# Qm problem

1. Mar 13, 2008

### neelakash

1. The problem statement, all variables and given/known data

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

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

Then prove that

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

and $$\ u_1 (\ x )$$ and $$\ u_2 (\ x )$$ are orthogonal to each other.

OR prove that

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

2. Relevant 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 $$\ u_1 (\ x )$$ and $$\ u_2 (\ x )$$
are non-degenerate eigenfunctions of this operator,then it is done.
But I wonder, how to show it?