Proof That Eigenvalues of Hermitian Operators Are Real

6-8 Proof That Eigenvalues of Hermitian Operators Are Real 

Let be a hermitian operator with a square-integrable eigenfunction 0. Then 

Each side of Eq. (6-16) must be expressible as a real and an imaginary part. The real parts must be equal to each other and so must the imaginary parts. Taking the complex 

We multiply Eq. (6-16) from the left by i/r and integrate over all spatial variables  

Since A is hermitian, the left-hand sides of Eqs. (6-18) and (6-19) are equal by definition (Eq. 6-10). Therefore, the right-hand sides are equal, and their difference is zero  

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