Additional Theorems in Quantum Mechanics

We now have seen two problems with exact solutions in quantum mechanics. Actually there are only a few such exact solutions remaining such as the harmonic oscUlator, the rigid rotor, the hydrogen atom, and the forced harmonic oscUlator, and we need to know some general principles 

This can also be stated in words that an operator A usually operates to the right on a function but it can also operate to the left on the complex adjoint i j = i j and the definition above means that if the operator acts to the left it must be the adjoint form of the operator. This is the characteristic of a Hermitian operator. This also means that in the matrix mechanics form of quantum mechanics a given matrix-element of a Hermitian matrix is related to another element on the other side of the (upper left to lower right) diagonal of the matrix by the relationship Amn = A. The following theorem shows why this definition is useful. 

Theorem 1 The eigenvalues of a Hermitian operator are real numbers. 

Proof Given Ai r = aijr, form the expectation value then operate to the left and the right. 

Start in the middle and operate inside the integral to the right and to the left using the adjoint rule. Now subtract the right side of the equation from the left to obtain a new condition 

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