# Operator in non-orthogonal basis

1. Oct 4, 2007

### j_dirac

Hi, is possible make up a operator in a non-orthogonal basis, if is possible how I can contruct the operator.

thanks

2. Oct 4, 2007

### quetzalcoatl9

why not form your operators as |b><a|

3. Oct 5, 2007

### j_dirac

which are the consequence of choice a basis non-orthogonal?

4. Oct 5, 2007

### A/4

Why do you want to form an operator in a non-orthogonal basis in the first place?

5. Oct 5, 2007

### nrqed

Of course.

All you need to know is the effect of the operator on all the basis states. So if you know all the values of $<a_i|A|a_j>$ then you know everything about the operator.

Alternatively, as quetzalcoatl9 pointed out, an arbitrary operators can be written as

$A = \sum c_{ij} |a_i><a_j|$

One consequence of having a non orthonogonal basis is that you can't read off directly from the above expression what is the effect of applying the operator to a basis state gives.

If the basis is orthogonal, then applying A to, say, $|a_3>$ will simply give $c_{13} |a_1> + c_{23} |a_2> + \ldots$ (I am assuming that the labels of the states are discrete and start at 1). If the basis is not orthogonal, the expression is of course more complicated.

6. Oct 5, 2007

### j_dirac

I can construct a basis depent of basis non-orthogonal, how might make up? and what happen with the eigenvalues and elements of the operator.

someone know if the situation present in some quantum system.