Simultaneously diagonalize two 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.

 

[micromass]

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.

 

[WannabeNewton]

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.

 

[kashokjayaram]

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

 

[micromass]

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".

 

[kashokjayaram]

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??

 

[sugeet]

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!