Theoretical Femtosecond Physics pp 1984  Cite as
TimeDependent Quantum Theory
Abstract
The focus of our interest will be the coupling of atomic and molecular systems to laser fields, whose maximal strength is of the order of the field experienced by an electron in the ground state of the hydrogen atom.
The focus of our interest will be the coupling of atomic and molecular systems to laser fields, whose maximal strength is of the order of the field experienced by an electron in the ground state of the hydrogen atom.^{1} This restriction allows us to describe the field matter interaction nonrelativistically by using the timedependent Schrödinger equation (TDSE) [1]. Analytical solutions of this linear partial differential equation are scarce, however, even in the case without external driving.
In this chapter, we continue laying the foundations for the later chapters by reviewing some basic properties of the timedependent Schrödinger equation and the corresponding timeevolution operator. After the discussion of two analytically solvable cases, we will consider various ways to rewrite and/or solve the timedependent Schrödinger equation. Formulating the solution with the help of the Feynman path integral will allow us to consider an intriguing approximate, socalled semiclassical approach to the solution of the timedependent Schrödinger equation by using classical trajectories. The last part of this chapter is dealing with numerical solution techniques for the timedependent Schrödinger equation that will be referred to in later chapters.
2.1 The TimeDependent Schrödinger Equation
In the heyday of quantum theory, Schrödinger postulated a differential equation for the wavefunction of a quantum particle. The properties of this partial differential equation of first order in time and the interpretation of the complex valued wavefunction are in the focus of this section. The importance of Gaussian wavepackets as (approximate) analytical solutions of the Schrödinger equation will show up for the first time by considering the socalled Gaussian wavepacket dynamics.
2.1.1 Introduction
2.1.
Derive the equation of continuity and prove that the norm is conserved in case that \(\varvec{j}\) falls to zero faster than \(1/r^2\) for \(r\rightarrow \infty \).
In the derivation above, we started from the timedependent Schrödinger equation in order to derive the timeindependent Schrödinger equation. Schrödinger, however, published them in reverse order. Furthermore, one can derive the timedependent from the timeindependent version, if one considers a composite system of many degrees of freedom and treats the “environmental” degrees of freedom classically, leading to the “emergence of time” for the subsystem [3].
2.1.2 TimeEvolution Operator
2.2.
2.3.
 (a)
Calculate \(\frac{\mathrm{d}\hat{U}}{\mathrm{d}t}\) by using Taylor expansion of the exponential function (keep in mind that, in general, an operator does not commute with its time derivative).
 (b)
Consider the special case \(\hat{B}(t)\equiv \frac{\mathrm{i}}{\hbar }\hat{H}_0t\) and give a closed form solution for \(\frac{\mathrm{d}\hat{U}}{\mathrm{d}t}\).
 (c)
Consider the special case \(\hat{B}(t)\equiv \frac{\mathrm{i}}{\hbar } \int _0^{t}\mathrm{d}t'\hat{H}(t')\) and convince yourself that a simple closed form expression for \(\frac{\mathrm{d}\hat{U}}{\mathrm{d}t}\) can not be given!
 (d)Show that the construction \(\hat{U}(t)=\hat{T}\exp [\hat{B}(t)]\) with the time ordering operator and the operator \(\hat{B}\) from part \(\mathrm {(c)}\) allows for a closed form solution of the timeevolution operator as well as of its time derivative by proving that the relationholds.$$\begin{aligned} \hat{T}\hat{B}^{n}= n!\left( \frac{\mathrm{i}}{\hbar }\right) ^n \int _{0}^{t}\mathrm{d}t_n\int _{0}^{t_n}\mathrm{d}t_{n1}\cdots \int _{0}^{t_{2}}\mathrm{d}t_{1} \hat{H}(t_n)\hat{H}(t_{n1})\cdots \hat{H}(t_{1}) \end{aligned}$$
2.1.3 Spectral Information
In the applications to be discussed in the following chapters, the initial state frequently is assumed to be the ground state of the undriven problem. In this section we will see how spectral information can, in general, be extracted form the propagator via Fourier transformation.
2.4.
Hint: Use the geometric series.
2.1.4 Analytical Solutions for Wavepackets
To conclude this introductory section, we review an Ansatz for the solution of the timedependent Schrödinger equation with the help of a Gaussian wavepacket and will derive equations of motion for the parameters of this wavepacket. We then continue with a review of the dynamics of a wavepacket in the box potential. Due to the specific nature of the spectrum in this case, insightful analytic predictions for the probability density as a function of time can be made.
2.1.4.1 Gaussian Wavepacket Dynamics (GWD)
Already in 1926, Schrödinger has stressed the central importance of Gaussian wavepackets in the transition from “micro” to “macromechanics” [7]. For this reason, we will now consider a wavefunction in the form of a Gaussian as the solution of the timedependent Schrödinger equation.
