Quantum Physics - hermitian and linear operators

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Description


1. Prove that operators i(d/dx) and d^2/dx^2 are Hermitian.


2. Operators A and B are defined by:

A$\psi$(x)=$\psi$(x)+x

B$\psi$(x)=$d\psi/dx$+2$\psi/dx$(x)

Check if they are linear.


The attempt at a solution


I noted the proof of the momentum operator '-ih/dx' being hermitian, should I just multiply all the terms involved in it by '-1/h'? I do not really know what should I do in the second exercise.

 

[dextercioby]

-ihbar d/dx is hermitean. You say you have the proof. Now dropping hbar which is a real (as opposed to an imaginary) constant, does it change the hermitean character or not ?

As for the second derivative operator, assuming wavefunctions dropping to 0 when going to infinity, can you show that it's hermitean by maneuvering the integrals ?

Consider the definition of linearity. It's not more complicated than that.