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Varlation Principal

The Schrödinger wave equation (or simply wave equation can be represented in shorthand form as

where H is the Hamiltonian operator which when performed on wave function Ψ, gives a value for the energy of the system. E, which is the total energy of that system
Equation can be rearranged to give equation

E =    Ψ*HΨ dτ

This is achieved by multiplying both sides in equation by Ψ * and then carrying out integration over all the coordinates, .e. over the entire space dτ.

Now if a correct wave function Ψ, were known, it would then be possible by means of equation to calculate the energy of a system. However, an exact form of wave function, Ψ, is never known. If one

supposes a wave function Ψ1 (which may not be the correct wave function, but is only an intelligent guess) then the corresponding energy. Ev is given by

E1  =  Ψ1*HΨ1dτ          

and its value may be very far off form the ground state energy of the system, E0. If one is dissatisfied with this particular wave function, then some other wave function, Ψ2 can be chosen and this would give energy corresponding to E2. There is no limit to the number of times the wave function can be “varied” in this manner. The Variation Principle state that the calculated energy E or E1 or E2 using wave functions Ψ or Ψ1 or Ψ2 is always greater then E0 the ground state energy of the system. The best possible wave functions then eh one which givens value of the energy closest to the ground state energy of the system. The Variation Principle may, therefore, be expressed as:

E <  E =   Ψ*HΨ dτ

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