In the cases of the free particle and of the harmonic oscillator, all equations of motion can be solved exactly analytically and, because the Taylor expansion is also exact in these cases, the procedure above leads to an exact analytic solution of the timedependent Schrödinger equation.
2.5.
Use the GWDAnsatz to solve the TDSE.
 (a)
Use the differential equations for \(q_t,p_t,\alpha _t,\delta _t\) in order to show that the Gaussian wavepacket fulfills the equation of continuity.
 (b)
Solve the differential equations for \(q_t,p_t,\alpha _t,\delta _t\) for the free particle case \(V(x)=0\).
 (c)
Solve the differential equations for \(q_t,p_t,\alpha _t,\delta _t\) for the harmonic oscillator case \(V(x)=\frac{1}{2}m\omega _\mathrm{e}^2x^2.\)
 (d)
Calculate \(\langle \hat{x}\rangle ,\langle \hat{p}\rangle \), and \(\Delta x=\sqrt{\langle \hat{x}^2\rangle \langle \hat{x}\rangle ^2}\), \(\Delta p=\sqrt{\langle \hat{p}^2\rangle \langle \hat{p}\rangle ^2}\) and from this the uncertainty product using the general Gaussian wavepacket. Discuss the special cases from \(\mathrm {(b)}\) and \(\mathrm {(c)}\). What do the results for the harmonic oscillator simplify to in the case \(\alpha _{t=0}=m\omega _\mathrm{e}/(2\hbar )\)?
The GWD method can also be applied to nonlinear classical problems, however, where it is typically valid only for short times. In this context one often uses the notion of the Ehrenfest time, after which nonGaussian distortions of a wavepacket become manifest. In general, the solutions of the equations of motion have to be determined numerically in the nonlinear case (see Sect. 2.3.4), and the Taylor expansion and thus the final result is only an approximation.
2.1.4.2 Particle in an Infinite Square Well
As another example for an exactly solvable problem in quantum dynamics, we now consider the evolution of an initial wavepacket in an infinite square well (box potential). This problem has been presented in the December 1995 issue of “Physikalische Blätter” [9] and leads to aesthetically pleasing spacetime pictures, which are sometimes referred to as “quantum carpets”.
2.6.
A particle shall be in the eigenstate \(\psi _n\) with energy \(E_n\) of an infinite potential well of width L (\(0\le x \le L\)). Let us assume that the width of the well is suddenly doubled.
 (a)
Calculate the probability to find a particle in the eigenstate \(\psi _m'\) with energy \(E_m'\) of the new well.
 (b)
Calculate the probability to find a particle in state \(\psi _m'\) whose energy \(E_m'\) is equal to \(E_n\).
 (c)
Consider the time evolution for \(n=1\), i.e. at \(t=0\) the wavefunction is the lowest eigenfunction of the small well \(\varPsi (x,0)=\psi _1(x)\). Calculate the smallest time \(t_\mathrm{min}\) for which \(\varPsi (x,t_\mathrm{min})=\varPsi (x,0)\).
 (d)
Draw a picture of the wavefunction \(\varPsi (x,t)\) at \(t=t_\mathrm{min}/2\).
2.2 Analytical Approaches to Solve the TDSE
A review of some exactly analytically solvable cases in nonrelativistic quantum dynamics is given in [11], and we have seen two examples for wavepacket solutions at the end of the previous section. For most problems of interest, however, an exact analytical solution of the timedependent Schrödinger equation cannot be found. It is therefore of quite some interest to devise alternative approaches to quantum dynamics and/or some approximate or exact Ansätze that are generally applicable and lead to viable approximate and/or numerical schemes.
A notable reformulation of the timedependent Schrödinger equation is the Feynman path integral expression for the propagator [12]. This is of utmost importance in the following, because from the path integral a much used approximation can be derived: the timedependent semiclassical formulation of quantum theory. Furthermore, in the case of small external perturbations, timedependent perturbation theory may be the method of choice for the solution of the timedependent Schrödinger equation. Moreover, for systems with many degrees of freedom, as a first approximation, the wavefunction can be factorized. We thus discuss the socalled Hartree Ansatz and for the first time also the BornOppenheimer method in this chapter. Finally, the exact analytical Floquet Ansatz for the treatment of periodically driven quantum systems is reviewed.
The discussion of the numerical implementation of some of these concepts will be postponed to Sect. 2.3.
2.2.1 Feynman’s Path Integral
For timedependent quantum problems, which occur naturally if we want to describe the interaction of a system with a laser field, as we will see in Chap. 3, an approach that deals with the propagator is very well suited. With the propagator at hand, we can calculate the time evolution of every wavefunction according to (2.36).
2.2.1.1 The Propagator as a Path Integral
 Postulate 1:

