# Simultaneously diagonalize two operators

• kashokjayaram
In summary: Thank you.In summary, the fundamental quantum mechanics relies upon finding some operators \hat{X} that commutes with hamiltonian and is able to simultaneously diagonalize \hat{X} and hamiltonian. However, the unitary matrix used in this process is chosen based on the similarity between the operators.
kashokjayaram
Most part of the fundamental quantum mechanics rely upon finding some operators $\hat{X}$ that commutes with hamiltonian and is able to simultaneously diagonalize $\hat{X}$ and hamiltonian.

Actually what do you mean by diagonalize simultaneously??

Is there any relation with diagonalize the corresponding matrix..??

Can anybody explain me the concept using some example?

Thank you.

For matrices, it just means to find a basis of the vector space which are eigenbases for both matrices.

For example, take the matrices

$$A=\left(\begin{array}{cccc} 1 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 & 1 \end{array}\right) ~\text{and}~ A=\left(\begin{array}{cccc} 2 & 1 & 1\\ 1 & 0 & -1\\ 1 & -1 & 2 \end{array}\right)$$

Then if we choose the basis
$$(1,0,1),~(-1,-1,1),~(1,-2,-1)$$
then we see that these vectors are eigenvectors for both matrices. So we have found a common eigenbasis for both matrices.

If we then change our basis to this eigenbasis, then both matrices will appear to be in diagonal form.

kashokjayaram said:
Actually what do you mean by diagonalize simultaneously??

It just means we can find a complete set of simultaneous eigenvectors of both operators. For example if we look at the total angular momentum operator ##J^2## and relative to some coordinate system the ##z## component of angular momentum ##J_z## then we know ##[J^2,J_z] = 0##. We go through the usual raising and lowering operator jazz to get the complete set of simultaneous eigenvectors of these operators that we usually label ##\{|j,m \rangle \}## and the matrices ##\langle j',m' |J^2|j,m \rangle ## and ##\langle j',m' |J_z|j,m \rangle ## will both be diagonal matrices.

micromass said:
If we then change our basis to this eigenbasis, then both matrices will appear to be in diagonal form.

I understood what you've told upto this line. But can you elaborate this line.
How can I represent an operator in this eigenbasis..? And can you tell me the diagonalized form of anyone of it?

Thank you

kashokjayaram said:
I understood what you've told upto this line. But can you elaborate this line.
How can I represent an operator in this eigenbasis..? And can you tell me the diagonalized form of anyone of it?

Thank you

See http://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/ under the heading "unitary transformations".

1 person
micromass said:
See http://www.quantumsciencephilippines.com/216/simultaneous-diagonalization-hermitian-matrices/ under the heading "unitary transformations".

Here, how did you choose the unitary matrix?
Or is it that the matrix you chosen is the eigen vectors written one after the other..?? What is the mathematics behind it??

kashokjayaram said:
Or is it that the matrix you chosen is the eigen vectors written one after the other..?? What is the mathematics behind it??
you are right, you could read abt similarity transformation!

## What does it mean to simultaneously diagonalize two operators?

Simultaneous diagonalization is a concept in linear algebra where two operators are represented by matrices that can be diagonalized simultaneously, meaning they have the same set of eigenvectors.

## Why is simultaneous diagonalization important in science?

Simultaneous diagonalization is important because it allows for easier analysis and manipulation of the operators and their corresponding matrices. It also simplifies calculations and makes it easier to understand the relationship between the two operators.

## Can any two operators be simultaneously diagonalized?

No, not all operators can be simultaneously diagonalized. For two operators to be simultaneously diagonalizable, they must commute, meaning that their order of operations does not affect the outcome.

## What is the significance of the eigenvalues in simultaneous diagonalization?

The eigenvalues of the two operators represent the measurements of observable quantities in quantum mechanics. Simultaneous diagonalization allows for the determination of these eigenvalues and provides a basis for understanding the behavior of the operators.

## How is simultaneous diagonalization used in practical applications?

Simultaneous diagonalization has practical applications in many fields, such as physics, chemistry, and engineering. It is used to simplify calculations and understand the behavior of operators in quantum mechanics, signal processing, and control systems.

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