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Approximation by Variational Method:

The variational method determines the most accurate energy eigenvalue for a system by choosing trial wavefunctions from a set of wavefunctions. The one wavefunction from the set which produces the lowest energy for the system is most similar to the actual but unknown wavefunction and therefore produces the most accurate energy eigenvalue. The energy eigenvalue determined by a trial wavefunction is called the variational energy and is denoted by W. The goal with this method is to minimize W.

The proof that W will always be either higher than or equal to the true energy value is as follows:

The variational energy of a system is determined by:

eq.1

Y ' represents a trial wavefunction. The variational energy of the system can be represented as the sum of the actual ground-state energy and some value caused by the inaccuracy of the trial wavefunction, denoted e :

eq.2

In order to complete this proof, it will be shown e is always a positive value. Rearranging:

eq.3

To continue, the trial wavefunction is assumed to be a linear combination of the actual set of wavefunctions for the system, n denoting the particular state of the system:

eq.4

Implementing this in eq. 3:

eq.5

Rewritting the expression as one summation:

eq.6

Realizing Y _n is an eigenfunction of the hamiltonian:

eq.7

Simplifying:

eq.8

From this expression, one is able to determine:

eq.9

Therefore, the proof is finished by stating:

eq.10

Having proved W is always greater than or equal to the correct ground-state energy value, one can utilize the alternative form of eq. 1 to find the lowest value of W, the best approximate energy eigenvalue for the system in question:

eq.11