If, in the particle picture, an event (e.g., an electron hitting the screen after passing a double slit) can have occurred in two mutually exclusive ways, the corresponding amplitudes have to be added to find the total amplitude
 Postulate 2:

If an event consists of two successive events, the corresponding amplitudes do multiply.
2.7.
Study the shorttime propagator and use it to derive the TDSE.
 (a)Derive the shorttime propagator starting fromfor the infinitesimal timeevolution operator.$$\begin{aligned} \hat{U}(\Delta t)=\exp \{\mathrm{i}\hat{H}\Delta t/\hbar \} \end{aligned}$$
Hint: Use first order Taylor expansion of the exponential function.
 (b)Use the shorttime propagator in order to propagate an arbitrary wavefunction \(\varPsi (x,t)\) over an infinitesimal time interval \(\Delta t\) viaand derive the TDSE!$$\begin{aligned} \varPsi (x,t+\Delta t)=\int \mathrm{d}y K(x,\Delta t;y,0)\varPsi (y,t) \end{aligned}$$
Hint: To this integral only a small intervall of y centered around x is contributing. Expansion of the expression above to first order in \(\Delta t\) and up to second order in \(\eta =yx\) leads to a linear partial differential equation for \(\varPsi (x,t)\).
2.2.2 Stationary Phase Approximation
By inspection of (2.75) it is obvious that the calculation of the propagator for arbitrary potentials becomes arbitrarily complicated. In the case of maximally quadratic potentials all integrals are Gaussian integrals, however, and thus can be done exactly analytically. There are some additional examples, for which exact analytic results for the path integral are known. These are collected in the supplement section of the Dover edition of the textbook by Schulman [14].
In the next subsection, the notion of stationary phase integration will be extended to the path integral, being an infinite dimensional “normal” integral. Before doing so, a remark on the direct numerical approach to the path integral is in order. As can be seen already by looking at the integrand of our 1D toy problem, a numerical attack to calculate the integral of a highly oscillatory function will be problematic due to the near cancellation of terms. This is even more true for the full fledged path integral and the associated problem is sometimes referred to as the sign problem, which is a topic at the forefront of present day research. Much more wellbehaved with respect to numerical treatment are imaginary time path integrals, which will not be dealt with herein, however.
2.2.3 Semiclassical Approximation
The semiclassical approximation of the propagator goes back to van Vleck [15]. Its direct derivation from the path integral followed many years later, however, and finally led to the generalization of the van Vleck formula by Gutzwiller [16]. We will lateron use semiclassical arguments quite frequently, because they allow for a qualitative and often also for a quantitative explanation of many interesting quantum phenomena. For this reason we will go through the derivation of the socalled van VleckGutzwiller (VVG) propagator in some detail.
This final expression has interference effects built in, because of the summation over classical trajectories and is very elegant and intuitive, because it relies solely on classical trajectories. However, it has also a major drawback, especially for systems with several degrees of freedom. Then the underlying root search problem becomes extremely hard to solve and a semiclassical propagator using the solution of classical initial value problems would be much needed. Such a reformulation of the semiclassical expression is possible and will be discussed in Sect. 2.3.4 on numerical methods.
2.2.4 Pictures of Quantum Mechanics and TimeDependent Perturbation Theory
2.8.
Verify that the time evolution operator in the interaction picture \(\hat{U}_\mathrm{I}(t,0)=\hat{U}_0^+(t,0)\hat{U}(t,0)\) fulfills the appropriate differential equation.

\(\hat{H}=\hat{H}_0+\hat{W}\) leads to the interaction picture

\(\hat{H}_0=0\) und \(\hat{W}=\hat{H}\) leads to the Schrödinger picture

\(\hat{H}_0=\hat{H}\) and \(\hat{W}=0\) leads to the Heisenberg picture
Relations between the wavefunctions in the different pictures of quantum mechanics
\(\varPsi _\mathrm{S}(t)\rangle \)  \(\varPsi _\mathrm{H}\rangle \)  \(\varPsi _\mathrm{I}(t)\rangle \)  

