Operator in non-orthogonal basis

[j_dirac]

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

thanks

 

[quetzalcoatl9]

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

 

[j_dirac]

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

 

[A/4]

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

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

 

[nrqed]

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

thanks

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.

 

[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.