Unitary operator/matrix

facenian

I have a very basic question. I'm confused because I've read in a text that the matrix representation of a unitary operator is a unitary matrix if the basis is orthogonal, however I believe that the matrix is unitary whatever basis one uses. I'd appretiate any comments on this.

malawi_glenn

which text?

facenian

Quantum Mechanics(third edition) E. Merzbacher-Chapter 17 Page 418,I quote :"Since the operators U_a were assumed to be unitary, the representation matrices are also unitary if the basis is orthonormal"

George Jones

Try a simple example. Take the identity matrix on a 2-dimensional space, which is clearly unitary. Use linearity to compute the matrix elements with respect to the basis [itex]e_{1}' = e_1[/itex] and [itex]e_{2}' = e_1 + e_2[/itex], where [itex]e_1[/itex] and [itex]e_1[/itex] make up an orthonormal basis.

Does this give a unitary matrix?

Fredrik

A unitary matrix U satisfies [itex]\sum_j U^*_{ji}U_{jk}=\delta_{ik}[/itex]. Is this satisfied by the matrix representation of a unitary operator?

[tex]\sum_j U^*_{ji}U_{jk}=\sum_j\langle j|U|i\rangle^*\langle j|U|k\rangle=\sum_j\langle i|U^\dagger|j\rangle\langle j|U|k\rangle=\langle i|U^\dagger\Big(\sum_j|j\rangle\langle j|\Big)U|k\rangle[/tex]

This reduces to [itex]\delta_{ij}[/itex] if the parenthesis is the identity operator. I can prove that it is, if I use that the basis is orthonormal, but not without that assumption. So it looks like your book is right. What makes you think it's wrong?

facenian

thank you people. The orgin of my mistake goes like this : Let T be a untiary operator and |a_i> (i=1,...n) a basis then the matrix elements satisfy,
<a_i|T|a_k>=<a_k|T^{\dag}|a_i>*=<a_k|T^{-1}|a_i>*
what a did not realize was that the matriz elementes in the basis |a_i> are <a_i|T|a_k> only if the basis is orthonomal.

Fredrik

Oops, I didn't realize that myself.