\(\varPsi _\mathrm{S}(t)\rangle \)  \(\hat{U}(t,0)\varPsi _\mathrm{H}\rangle \)  \(\hat{U}_0(t,0)\varPsi _\mathrm{I}(t)\rangle \)  
\(\varPsi _\mathrm{H}\rangle \)  \(\hat{U}^\dagger (t,0)\varPsi _\mathrm{S}(t)\rangle \)  \(\hat{U}_\mathrm{I}^\dagger (t,0)\varPsi _\mathrm{I}(t)\rangle \)  
\(\varPsi _\mathrm{I}(t)\rangle \)  \(\hat{U}_0^\dagger (t,0)\varPsi _\mathrm{S}(t)\rangle \)  \(\hat{U}_\mathrm{I}(t,0)\varPsi _\mathrm{H}\rangle \) 
Relations between the operators in the different pictures of quantum mechanics
\(\hat{A}_\mathrm{S}\)  \(\hat{A}_\mathrm{H}(t)\)  \(\hat{A}_\mathrm{I}(t)\)  

\(\hat{A}_\mathrm{S}\)  \(\hat{U}(t,0)\hat{A}_\mathrm{H}(t)\hat{U}^\dagger (t,0)\)  \(\hat{U}_0(t,0)\hat{A}_\mathrm{I}(t)\hat{U}_0^\dagger (t,0)\)  
\(\hat{A}_\mathrm{H}(t)\)  \(\hat{U}^\dagger (t,0)\hat{A}_\mathrm{S}\hat{U}(t,0)\)  \(\hat{U}_\mathrm{I}^\dagger (t,0)\hat{A}_\mathrm{I}(t)\hat{U}_\mathrm{I}(t,0)\)  
\(\hat{A}_\mathrm{I}(t)\)  \(\hat{U}_0^\dagger (t,0)\hat{A}_\mathrm{S}\hat{U}_0(t,0)\)  \(\hat{U}_\mathrm{I}(t,0)\hat{A}_\mathrm{H}(t)\hat{U}_\mathrm{I}^\dagger (t,0)\) 
2.2.5 Magnus Expansion
2.9.
Verify the second order expression \(\hat{\mathrm{H}}_2\) of the Magnus expansion in the exponent of the timeevolution operator in the interaction picture.
The main advantage of the expression in (2.109) is that, in principle, it is an exact result and that it does not contain the timeordering operator any more. In numerical applications the summation in the exponent will be terminated at finite n, however, and leads to a unitary propagation scheme at any order. If one would truncate the expansion after \(n=1\), then the timeordering operator in (2.109) would have been ignored altogether. Although this seems to be rather a crude approximation, in Chap. 4 we will see that it leads to reasonable results in the case of atoms subject to extremely short pulses. Furthermore, it has turned out that in the interaction picture with a suitable choice of \(\hat{H}_0\), truncating the Magnus expansion is a successful numerical approach [19].
2.2.6 TimeDependent Hartree Method
Especially in Chap. 5, we will investigate systems with several coupled degrees of freedom. The factorization of the total wavefunction is a first very crude approximative way to solve the timedependent Schrödinger equation for such systems. It shall therefore be discussed here for the simplest case of two degrees of freedom corresponding to distinguishable particles.
The particles move in effective “mean” fields that are determined by the dynamics of the other particle. The coupled equations have to be solved selfconsistently. This is the reason that the Hartree method sometimes is called a TDSCF (timedependent self consistent field) method. The multiconfiguration timedependent Hartree (MCTDH) method [20] goes far beyond what has been presented here and in principle allows for an exact numerical solution of the timedependent Schrödinger equation.
2.2.7 QuantumClassical Methods
In quantumclassical methods, the degrees of freedom are separated into a subset that shall be dealt with on the basis of classical mechanics and a subset to be described fully quantum mechanically. Analogously to the Hartree method, the classical degrees of freedom will move in an effective potential that is determined by the solution of the quantum problem.
2.10.
Verify the fundamental equations of the Ehrenfest method.
 (a)
First prove the validity of the coupled differential equations for the coefficients \(c_j\) (Use the product and the chain rule of differentiation).
 (b)
Calculate the effective force by using \(d_{kj}=d_{jk}\) (Proof?)
The first term in the expression of the force is the socalled external force, whereas the second one describes adiabatic and the third one nonadiabatic dynamics.^{10} The last two terms have to be determined by solving the quantum problem of the light particle. An alternative quantumclassical approach is the socalled surface hopping technique. Its relation to the Ehrenfest approach, and which method is suited under which circumstances is discussed in [21].
