14 t 0$ (dashed line), and $t = 0.28 t 0$ (double-dashed dotted line). Note that $a ρ q m ( 0 , 0.14 )$ is larger than $a ρ q m ( 0 , 0 )$.

Fig. 2.

Probability density ρqm times a as a function of x/a for four different times: T = 0 (solid line), $t = 0.07 t 0$ (dashed-dotted line), $t = 0.14 t 0$ (dashed line), and $t = 0.28 t 0$ (double-dashed dotted line). Note that $a ρ q m ( 0 , 0.14 )$ is larger than $a ρ q m ( 0 , 0 )$.

Close modal

To elucidate this surprising effect in more detail, $a ρ q m$ is shown for x = 0 as a function of $t / t 0$ for three different values of $k 0 a$ in Fig. 3. The short time oscillatory behavior is more pronounced when the well is deeper.

Fig. 3.

Probability density ρqm times a for x = 0 as a function of $t / t 0$ for three different values of $k 0 a$: $k 0 a = π / 2$, which corresponds to the infinitely deep well (solid line), $k 0 a = π / 2.5$ (dashed line), and $k 0 a = π / 3$ (dashed-dotted line).

Fig. 3.

Probability density ρqm times a for x = 0 as a function of $t / t 0$ for three different values of $k 0 a$: $k 0 a = π / 2$, which corresponds to the infinitely deep well (solid line), $k 0 a = π / 2.5$ (dashed line), and $k 0 a = π / 3$ (dashed-dotted line).

Close modal
The fact that the probability to find the particle at the origin for a t > 0 is larger than in the initial state is a purely quantum mechanical effect. It is easy to see that this cannot happen in the classical case when the probability to find the particle at t = 0 has its maximum at $x 0 = 0$. For a free particle, the trajectory is given by $x x 0 p 0 ( t ) = x 0 + p 0 t / m$ and using Eq. (4), one obtains
(35)
In order to be as close as possible to the quantum mechanical case for a general initial state $| ψ ⟩$, the phase space density $ρ 0 ( x 0 , p 0 )$ should yield the quantum mechanical space probability $| ⟨ x 0 | ψ ⟩ | 2$ by integration over p0 and the momentum probability $| ⟨ p 0 | ψ ⟩ | 2$ by integration over x0. By choosing
(36)
this is obviously fulfilled. We call this the “classical approximation” in quotation marks as the dynamics using Eq. (35) is classical, but this initial condition involves quantum mechanical probability densities.

For $| ψ ⟩$, the ground state of the square well potential $| ⟨ x 0 | ψ ⟩ | 2$ has its maximum at $x 0 = 0$. For t > 0, one has $ρ 0 ( − p 0 t / m , p 0 ) < ρ 0 ( 0 , p 0 )$, which using Eq. (35) implies $ρ c l ( 0 , t ) < ρ c l ( 0 , 0 )$.

It turns out that the “classical approximation” works rather well for larger times. This is shown in Fig. 4 for $k 0 a = π / 2$.

Fig. 4.

Probability density ρqm times a for $k 0 a = π / 2$ as a function of $t / t 0$ for three different values of x/a (solid lines), compared to the “classical approximation” described in the text (dashed lines).

Fig. 4.

Probability density ρqm times a for $k 0 a = π / 2$ as a function of $t / t 0$ for three different values of x/a (solid lines), compared to the “classical approximation” described in the text (dashed lines).

Close modal
For a better understanding of this, we show that the “classical approximation” gives the exact result for the time dependence of the quantum mechanical width of the wave packet. With the initial condition in Eq. (36), one obtains $⟨ ( x ( 0 ) ) n ⟩ c l = ⟨ ( x ̂ ( 0 ) ) n ⟩ q m$ and $⟨ ( p ( 0 ) ) n ⟩ c l = ⟨ ( p ̂ ( 0 ) ) n ⟩ q m$. As we consider initial states $| ψ ⟩$ with even wave functions $⟨ x ( 0 ) ⟩ c l = 0$ and $⟨ ( x ( 0 ) p ( 0 ) ⟩ c l = 0$ holds. This leads to
(37)
To calculate the expectation value $⟨ x ̂ p ̂ ⟩ + ⟨ p ̂ x ̂ ⟩$ for a square integrable wave function $ψ ( x )$, one can use that the integral over the derivative of $x | ψ ( x ) | 2$ vanishes as $x | ψ ( x ) | 2 → 0 for x → ± ∞$. For $ψ ( x ) = f ( x ) e i ϕ ( x )$ with f and $ϕ$ real functions, one obtains (exercise)
(38)
Therefore, the sum vanishes for real $ψ ( x )$. This leads to
(39)
In order to discuss the long time behavior of $ρ c l ( x , t )$, one can alternatively perform the p0 integration in Eq. (4) first or change the integration variable in Eq. (35), both leading to
(40)
If $| ψ ( x 0 ) | 2$ in Eq. (36) decays fast enough like for the square well potential Eq. (32), the term $m x 0 / t$ in the integrand can be neglected in the long time limit leading to
(41)
for finite x/t.

