Matrix representation of an operator in a new basis

[peripatein]

Homework Statement

Let A_mn be a matrix representation of some operator A in the basis |φ_n> and let U_nj be a unitary operator that changes the basis |φ_n> to a new basis |ψ_j>. I am asked to write down the matrix representation of A in the new basis.

Homework Equations

The Attempt at a Solution

Should I try to evaluate <m|U_njA|n>? Is that how this should be approached?
I know that U_nj|φ_n>=|ψ_j>.

 

[Orodruin]

No. The definition of the matrix elements in the new basis is: A_ij = <ψ_i| A |ψ_j>. This is what you have to evaluate.

 

[peripatein]

I realize that, which is precisely why I asked whether I should evaluate the effect Unj has on the basis elements, namely through evaluating <m|UnjA|n>. Or thus I presumed. Am I totally off?

 

[Orodruin]

Not totally, but a bit. What is |ψj> and what is <ψi|? What happens when you insert these expressions into A_ij = <ψi| A |ψj>?

 

[peripatein]

Row and column vectors?
Aij are the elements of matrix A, I think.

 

[Orodruin]

I mean in terms of the original basis vectors. What happens when you insert this into what you want to compute?

 

[peripatein]

I am not sure I understand what you mean by "inserting that into what I wish to compute". Please clarify.

 

[Orodruin]

The definition of the matrix elements in the new basis is: Aij = <ψi| A |ψj>

I realize that, which is precisely why I asked whether I should evaluate the effect Unj has on the basis elements, namely through evaluating <m|UnjA|n>.

If you insert the definition of the |ψj>, this is not what you get. The definition is:

I know that Unj|φn>=|ψj>.

 

[peripatein]

Let's try this. I know that the i-th column is obtained through U|i>=|Ai> and <Ai|=<i|U^†. Is this somewhat closer to what you intended?