2.2.8 Floquet Theory
2.11.
2.3 Numerical Methods
Apart from special twolevel problems that will be dealt with in the next chapter and systems with maximally quadratic potentials (and problems that can be mapped onto such cases) there are only a few exactly analytically solvable problems in quantum dynamics, as can be seen by studying the review by Kleber [11].
Almost all interesting problems of atomic and molecular physics with and without the presence of laser fields classically display nonlinear dynamics, however, and the Gaussian wavepacket dynamics of Sect. 2.1.4 will be valid only for short times. Exact numerical solutions of the quantum dynamics are therefore sought for. Apart from timedependent information that is, e.g., needed for the description of pumpprobe experiments, to be discussed in Chap. 5, also spectral information for systems with autonomous Hamiltonians can be gained from time series, as was shown in Sect. 2.1.2.
In the following, different ways to solve the timedependent Schrödinger equation numerically will be described. First, we will review some numerically exact methods, and in the end the implementation of the semiclassical theory lined out in Sect. 2.2.3 by socalled initial value methods will be discussed, thereby also touching the numerical solution of the underlying classical equations of motion.
 (a)
Which (finite) basis is used to represent the wavefunction?
 (b)
In which (approximate) way is the timeevolution performed?
We will distinguish the methods according to their different approach to the solution of the problems above.
2.3.1 Orthogonal Basis Expansion
 (a)
The basis problem is solved by truncating the expansion at a large \(l=L1\), which is determined by the initial state that shall be described. One thus uses a “Finite Basis Representation”. Convergence of the results can be checked by increasing the size L of the finite basis.
 (b)
The numerical integration of the linear system of differential equations could be performed with the help of a suitable integration routine like the RungeKutta method [25].
2.3.1.1 The Floquet Matrix
In general the basis function expansion method is only a viable approach if the matrix elements of the Hamiltonian can be calculated easily. If the basis is the harmonic oscillator one, this is the case if the potential is given by a polynomial of low order. In other cases or if the potential is multidimensional, socalled “discrete variable representations” (DVR) [27] are frequently used. Finally, it should be noted that the diagonalization of the Floquet matrix becomes much more difficult, if the system under consideration contains a continuum of states. Then the method of complex rotation can be employed [28].
2.3.2 SplitOperator Method
2.12.
Show that the Strang splitting of the timeevolution operator leads to a second order method.
Hint: use the Zassenhaus as well as the BCH formula
 1.
Represent the initial wavefunction on a position space grid
 2.
Apply the operator \(\mathrm{e}^{\mathrm{i}\hat{V}\Delta t/(2\hbar )}\)
 3.
Perform a FFT into momentum space
 4.
Apply the operator \(\mathrm{e}^{\mathrm{i}\hat{T}_\mathrm{k}\Delta t/\hbar }\)
 5.
Perform an inverse FFT back into position space
 6.
Apply the operator \(\mathrm{e}^{\mathrm{i}\hat{V}\Delta t/(2\hbar )}\).
This procedure is applied for the propagation over a small time step. For the propagation over long times it will be repeated frequently and if the intermediate values of the wavefunction are not needed, the two half time steps of potential propagation can be combined (apart from the first and the last one). Furthermore, we stress that to propagate the wavefunction over the next time step, we will need its value not only at \(x_n\) but at all values of x. This is reflecting the nonlocal nature of quantum theory. For the calculation of the new wavefunction the old one is needed everywhere. This is in contrast to classical mechanics. A trajectory only depends on its own initial conditions; classical mechanics is a local theory.

