Phys 498CQM Lecture 5

Outline

• Time-Dependent Schrodinger Equations
• Example of initial value problem; parabolic PDE

See Thijssen, Appendix A7.2.1; Koonin, Ch. 7

where the hamiltonian operator is

Here we have allowed the potential to be an explicit function of time. There are two basic approaches:
Fourier Transform in time to get equations for u(x,omega), where omega is frequency.
Explicit evolution in time. The one can Fourier transform if desired. We will follow the second approach as an example of numerical methods, and the associated stability and error analysis. Also it has become popular and used in much recent work using "time-dependent density functional theory" or "time-dependent Hartree-Fock". [This has been widely used in nuclear physics (e.g. Koonin) and recent in application to clusters (e.g. K. Yabana and G. F. Bertsch, Phys REv B54, 4484 (1996).] Although we have not introduced these theories yet, we can note that they all involve some form of independent particle equations with some approximation for V(x,t). This is an initial value problem requiring the wavefunction to be given at one time. Then the evolution in time is fixed. Because of the factor or i in the equation, the evolution of u(x,t) is manifestly unitary, i.e. the norm of u(x,t) is conserved, which is equivalent to conservation of particle number. As discussed in Thijssen and Koonin, the simplest approximation is to discretize time in steps t = n (delta t) Expanding the exponential leads to:

The error analysis in Thijssen shows this is ALWAYS unstable. A simple modification (which however makes it an implicit method requiring an inversion is to make the substitution of uⁿ by uⁿ⁺¹ on the righ hand side. This leads to

This form is stable, BUT it is not unitary (as is obvious for the form). An explicitly unitary algorithm can be made by using the operator

which is known as the Crank-Nicholson method. The inverse means that this algorithm requires explicit inversion of an operator. [This approach is used in recent work by Argyrios Tsolodikis, U of Illinois, unpublished.] This is possible if the operator can be represented as a small matrix (e.g. quantum system with a restricted Hilbert space) and it can be done for very large bases, e.g., large grids if the operator is diagonal or nearly diagonal. This is the case for grids with the kinetic energy operator in H approximated by a finite difference formula.

Another approach is to use the "Trotter decomposition"

where H = T + V. One can use this (unitary) operator to evolve the wavefunction by time delta t. The exponential of operators is in general not simple, but in the case of diagonal operators, one has simply = exp(). How can this be done, since T and V do not commute? By a Fast Fourier Transform (FFT) one can go back and forth between real space (where V is diagonal) to reciprocal space (where T is diagonal). [This is done, e.g., in M. Suzuki, J. Phys. Soc. Jpn. 61, L3015 (1992).]

Alternatively, one can expand the exponential to higher powers and have algorithms that are sufficiently stable and unitary for desired purposes. [An analysis is given in Iitaka, et al, Comp. Phys. Commun. 90, 251 (1995) for symmetric multistep algorithms.]

Example of time dependent Schrodinger Eq. using C++ codes provided by Todd Martinez.