Matrix representation of an operator in a new basis

In summary, the author is trying to figure out if they should evaluate the matrix element <m|UnjA|n>. After trying to understand the situation, the author realizes that what they are trying to do is not what is desired.
  • #1
peripatein
880
0

Homework Statement


Let Amn be a matrix representation of some operator A in the basis |φn> and let Unj be a unitary operator that changes the basis |φn> to a new basis |ψj>. 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|UnjA|n>? Is that how this should be approached?
I know that Unjn>=|ψj>.
 
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  • #2
No. The definition of the matrix elements in the new basis is: Aij = <ψi| A |ψj>. This is what you have to evaluate.
 
  • #3
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?
 
  • #4
Not totally, but a bit. What is |ψj> and what is <ψi|? What happens when you insert these expressions into Aij = <ψi| A |ψj>?
 
  • #5
Row and column vectors?
Aij are the elements of matrix A, I think.
 
  • #6
I mean in terms of the original basis vectors. What happens when you insert this into what you want to compute?
 
  • #7
I am not sure I understand what you mean by "inserting that into what I wish to compute". Please clarify.
 
  • #8
Orodruin said:
The definition of the matrix elements in the new basis is: Aij = <ψi| A |ψj>
peripatein said:
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:
peripatein said:
I know that Unj|φn>=|ψj>.
 
  • #9
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?
 

1. What is the purpose of representing an operator in a new basis?

The purpose of representing an operator in a new basis is to simplify calculations and analysis of the operator's behavior. By changing to a different basis, the operator may have a simpler or more intuitive form, making it easier to understand and manipulate.

2. How is the matrix representation of an operator in a new basis calculated?

The matrix representation of an operator in a new basis is calculated by first determining the transformation matrix that maps the original basis to the new basis. This transformation matrix is then used to transform the original operator's matrix representation to the new basis.

3. Can any operator be represented in any basis?

Yes, any operator can be represented in any basis. However, the resulting matrix may be more complex or difficult to work with depending on the choice of basis.

4. What is the significance of the eigenvalues and eigenvectors in the matrix representation of an operator in a new basis?

The eigenvalues and eigenvectors in the matrix representation of an operator in a new basis represent the behavior of the operator in that basis. The eigenvalues determine the possible outcomes of measurements, while the eigenvectors represent the corresponding states.

5. How does changing the basis affect the properties of an operator?

Changing the basis of an operator does not change its fundamental properties, such as its eigenvalues or trace. However, it may change the specific values of these properties and make them easier to calculate or analyze.

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