\(N=2^j\) has to be an integer power of 2

The grid length is \(X=x_{max}x_{min}\) and \(x_n\) are equidistant with \(\Delta x=X/N\)

The numerical effort scales with \(N \ln N\) [25]
 The maximal momentum that can be described isand \(p_{min}=p_{max}\)$$p_{max}=h/(2\Delta x)=Nh/(2X),$$

The covered phase space volume is \(V_P=2Xp_{max}=Nh\)

The time step should fulfill \(\Delta t<\hbar \pi /(3V_\mathrm{max})\), with \(V_\mathrm{max}\) the maximal excursion of the potential [31]. For very long propagation times, see also [32].

If calculated using (2.45), energy resolution is given by \(\Delta E_\mathrm{min}=\hbar \pi /T_\mathrm{t}\), with \(T_\mathrm{t}\) the total propagation time.
There are more recent implementations of FFT which do not have the restriction to integer powers of 2 and which, through adaption to the platform that is used for the calculations can have considerable advantages in speed (FFTW: fastest Fourier transformation in the West [33]).
The usage of the timeevolution operator for constant Hamiltonians at the beginning of our discussion is no restriction of the presented methodology to timeindependent Hamiltonians. As in the case of the infinitesimal timeevolution operator (2.32), one can use a constant Hamiltonian for the propagation over a small time interval \(\Delta t\). At the beginning of the next time step a slightly changed Hamiltonian is employed.
Finally, one drawback of the method that will not come into play in the present book, however, shall be mentioned. The splitoperator idea only succeeds in producing simple multiplicative exponentials if there are no products of \(\hat{p}\) and \(\hat{x}\) in the Hamiltonian. These would appear in the treatment of dissipative quantum problems, which are outside the scope of this presentation.
2.3.2.1 Negative Imaginary Absorbing Potentials
The choice of the functional form of f(x) in (2.197) is crucial. It turns out that the potential has to rise smoothly and rather slowly in order that there do not occur unphysical reflections of the wavefunction induced by the negative imaginary potential. A detailed study of several different functional forms of the imaginary potential can be found in [34].
2.3.3 Alternative Methods of TimeEvolution
In the material presented so far we have dealt both, with the solution of problem (a) as well as problem (b). In the following some alternative ways of treating the timeevolution, i.e., problem (b) shall be reviewed.
2.3.3.1 Method of Finite Differences
2.13.
 (a)
\( \mathrm{Re}\langle \varPsi (t\Delta t)\varPsi (t)\rangle =\mathrm{Re}\langle \varPsi (t)\varPsi (t+\Delta t)\rangle =\mathrm{const} \)
 (b)
\( \mathrm{Re}\langle \varPsi (t\Delta t)\hat{H}\varPsi (t)\rangle =\mathrm{Re}\langle \varPsi (t)\hat{H}\varPsi (t+\Delta t)\rangle =\mathrm{const} \)
 (c)
Interprete the results gained above.
 (d)
Consider the timeevolution of an eigenstate \(\psi \) of the Hamiltonian with eigenvalue E and derive a criterion for the maximally allowed time step \(\Delta t\).
Hint: Insert the exact timeevolution into the SOD scheme and distinguish the exact eigenvalue from the approximate \(E_\mathrm{app}\) due to SOD time evolution.
2.3.3.2 CrankNicolson Method
2.3.3.3 Polynomial Methods

In the Chebyshev method, the polynomials are fixed to be the complex valued Chebyshev ones. A first application to the problem of wavefunction propagation has been presented by TalEzer and Kosloff [37]. These authors have shown that the approach is up to six times more efficient than the SOD method, presented above. It allows for evolution over relatively long time steps. Drawbacks are that intermediate time information is not readily available and, even worse in the present context, that timedependent Hamiltonians cannot be treated.

In contrast to the first approach, in the Lanczos method , the polynomials are not fixed but are generated in the course of the propagation. A very profound introduction to the commonly applied short iterative Lanczos method can be found in [38].
2.3.4 Semiclassical Initial Value Representations
As the final prerequisite before we deal with the physics of lasermatter interaction, a reformulation of the semiclassical van VleckGutzwiller propagator presented in Sect. 2.2.3 shall be discussed. We have already mentioned that the VVG method is based on the solution of classical boundary value (or root search) problems, which makes it hard to implement. A much more user friendly approach would be based on classical initial value solutions and is therefore termed initial value representation.
We start the discussion of a specific initial value representation of the semiclassical propagator with a short introduction to commonly used symplectic integration procedures for the solution of the underlying classical dynamics.
2.3.4.1 Symplectic Integration
2.14.
By expanding up to second order in \(\Delta t\) show that there is a difference between \(\exp \{\Delta t\hat{H}\}\) and \(\exp \{\Delta t\hat{T}_\mathrm{k}\} \exp \{\Delta t\hat{V}\}\).
Hint: The Jacobi identity \(\{A,\{B,C\}\}+\{B,\{C,A\}\}+\{C,\{A,B\}\}=0\) might be helpful.
Coefficients for some symplectic integration methods of increasing order
Ruth’s leap frog (position Verlet)  \(a_1=1/2\) \(a_2=1/2\)  \(b_1=0\) \(b_2=1\) 