In Fig. 5, we show numerical results for the quantum mechanical long time behavior using Eqs. (30) and (34). The function $( t / t 0 ) a ρ q m$ is shown for various values of $t / t 0$ as a function of $( x / a ) / ( t / t 0 )$. It quickly approaches $ℏ | ψ ̃ ( ( x / a ) / ( t / t 0 ) ) | 2 / a$, in the agreement with Eq. (41).

Fig. 5.

Long time behavior of $( t / t 0 ) a ρ q m$ for the infinitely deep square well as a function of $( x / a ) / ( t / t 0 )$ for different values of $t / t 0$. It quickly approaches $ℏ | ψ ̃ ( ( x / a ) / ( t / t 0 ) ) | 2 / a$ (long dashed curve).

Fig. 5.

Long time behavior of $( t / t 0 ) a ρ q m$ for the infinitely deep square well as a function of $( x / a ) / ( t / t 0 )$ for different values of $t / t 0$. It quickly approaches $ℏ | ψ ̃ ( ( x / a ) / ( t / t 0 ) ) | 2 / a$ (long dashed curve).

Close modal

Before we present an explanation why the quantum mechanical and the “classical” long time behavior agree we switch to Eq. (2) to calculate $ρ q m ( x , t )$ without computing $⟨ x | ψ ( t ) ⟩$ first.

For the later treatment of the free particle time dependence, it turns out to be useful to factorize $e i k x ̂ = e i k x ̂ / 2 e i k x ̂ / 2$ and insert the unit operator expressed in terms of the momentum states in between. Using $e i k x ̂ / 2 | p ⟩ = | p + ℏ k / 2 ⟩$ as can be proven by multiplying both sides by $⟨ x |$, this leads to
(42)
Putting this into Eq. (2) and changing the integration variable $k = p ̃ / ℏ$, one obtains
(43)
with
(44)
This is a real function as seen by changing the integration variable $p ̃ → − p ̃$. Integration over x gives a factor $2 π δ ( p ̃ / ℏ )$ leading to
(45)
It is left as an exercise to show that after inserting unit operators in terms of position states in the integral in Eq. (44), the function $W ( x , p , t )$ can also be written as
(46)
In 1932, Wigner3 introduced the quantum phase-space distribution $W ( x , p , t )$ for arbitrary time dependent states $| ψ ( t ) ⟩$, which has similarities to the classical phase-space distribution,2 but allows to obtain exact quantum mechanical results, as shown above. The higher dimensional generalization of the Wigner function is widely used as a tool in various areas, e.g., quantum optics.12,13 However, it is introduced only in a few quantum mechanics textbooks, e.g., Refs. 5 and 6. A “pedestrian” introduction has been published in this journal.4

For $W ( x , p , 0 )$ to be a probability density, i.e., it is positive everywhere, $ψ ( x )$ has to be the exponential of a quadratic polynomial.14 A well-known example is a real-valued Gaussian. The integration in Eq. (46) can then be easily performed and one obtains $W ( x , p , 0 ) = ψ ( x ) 2 ψ ̃ ( p ) 2$.

The form of the Wigner function in Eq. (44) is well suited to determine its time dependence for free particles discussed in this section. The momentum states are the eigenstates for $H ̂ f = T ̂$. With $ϵ p + p ̃ / 2 − ϵ p − p ̃ / 2 = p ̃ p / m$, one obtains
(47)
If this is inserted into Eq. (43), it leads to
(48)
This result for $ρ q m ( x , t )$ has the form as in the classical case in Eq. (35), with $W ( x , p , 0 )$ instead of $ρ 0 ( x , p )$. Therefore, the arguments leading to Eq. (41) can be generalized to obtain
(49)
if $W ( x , p , 0 )$ has a well-localized x-dependence, which is the case for the ground state of a deep square well potential.

The long time behavior of the broadening of wave packets has been discussed previously in two papers in this journal.7,15 In Ref. 7, the result Eq. (49) is presented using the free particle propagator, shortly discussed in the  Appendix. The statement in Ref. 15 that all wave packets become “approximately Gaussian” after a long enough time is critically discussed.

