Question about the hermiticity of the momentum operator

In summary, the Hermitian property of the momentum operator, represented by \frac{\hbar}{i}\frac{\partial}{\partial\ x }, is independent of whether the wave function is normalized or not. This is because when proving its Hermiticity, the normalization of the wave function is not taken into account. The analysis on the momentum operator should be carried out in different configuration schemes, and the important factor is the couple (A,D(A)) rather than the normalization of the wave function. Normalization of a function does not affect the physical meaning of the operator.
  • #1
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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?
 
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  • #3
So, hermiticity of the momentum operator is independent of whether the wave function is normalized or not----right?

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 box-normalized.
 
  • #4
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
neelakash said:
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?

Actually the analysis on P must be carry out in the differnet configurations scheme. I mean that its hermitianity depend upon its domain D(P).
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
 

1. What does it mean for a momentum operator to be Hermitian?

A Hermitian operator is one that is equal to its own conjugate transpose. In the context of quantum mechanics, this means that the momentum operator will have the same effect on a particle regardless of whether it is moving in the positive or negative direction.

2. Why is Hermiticity important in quantum mechanics?

Hermiticity is important because it ensures that the eigenvalues of a Hermitian operator are real, which is necessary for the physical interpretation of quantum mechanics. It also guarantees that observables, such as momentum, will have real and measurable values.

3. How is the Hermiticity of the momentum operator related to the uncertainty principle?

The Hermiticity of the momentum operator is related to the uncertainty principle through the commutation relation between position and momentum. The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. The Hermiticity of the momentum operator ensures that the uncertainty principle holds true.

4. Can the momentum operator ever be non-Hermitian?

No, the momentum operator must always be Hermitian in quantum mechanics. This is a fundamental property of the mathematical framework of quantum mechanics and is necessary for the physical interpretation of the theory.

5. How is the Hermiticity of the momentum operator related to conservation of momentum?

The Hermiticity of the momentum operator is related to conservation of momentum through the fact that Hermitian operators have real eigenvalues. In quantum mechanics, the eigenvalues of the momentum operator represent the possible values of momentum that a particle can have. Conservation of momentum requires that the total momentum of a system remains constant, and the Hermiticity of the momentum operator ensures that this is the case.

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