Fourthorder Gray [41]  \(a_1=(1\sqrt{1/3})/2\)  \(b_1=0\) 
\(a_2=\sqrt{1/3}\)  \(b_2=(1/2+\sqrt{1/3})/2\)  
\(a_3=a_2\)  \(b_3=1/2\)  
\(a_4=(1+\sqrt{1/3})/2\)  \(b_4=(1/2\sqrt{1/3})/2\)  
Sixthorder Yoshida [42]  \(a_1=0.78451361047756\)  \(b_1=0.39225680523878\) 
\(a_2=0.23557321335936\)  \(b_2=0.51004341191846\)  
\(a_3=1.1776799841789\)  \(b_3=0.47105338540976\)  
\(a_4=1.3151863206839\)  \(b_4=0.068753168252520\)  
\(a_5=a_3\)  \(b_5=b_4\)  
\(a_6=a_2\)  \(b_6=b_3\)  
\(a_7=a_1\)  \(b_7=b_2\)  
\(a_8=0\)  \(b_8=b_1\) 
In a thorough study of the numerical accuracy of symplectic integrators it has been found that they yield very stable trajectories and that, for autonomous Hamiltonians, the standard deviation of the energy and the energy drift are comparatively small [39].
2.3.4.2 Coherent States
2.15.
Restricting the discussion to \(N=1\), prove that the coherent states form a complete set by expressing them as a sum over harmonic oscillator eigenstates.
2.3.4.3 HermanKluk Propagator
In contrast to the VVG prefactor, the expression (2.231) does not exhibit singularities at caustics. Recently, it has been proven that the HermanKluk method is a uniform semiclassical method [49]. Furthermore, for the numerics it is important that the square root in the prefactor has to be taken in such a fashion that the result is continuous as a function of time [50]. This is reminiscent of the Maslov phase in the van VleckGutzwiller expression (2.91), which does not have to be calculated explicitly, however.
A final remark on the connection between the two semiclassical expressions for the propagator, which we have discussed so far, shall be made. After performing the integration over the initial phase space variables in (2.229) in the stationary phase approximation, the van VleckGutzwiller expression will emerge. One can also turn around that reasoning and derive the HermanKluk prefactor, by demanding that the SPA applied to the phase space integral yields the van VleckGutzwiller expression [50]. For the derivation of a more general form of the prefactor in this way, see [51]. An even simpler way to derive the VVG propagator from the HermanKluk expression by taking the limit \(\gamma \rightarrow \infty \) is explicitly given in Appendix 2.D.
2.3.4.4 Semiclassical Propagation of Gaussian Wavepackets
Finally, it is worthwhile to check that the timedependent parameter \(\gamma _t\) fulfills a nonlinear Riccati differential equation similar to (2.61). To this end, in Exercise 2.16, the equations of motion of the stability matrix elements given in Appendix 2.C should be used.
2.16.
The TGWD inverse width parameter allows for a reformulation of the HK prefactor. For reasons of simplicity, consider the 1D case.
 (a)Show that the nonlinear Riccati differential equationis fulfilled by the inverse width parameter \(\gamma _t\).$$\begin{aligned} \dot{\gamma }_t=\frac{\mathrm{i}\hbar }{m}\gamma _t^2\frac{1}{\mathrm{i}\hbar }V'' \end{aligned}$$
 (b)Writing the inverse width parameter in the logderivative formwith \(Q=m_{22}+\mathrm{i}\hbar \gamma m_{21}\), show that the complex conjugate of the HK prefactor can be entirely formulated in terms of \(\gamma _t\) via$$\begin{aligned} \gamma _t=\frac{m}{\mathrm{i}\hbar }\frac{\dot{Q}}{Q}, \end{aligned}$$$$\begin{aligned} R^*=\sqrt{\frac{1}{2}(1+\gamma _t/\gamma )}\exp \left\{ \frac{1}{2}\int _0^t\mathrm{d}t'\frac{\mathrm{i}\hbar }{m}\gamma _{t'}\right\} . \end{aligned}$$
2.4 Notes and Further Reading
TDSE and TimeEvolution Operator
The restriction to the use of the nonrelativistic TDSE for the description of lasermatter interaction is due to the focus of this book on intermediate field strengths, maximally of the order of an atomic unit (see also Appendix 4.A), and wavelengths around the visible range. If stronger fields are considered (and for longer wavelengths), in the case of electrons, the Dirac equation has to be solved. A concise discussion of the Dirac equation and relativistic corrections to the Schrödinger equation can be found in Appendix 7 of [56]. The spinor character of the wavefunction leads to a considerable complexification of the problem, which may not be necessary if the dipole approximation (see Chap. 3) is still applicable [57].
The very detailed book by Schleich [58] contains a lot of additional information on Schrödinger type timeevolution operators and also the timeevolution of the density operator is discussed therein. More on timedependent and energydependent Green’s function can be found in Economou’s book [59]. The extraction of spectral information from timedependent quantum information goes back to work of Heller, which is reviewed, e.g., in [44] and is covered also in the book by Tannor [38]. In both previous citations many additional references concerning Gaussian wavepacket dynamics can be found. A recent book by Schuch is focusing on the nonlinear Riccati equation, appearing in the GWD, with applications to quantum theory and irreversible processes [60].
Analytical Methods
