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Qm problem

  • Thread starter neelakash
  • Start date
511
1
1. Homework Statement

I found the following problem in two places.But I doubt the first one is wrong.



Let [tex]\ u_1(\ x )[/tex] and [tex]\ u_2(\ x )[/tex] are two degenerate eigenfunctions of the hamiltonian [tex]\ H =\frac{\ p^2 }{2\ m }\ + \ V (\ x )[/tex]



Then prove that

[tex]
\int u_1(x)\left(x\frac{\hbar}{i}\frac{\partial}{\partial x}-\frac{\hbar}{i}\frac{\partial}{\partial x}x\right)u_2(x)dx=0.
[/tex]

and [tex]\ u_1 (\ x )[/tex] and [tex]\ u_2 (\ x )[/tex] are orthogonal to each other.

OR prove that

[tex]
\int u_1*(x)\left(x\frac{\hbar}{i}\frac{\partial}{\parti al x}+\frac{\hbar}{i}\frac{\partial}{\partial x}x\right)u_2(x)dx=0.
[/tex]



2. Homework Equations



3. The Attempt at a Solution

Now I do not know what is correct expression.I found them in different places.May be both are correct.
But in the first one the sandwitched operator (xp-px) is skew hermitian.Hence, its eigenvalues are imaginary!!!So I doubt about its validity.

Now how to prove?As xp+px is hermitian, it is tempting to think that if [tex]\ u_1 (\ x )[/tex] and [tex]\ u_2 (\ x )[/tex]
are non-degenerate eigenfunctions of this operator,then it is done.
But I wonder, how to show it?
 

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