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Question about the hermiticity of the momentum operator 
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#1
Feb2208, 07:56 AM

P: 511

Is momentum operator [tex]\frac{\hbar}{i}\frac{\partial}{\partial\ x }[/tex] is Hermitian only for a normalized wave function?What is the case for the box normalization as done for a free particle?
Actually when we prove the Hermiticity of the momentum operator, we do simple by parts integartion and use the scalar products.I never bothered about whether the wave function is normalized or not. Can anyone suggest anything? 


#2
Feb2208, 09:40 AM

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P: 4,738

http://mathworld.wolfram.com/HermitianOperator.html
See eq (3) Thats how you can do it. You can use any wave function. 


#3
Feb2208, 10:02 AM

P: 511

So, hermiticity of the momentum operator is independent of whether the wave function is normalized or notright?
I faced this question in university question book and got astonished.They asked to prove this and also asked what would it be if the wave function is boxnormalized. 


#4
Feb2208, 10:07 AM

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P: 4,738

Question about the hermiticity of the momentum operator
It should not matter.
Look at eq (13) and below, never is it stated that the wave function is normalized. Normalized means that <psi psi > = 1, so it is just some multiplicatible constants that is added. And constants never affects what your operator does with the wave function. A is an operator, b is constant: [A,b] = 0, they commute. etc. 


#5
Feb2208, 11:30 AM

P: 173

In fact mathematically speaking what is really important is the couple (A,D(A)) to analize all the spectrum of an operator. In some circumstances can happen that P does posses residue spectrum. Obviously they have no physical meaning. Normalization on a function doesn't mean anything. f(x)=100g(x) its ok either. regards marco 


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