A lot of additional material on the path integral formulation of quantum mechanics is contained in the tutorial article by Ingold [61] and the books by Feynman and Hibbs [6] and by Schulman [14]. The second book has more details on variational calculus and on exactly solvable path integrals (especially in the new Dover edition). It has been pointed out by Makri and Miller that a simple shorttime propagator, based on the trapezoidal rule for the discretization of the potential part of the Lagrangian (or taking a midpoint rule), is not correct through first order in \(\varDelta t\), even in the case of the harmonic oscillator [62], although it leads to the correct propagator (2.75). Improved shorttime propagators have been proposed there.
The significance of generating functions of canonical transformations in the (semi)classical limit of quantum mechanics is discussed in depth by Miller [63]. The book by Reichl [64] contains chapters on semiclassical methods and on timeperiodic systems, dealing with Floquet theory. These methods are then used in the context of quantum chaology. The book by Billing contains more material on semiclassics and on mixed quantum classical methods [65]. A book devoted to the semiclassical approach to the solution of the TDSE, as well as to the understanding of quantum mechanics, using this approach, is the one by Heller [66].
The Magnus expansion of the timeevolution operator is formally closely analogous to the socalled cumulant expansion, known from statistical physics [67].
Numerical Methods
The book by Tannor [38] contains more information on methodological and numerical approaches to solve the timedependent Schrödinger equation. Polynomial and DVR methods are dealt with in detail there. A book that is fun to read, although it covers a seemingly dry topic is the classic “Numerical Recipes” [25]. Among many other things, more details on FFT and on finite difference methods to solve the TDSE can be found therein. Methods for the solution of the TDSE in the case of strong field driving are discussed in [68]. More recently, a phase space approach has been devised that is taylored for the solution of the TDSE for laserdriven electronic wavepacket propagation [69]. In [70], molecular quantum dynamics is discussed from the viewpoint of the MCTDH method.
A review of different semiclassical approximations based on Gaussian wavepackets is given in [55], whereas a combination of the HermanKluk method with thawed GWD for correlated manyparticle systems, termed semiclassical hybrid dynamcis (SCHD), is laid out in [51]. The coupled coherent states (CCS) method of Shalashilin and Child [71] allows in principle for an exact numerical solution of the TDSE and has the HermanKluk method as a limiting case, as shown very elegantly in [72]. Finally, we have not discussed Bohmian mechanics that is recently being used not only as an interpretational tool, but in a synthetic way, in order to generate solutions of the TDSE using (nonclassical) trajectories [73]. For another very good discussion of this topic, see Chap. 4 of [38]. The nonlocality of quantum mechanics is especially apparent in Bohmian mechanics, due to the presence of the socalled quantum potential that couples the motion of individual trajectories.
Footnotes
 1.
This is the atomic unit for the electric field, given by \(\mathcal{E}=5.14\times 10^9\) V/cm.
 2.
We stress that the integral form given in (2.28) is equivalent to the differential form of the timedependent Schrödinger equation (as can be shown by differentiation) and in addition it has the initial condition “built in”.
 3.
This is how Dyson proceeded in [4].
 4.
At \(t_1=t_2\) and for \(\hat{A}\ne \hat{B}\) additional assumptions on ordering would have to be made.
 5.
In cases where no inverse group element exists this is called semigroup property.
 6.
In order to ensure convergence of the integral, one adds a small positive imaginary part to the energy, \(z=E+\mathrm{i}\epsilon \).
 7.
This is a Fresnel integral, i.e., the Gaussian integral of ( 1.29) with purely imaginary a.
 8.
This Ansatz is also the starting point of socalled Bohmian mechanics approaches to quantum dynamics.
 9.
In general, \(x_f\) and \(x_i\) have to be replaced by vectors!
 10.
The explanation of these terms follows in Chap. 5.
 11.
We have used \(\hat{U}(t+T,t)\hat{\mathcal{H}}(t)\hat{U}^{1}(t+T,t)= \hat{\mathcal{H}}(t+T)=\hat{H}(t+T)\mathrm{i}\hbar \partial _{t+T} =\hat{H}(t)\mathrm{i}\hbar \partial _t\).
 12.
Formally this Ansatz is equivalent to the Bloch theorem of solid state physics.
 13.
Note that k has to be an integer in order for the modified quasieigenfunction to be periodic.
 14.
This would be true, if the energies were exact, which is prohibited by problem (a).
 15.
The BakerCampbellHaussdorff (BCH) formula is the dual relation and reads \(\exp \{\hat{x}\}\exp \{\hat{y}\}=\exp \{\hat{x}+\hat{y}+1/2[\hat{x},\hat{y}]+1/12( [\hat{x},[\hat{x},\hat{y}]]+[\hat{y},[\hat{y},\hat{x}]])+\cdots \}\).
 16.
Originally this approach was proposed by Fleck, Morris and Feit for the solution of the Maxwell waveequation [30].
 17.
For notational convenience, we assume the Hamiltonian to be timeindependent; the following results are also valid in the general case of a timedependent Hamiltonian, however.
 18.
Note that this procedure is more approximative than a stationary phase approximation.
 19.
In Sect. 5.1.2 the physical background of the Morse oscillator will be elucidated.
 20.
Be careful to use the formula \(\det \mathbf{M}=\det \mathbf{m_{22}}\det (\mathbf{m_{11}m_{12}m_{22}^{1}m_{21}})\) valid for block matrices!
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