Understanding Shankar's Principles of QM: Changing Basis of Operators

[Dead Boss]

Hi,

I'm reading Shankar's Principles of QM and I find it not very clear on how exactly should I change basis of operator. I know how to change basis of a vector so can I treat the columns of operator matrix as vectors and change them? Or is it something else?

 

[jasonRF]

It is something a little different. Let $$v_1$$ denote a vector represented in basis 1. Then to represent this same vector in terms of a different basis, basis 2, we need to find a matrix $$T_{1:2}$$ that maps any vector representation from basis 1 to basis 2. Thus if we let $$v_2$$ denote that vector represented in basis 2, then
$$v_2 = T_{1:2} \, v_1$$.
This means that
$$v_1 = T_{1:2}^{-1} \, v_2$$
so the matrix that maps a vector representation from basis 2 to basis one is
$$T_{2:1}=T_{1:2}^{-1}$$.

Now, if we have a matrix representation of an operator in basis 1, say $$A_1$$, then it takes a vector represented in basis 1 and maps it to a different vector represented in basis 1. For our example let
$$y_1 = A_1 v_1$$.
So if we want to represent y in basis 2 we have,
$$y_2 = T_{1:2} y_1 = T_{1:2} A_1 v_1 = T_{1:2} A_1 T_{2:1} v_2$$.
Hence, if we want to represent the operator in basis 2, the matrix representation must be
$$A_2 = T_{1:2} A_1 T_{2:1} = T_{1:2} A_1 T^{-1}_{1:2}$$,
and we have
$$y_2 = A_2 v_2$$
as required. If you think about what is happening, it should be easy to remember.

Note that most linear algebra books will cover this.

jason

 

[Dead Boss]

Thank you very much. Maybe I should do some linear algebra book first and then return to Shankar. Can you advise some good books about the subject?

 

[Fredrik]

"Linear algebra done right", by Sheldon Axler.

But you should start with this post about the relationship between linear operators and matrices.

 

[blue_raver22]

I understand your confusion about changing the basis of operators in Shankar's Principles of QM. The process of changing the basis of operators can be a bit complex, but it is an essential concept in quantum mechanics. To answer your question, yes, you can treat the columns of an operator matrix as vectors and change their basis in the same way you would for a vector. However, it is important to note that the operator itself is not a vector, but rather a mathematical representation of a physical quantity. Therefore, the process of changing its basis involves more than just changing the columns of its matrix. It also involves transforming the mathematical representation of the operator to reflect its new basis. This can be done using techniques such as unitary transformations or diagonalization. I recommend reviewing these concepts in Shankar's book or seeking additional resources to fully understand the process of changing basis for operators in quantum mechanics.