Hermiticity of permutation operator

[issacnewton]

Hi

here's a problem I am having.

Consider the hilbert space of two-variable complex functions $$\psi (x,y)$$.

A permutation operator is defined by its action on $$\psi (x,y)$$ as follows.

$$\hat{\pi} \psi (x,y) = \psi (y,x)$$

a) Verify that operator is linear and hermitian.

b) Show that

$$\hat{\pi}^2 = \hat{I}$$

find the eigenvalues and show that the eigenfunctions of $$\hat{\pi}$$ are given by

$$\psi_{+} (x,y)= \frac{1}{2}\left[ \psi (x,y) +\psi (y,x) \right]$$

and

$$\psi_{-} (x,y)= \frac{1}{2}\left[ \psi (x,y) -\psi (y,x) \right]$$

I could show that the operator is linear and also that its square is unity operator I . I did
find out the eigenvalues too. I am having trouble showing that its hermitian and the
part b about its eigenfunctions.

Any help ?

 

[mathfeel]

After you already show that $\hat{\pi}$ is linear. Take a look at $\hat{\pi}^2$. What is it? From this what can you infer about the possible eigenvalues for $\hat{pi}$? What properties for the eigenvalues for an operator implies Hermiticity?

 

[issacnewton]

Hi , eigenvalues are $$\pm 1$$ they are real. I know that hermitian operators have real eigenvalues but does it mean that if operator has real eigenvalues then its a hermitian
operator ( this is converse statement , so not necessarily true) ?

 

[mathfeel]

Hi , eigenvalues are $$\pm 1$$ they are real. I know that hermitian operators have real eigenvalues but does it mean that if operator has real eigenvalues then its a hermitian
operator ( this is converse statement , so not necessarily true) ?

If all the eigenvalues of an operators are real.

More explicitly, you are trying to show that:
$$\int_{-\infty}^{\infty} \psi(r)^{*} \phi(-r) dr = \int_{-\infty}^{\infty} \psi(-r)^{*} \phi(r) dr$$
which is easy by simple change of variable.

 

[issacnewton]

So since all eigenvalues of $\hat{\pi}$ are real , its hermitian. fine that's solved.
what about the second part about the eigenfunctions ? can you help ?

 

[vela]

Just apply the operator to the functions and show that you get an eigenvalue times the function.

 

[issacnewton]

vela, yes that's one approach. but how do I derive these eigenfunctions in the first place ?

 

[mathfeel]

vela, yes that's one approach. but how do I derive these eigenfunctions in the first place ?

There are really no general approach for arbitrary operator. A good guess is a start.

 

[vela]

I suppose you could make a hand-waving argument that the Hilbert space can be written as a direct sum of subspaces, where each subspace is the span of Φ(x,y) and Φ(y,x) for a particular Φ(x,y). Then find the matrix representing $\hat{\pi}$ in one of these subspaces and diagonalize it. Blah, blah, blah...

It's more useful, though, to recognize this pattern of combining elements to achieve a certain symmetry.

 

[issacnewton]

thanks fellas. makes some sense now...