Time-independent Schrodinger equation TISE

The above problem can be simplified by separating the fast electronic motion from the slow nuclear motion. One first defines an electronic Hamiltonian, also called clamped nucleus Hamiltonian Hgiir R) = Tgiir) + V r R). This electronic Hamiltonian acts in the electronic space and depends parametrically on the nuclear coordinates R, as indicated by the semicolon in the coordinate dependence of the operators. The eigenfunctions and eigenvalues of the associated time-independent Schrodinger equation (TISE)... 

This is known as the time-independent Schrodinger equation (TISE). The quantity inside the square brackets is called the hamiltonian, TL. In the TISE the hamiltonian corresponds to the energy operator. Notice that the energy e has now become the eigenvalue of the wavefunction

second-order differential equation represented by the TISE. 

The objective of WFT is the exact solution of the time-independent Schrodinger equation (TISE), H V = ET, for a system of interest. (We recall that in quantum mechanics, associated with each measurable parameter in a physical system is an operator, and the operator associated with the energy of a system is called the Hamiltonian H. The Hamiltonian contains the operations associated with the kinetic and potential energies of aU particles that comprise a system. We further note that the terms function, operator, and functional are to be understood such that a function is a prescription which maps one or more numbers to another number, an operator is a prescription which maps one function to another function, and a functional takes a function and provides a number.) The solution to the TISE yields the wave function T as weU as the energy E for the system of interest. In a systematic, variational search one looks for the wave function that produces the lowest energy, and arrives at a description for the system in its ground state. 

The TDSE is a generalization of the time-independent Schrodinger equation (TISE), but it does not invalidate the TISE. This is because tire TDSE is always a differential equation that is separable in space and time in those cases where the Hamiltonian has no explicit time dependence (which are the only kinds of cases considered until now). We demonstrate this separability in the following steps with only one spatial coordinate, x, used for simplicity. The test for separability is to assume a product form for the wavefunction and then determine if such a product form can satisfy the TDSE. 

TDSE - time-dependent Schrodinger equation TISE - time-independent Schrodinger equation... 

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