- #1
issacnewton
- 1,016
- 35
Hi
here's a problem I am having.
Consider the hilbert space of two-variable complex functions [tex]\psi (x,y)[/tex].
A permutation operator is defined by its action on [tex]\psi (x,y)[/tex] as follows.
[tex]\hat{\pi} \psi (x,y) = \psi (y,x) [/tex]
a) Verify that operator is linear and hermitian.
b) Show that
[tex]\hat{\pi}^2 = \hat{I}[/tex]
find the eigenvalues and show that the eigenfunctions of [tex]\hat{\pi}[/tex] are given by
[tex]\psi_{+} (x,y)= \frac{1}{2}\left[ \psi (x,y) +\psi (y,x) \right] [/tex]
and
[tex]\psi_{-} (x,y)= \frac{1}{2}\left[ \psi (x,y) -\psi (y,x) \right] [/tex]
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 ?
here's a problem I am having.
Consider the hilbert space of two-variable complex functions [tex]\psi (x,y)[/tex].
A permutation operator is defined by its action on [tex]\psi (x,y)[/tex] as follows.
[tex]\hat{\pi} \psi (x,y) = \psi (y,x) [/tex]
a) Verify that operator is linear and hermitian.
b) Show that
[tex]\hat{\pi}^2 = \hat{I}[/tex]
find the eigenvalues and show that the eigenfunctions of [tex]\hat{\pi}[/tex] are given by
[tex]\psi_{+} (x,y)= \frac{1}{2}\left[ \psi (x,y) +\psi (y,x) \right] [/tex]
and
[tex]\psi_{-} (x,y)= \frac{1}{2}\left[ \psi (x,y) -\psi (y,x) \right] [/tex]
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 ?