The Wigner function for the ground state of the infinitely deep square well can be obtained by simple integration using Eq. (46),16
(50)
It is shown in Fig. 6 as a function of momentum for two different values of x. One can see that W is negative in some intervals. This is different for $| ψ ( 0 ) | 2 | ψ ̃ ( p ) | 2$ shown as the dashed curve.
Fig. 6.

Wigner function $ℏ W ( x , ℏ k , 0 )$ as a function of ak for x = 0 and $x = a / 2$. The dashed curve is the positive function $ℏ | ψ ( 0 ) | 2 | ψ ̃ ( ℏ k ) | 2$.

Fig. 6.

Wigner function $ℏ W ( x , ℏ k , 0 )$ as a function of ak for x = 0 and $x = a / 2$. The dashed curve is the positive function $ℏ | ψ ( 0 ) | 2 | ψ ̃ ( ℏ k ) | 2$.

Close modal

The oscillatory behavior of $ρ q m ( 0 , t )$ shown in Fig. 3 can be understood without performing a numerical integration by plotting the integrand in Eq. (48) $W ( − ℏ k t / m , ℏ k , 0 ) = W ( k a 2 t / t 0 , ℏ k , 0 )$ (symmetric in k). In Fig. 7, this is shown for the times of the deepest minimum ($t / t 0 ≈ 0.071$) and highest maximum ($t / t 0 ≈ 0.128$) of the full line in Fig. 3. One clearly sees that the integral over the full curve yields a result larger than that of the dotted curve.

Fig. 7.

Integrand $W ( − ℏ k t / m , ℏ k , 0 )$ for the calculation of $ρ q m ( 0 , t )$ using Eq. (48) for $t = 0.14 t 0$ (full curve) and $t = 0.07 t 0$ (dashed curve).

Fig. 7.

Integrand $W ( − ℏ k t / m , ℏ k , 0 )$ for the calculation of $ρ q m ( 0 , t )$ using Eq. (48) for $t = 0.14 t 0$ (full curve) and $t = 0.07 t 0$ (dashed curve).

Close modal
What we called the “classical approximation” can be given a different interpretation. Comparing Eqs. (35) and (48), it can be viewed as the following approximation for the Wigner function:
(51)
i.e., a factorized probability distribution. As discussed above, this holds exactly only for a Gaussian wave packet.

It should be mentioned that various other wave packets have been studied which are not ground state wave functions of an initial potential, e.g., a rectangular initial wave function,7,17 which at short times leads to a strongly oscillatory behavior. In Ref. 17, this is discussed also using the Wigner function.

The broadening of a free Gaussian wave packet is one of the few time-dependent problems treated in courses on quantum mechanics. The wave function $⟨ x | ψ ( t ) ⟩$ can be calculated analytically to obtain the probability density $| ⟨ x | ψ ( t ) ⟩ | 2$. It is little known that the time dependence of other wave packets can differ considerably from the smooth Gaussian broadening.

It is important to include the treatment of time-dependent Hamiltonians when teaching quantum mechanics. A simple case is the sudden change from $H ̂ i$ to $H ̂ f$ treated in this paper.

It turned out that directly addressing $| ⟨ x | ψ ( t ) ⟩ | 2$ using Eq. (3) allowed a simple way to obtain results for harmonic oscillator systems. It is little known that the oscillatory behavior in Fig. 1 can also happen in the framework of classical statistical mechanics. For the free time evolution after the quench directly addressing $| ⟨ x | ψ ( t ) ⟩ | 2$ provides additional insight, as the introduction of the concept of the Wigner function is straightforward. This function allows an understanding of the oscillatory behavior of the broadening of the initial ground state of an infinitely deep well without performing a numerical integration. This is an argument for presenting the concept of the Wigner function in quantum mechanics courses.

The author would like to thank T. Dittrich, G. Hegerfeld, F. Heidrich-Meisner, H. Leschke, S. Manmana, and V. Meden for a critical reading of the manuscript and useful suggestions. Special thanks go to W. Schleich for bringing Refs. 8 and 17 to my attention and for a stimulating discussion about the use of Wigner functions.

The author has no conflicts to disclose.

In this  Appendix, we shortly present the alternative derivation7 of the result for the quantum mechanical long time limit Eq. (49) for the broadening of wave packets.

The time dependent wave function $⟨ x | ψ ( t ) ⟩$ is calculated by inserting position states
(A1)
The propagator $K ( x . x ′ , t )$ can be calculated inserting momentum eigenstates5
(A2)
If one writes
(A3)
the last term can be neglected in the long time limit for a sufficiently localized wave function $ψ ( x ′ )$ and one obtains
(A4)
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