The Variational Integral and its Properties

The Variational Integral and Its Properties.—We shall show1 in this section that the integral 

If we used for our function / , called the variation function, the true wave function of the lowest state, E would equal Wo that is, 

Substitution of this expansion in the integral for -leads to the equation 

Subtracting Wo, the lowest energy value, from both sides gives 

Since W is greater than or equal to Wo for all values of n and the coefficients a an are of course all positive or zero the right side of Equation 26-6 is positive or zero. We have therefore proved that E is always an upper limit to Wo that is, 

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