Proof of the Variation Principle This problem involves the proof of the variatio

Question

Proof of the Variation Principle

This problem involves the proof of the variational principle, Equation 7.4. Let be the problem of interest, and let be our approximation to Even though we do not know the we can express formally as where the cn are constants. (We can do this because the form a complete set of orthogonal functions.) Using the fact that the on are orthonormal (i.e. orthogonal and normalized), show that In the case at hand, we do not know the however, so the summation above is what we call a formal expansion. Now substitute the above summation into to obtain Subtract Eo from the left side of the above equation and from the right side to obtain Now explain why every term on the right side is positive, proving that

Explanation / Answer

Let energy at ground state be E0 and energy at nth state be En

Now, as we know the ground state energy is the lowest possible energy. Therefore, En>=E0

Hence, En-E0>=0

Which Proves that right hand side on above equation in question is always positive.

Thus proving E

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