Quantum versus classical quenches and the broadening of wave packets

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 H_f and the initial state, here taken as the ground state of H_i. 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.

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.

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 $⟨ p ̂ ⟩ = 0$⁠.

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 H_f. 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.

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 states^5,6 as a typical quantum mechanical behavior. The solution presented here without introducing these states is simpler.

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.

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 ρ_qm 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.

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.

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 $ψ 0 ( x ) = cos ( k 0 x ) Θ ( a − | x | ) / a$⁠.

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.

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 $k 0 a = π / 2$⁠.

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).

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 $W ( x , p , 0 )$ 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 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 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.

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.