The Time-Dependent Schrodinger Equation

The total Hamiltonian for a molecule with N electrons and M nuclei in the absence of fields is, in non-relativistic quantum mechanics, given as 

The first axiom of quantum mechanics states that the state of a molecule is completely described by the time-dependent wavefunction n, f), where 

Finally, the last postulate states that the time dependence of the wavefunction is governed by the time-dependent Schrodinger equation 

When the Hamiltonian does not depend explicitly on time like the one given in Eq. (2.1), we can apply the separation of variables technique and separate the time variable t from the spatial coordinates Rg and r. We write therefore the time-dependent wavefunction °)( Rif, rj, t) as the product of a time-independent wavefunction ° Rg, r ) and a time-dependent phase factor i9(t). 

Inserting this trial solution in Eq. (2.3), the time-dependent Schrodinger equation separates in two equations the time-independent Schrodinger equation 

Demonstrate explicitly that for the 1-D particle-in-a-box, P l is orthogonal to I 2-SOLUTION 

Substituting the limits 0 and a into this expression and evaluating  

Although the time-independent Schrodinger equation is heavily utilized in this chapter, it is not the fundamental form of the Schrodinger equation. Only stationary states—wavefunctions whose probability distributions do not vary over time— provide meaningful eigenvalues using the time-independent Schrodinger equation. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

There is a form of the Schrodinger equation that does include time. It is called the time-dependent Schrodinger equation, and has the form 

Although we have demonstrated (1.7) for only one experimental setup, its validity is general. No matter what attempts are made, the wave-particle duality of microscopic particles imposes a limit on our ability to measure simultaneously the position and momentum of such particles. The more precisely we determine the position, the less accurate is our determination of momentum. (In Fig. 1.1, sin a = A/w, so narrowing the slit increases the spread of the diffraction pattern.) This limitation is the uncertainty principle, discovered in 1927 by Werner Heisenberg. 

Because of the wave-particle duality, the act of measurement introduces an uncontrollable disturbance in the system being measured. We started with particles having a precise value of Pj (zero). By imposing the slit, we measured the x coordinate of the particles to an accuracy w, but this measurement introduced an uncertainty into the p c values of the particles. The measurement changed the state of the system. 

Classical mechanics applies only to macroscopic particles. For microscopic particles we require a new form of mechanics, called quantum mechanics. We now consider some of the contrasts between classical and quantum mechanics. For simplicity a one-particle, one-dimensional system will be discussed. 

In classical mechanics the motion of a particle is governed by Newton s second law  

Since we have two constants to determine, more information is needed. Differentiating (1.9), we have 

If we also know that at time fo the particle has velocity Vq, then we have the additional relation 

The classical-mechanical potential energy y of a particle moving in one dimension is defined to satisfy 

What would happen if one prepared the system in a given state which does not represent a stationary state For example, one may deform a molecule by using an electric field and then switch the field off. The molecule will suddenly turn out to be in state ip, that is not its stationary state. Then, according to quantum mechanics, the state of the molecule will start to change according to the time evolution equation (time-dependent Schrodinger equation) 

The equation plays a role analogous to Newton s equation of motion in classical mechanics. The position and momentum of a particle change according to Newton s equation. In the time-dependent Schrodinger equation the evolution proceeds in a complete different space - in the space of states or Hilbert space (cf. Appendix B, p. 895). 

Therefore, in quantum mechanics one has absolute determinism, but in the state space. Indeterminism begins only in our space, when one asks about the coordinates of a particle. 

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The difTerential equation above is known as the time-dependent Schrodinger equation. There is an interesting and... 

From the fact that f/conmuites with the operators Pj) h is possible to show that the linear momentum of a molecule in free space must be conserved. First we note that the time-dependent wavefiinction V(t) of a molecule fulfills the time-dependent Schrodinger equation... 

A1.6.2.1 WAVEPACKETS SOLUTIONS OF THE TIME-DEPENDENT SCHRODINGER EQUATION... 

The central equation of (non-relativistic) quantum mechanics, governing an isolated atom or molecule, is the time-dependent Schrodinger equation (TDSE) ... 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

In the diflfiision QMC (DMC) method [114. 119], the evolution of a trial wavefiinction (typically wavefiinctions of the Slater-Jastrow type, for example, obtained by VMC) proceeds in imaginary time, i = it, according to the time-dependent Schrodinger equation, which then becomes a drfifiision equation. All... 

Neuhauser D and Baer M 1989 The time dependent Schrodinger equation application of absorbing boundary conditions J. Chem. Phys. 90 4351... 

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schrodinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes. 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

De Raedt, H. Product formula algorithms for solving the time-dependent Schrodinger equation. Comput. Phys. Rep. 7 (1987) 1-72. 

If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrodinger equation... 

One then writes the time-dependent Schrodinger equation... 

RPA, and CPHF. Time-dependent Hartree-Fock (TDFIF) is the Flartree-Fock approximation for the time-dependent Schrodinger equation. CPFIF stands for coupled perturbed Flartree-Fock. The random-phase approximation (RPA) is also an equivalent formulation. There have also been time-dependent MCSCF formulations using the time-dependent gauge invariant approach (TDGI) that is equivalent to multiconfiguration RPA. All of the time-dependent methods go to the static calculation results in the v = 0 limit. 

The expression 17.35 is substituted into the time-dependent Schrodinger equation to give, after a little rearrangement. 

For a free electron Dirac proposed that the (time-dependent) Schrodinger equation should be replaced by... 

The Time Reversal Operator.—In this section we show that spatial operators are linear whereas the time reversal operator is antilinear.5 This may be seen by examining the eigenfunctions of the time dependent Schrodinger equation... 

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

Application of the time-dependent Schrodinger equation gives equation (26). This 3nelds two coupled equations which, solved for... 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

All of the methods for designing laser pulses to achieve a desired control of a molecular dynamical process require the solution of the time-dependent Schrodinger equation for the system interacting with the radiation field. Normally, this equation must be solved many times within an iterative loop. Different possible approaches to the solution of these equations are discussed in Section V. 

Equation (4.a) states that the wave function must obey the time-dependent Schrodinger equation with initial condition /(t = 0) = < ),. Equation (4.b) states that the undetermined Lagrange multiplier, x t), must obey the time-dependent Schrodinger equation with the boundary condition that x(T) = ( /(T))<1> at the end of the pulse, that is at f = T. As this boundary condition is given at the end of the pulse, we must integrate the Schrodinger equation backward in time to find X(f). The final of the three equations, Eq. (4.c), is really an equation for the time-dependent electric field, e(f). 

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