Time-dependent Schrodinger equation, real

The dynamics of electrons are defined by the time dependent Schrodinger equation, ih(dA>/dt) — HAs. The appearance of i — J 1 in this equation makes it clear that the wave func tion is a complex valued function, not a real valued function. 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

This equation is exact and constitutes an iterative equation equivalent to the time-dependent Schrodinger equation [185,186]. The iterative process itself does not involve the imaginary number i therefore, if h(f) and )( — x) were the real parts of the wavepacket, then (f + x) would also be real and would be the real part of the exact wavepacket at time (f + x). Thus, if )( ) is complex, we can use Eq. (4.68) to propagate the real part of 4>(f) forward in time without reference to the imaginary part. 

Taken together, the use of the real part of the wavepacket and the mapping of the time-dependent Schrodinger equation lead to a very significant reduction of the computational work needed to accompfish the calculation of reactive cross sections using wavepacket techniques. 

In practice one does not proceed as we did in the above derivation. Instead of calculating first all stationary wavefunctions and then constructing the wavepacket according to (4.3), one solves the time-dependent Schrodinger equation (4.1) with the initial condition (4.4) directly. Numerical propagation schemes will be discussed in the next section. Since 4 /(0) is real the autocorrelation function fulfills the symmetry relation... 

The time-dependent Schrodinger equation can, in principle, be used to predict the evolution of any physical system, but this method is not feasible in practice. First, the deterministic character of the Schrodinger equation forbids irreversible processes. Second, the many-body character of the Schrodinger equation, and the large number of degrees of freedom, such as lattice vibrations, complicate the description of real magnetic systems. 

Equations (10.47) or (10.50) do not have a mathematical or numerical advantage over Eqs (10.43), however, they show an interesting analogy with another physical system, a spin particle in a magnetic field. This is shown in Appendix 1 OA. A more important observation is that as they stand, Eqs (10.43) and their equivalents (10.47) and (10.50) do not contain information that was not available in the regular time-dependent Schrodinger equation whose solution for this problem was discussed in Section 2.2. The real advantage of the Liouville equation appears in the description... 

The easiest way to see this is simply to substitute a complex energy into the time-dependent Schrodinger equation the real part of the energy can then be associated with an oscillatory eigenfunction, while the imaginary part is associated with exponential decay. 

It is a real quantity, which follows from the fact that the time-dependent Schrodinger equation conserves the norm of the wave function. The phase-isolated wave... 

The diffusion Monte Carlo (DMC) method is based on the simulation of the diffusion equation. Let us start with the primitive DMC method. By replacing the real time t with the imaginary time r = it, the time-dependent Schrodinger equation is rewritten as... 

The time-reversal symmetry of the crystalline Hamiltonian introduces an additional energy-level degeneracy.Let the Hamiltonian operator H be real. The transition in the time-dependent Schrodinger equation to a complex-conjugate equation with simultaneous time-inversion substitution... 

The simulation of a real-time spectrum requires the solution of the time-dependent Schrodinger equation for a system of n potential-energy surfaces which are coupled by a laser field ... 

Because of its similarity to the time-dependent Schrodinger equation, Eq. [12] is often referred to as the Schrodinger equation in imaginary time. The analogy is formally correct, since solutions of the time-dependent Schrodinger equation have equivalent real and imaginary parts under steady state conditions. 

The description of quantum scattering closely parallels the classical formalism. In lieu of the classical orbit x t) satisfying Newton s equation, we now have a state vector ipt) satisfying the time-dependent Schrodinger equation (3.1). I pt) is any vector in the appropriate Hilbert space H.We shall adopt the classical terminology and refer to the solution U(t) ip) as an orbits although of course it is no longer an orbit in real space R. Every orbit U (t) ip) can be uniquely identified by the fixed vector ip), which is just the state vector at the instant =0. 

Formally these equations are not too different from the matrix formulation of the time dependent Schrodinger equation, except that the K and Y matrices are real matrices. The fundamental simplification arises through the order of the matrices. If the number of coarse grained levels is just defined by the energy in terms of the number of photons absorbed, then the order of K and Y would perhaps be around 50, typically (allowing up to 50 photons to be absorbed). If one allowed specification of another quantity by perhaps 10 different attributes, this would increase the order to 500. Even if many more attributes are specified, the order of the matrices remains within reasonable , perhaps rather large, limits. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

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