# Diagonal Matrix & Perturbation Theory in Quantum Mechanics

• M. next
In summary, a diagonal matrix in Quantum Mechanics refers to an operator where the basis vectors are all eigenvectors (eigenstates) and the matrix representation has non-zero entries only on the main diagonal. This allows for specific values of properties associated with the state to be determined. In the context of Perturbation theory, understanding how a matrix is diagonal in a specific basis can aid in understanding the degenerate case.
M. next
What does it mean for a matrix to be diagonal, especially in Quantum Mechanics, where we get to Perturbation theory (Degeneracy).
I don't get it. Please if you can explain in 'simple' language.

For finite dimensional vector spaces, a "diagonal matrix" is something like
$$\begin{bmatrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{bmatrix}$$
having non-zero entries only on the main diagona. But I suspect you already knew that!

More generally, a square matrix represents a linear operator on some vector space. If that that vector space has finite dimension, n, then we can represent the operator as a n by n matrix. If the vector space is infinite dimensional is, as is typically the case in Quantum theory, we can't really write it as a "matrix" but the same ideas work.

The matrix is "diagonal" in a particular basis, $\{v_1, v_2, ..., v_n\}$ then the basis vectors are eigenvectors: $Av_i= a_iv_i$ where $a_i$ is the number, on the diagonal, at the ith row and column. To say that an operator in Quantum Mechanics is "diagonal" also means that the basis vectors are all eigenvectors (eigenstates). Physically, "eigenstates" are those that give specific values to whatever the property is associated to the state while vectors that are not eigenstates can be written as linear combinations of the eigenstates and then give "mixtures" of those values.

Your question is pretty vague, so it's going to be hard to offer any concrete help. Here is what a diagonal matrix is in general: http://en.wikipedia.org/wiki/Diagonal_matrix

In QM, we say an operator ##\hat{A}## is diagonal in some basis of states ##| \psi_i\rangle## (where i is some index labeling the states) if ##\langle \psi_i | \hat{A} | \psi_j \rangle## is only nonzero when ##i = j##.

Thank you, but I suppose it was really vague. I am having a hard time understanding Perturbation theory in its degenerate case.
Anything on that matter could help especially the usage of Matrix in that section..

## 1. What is a diagonal matrix?

A diagonal matrix is a special type of square matrix where all of the entries outside of the main diagonal (the diagonal from the top left to the bottom right) are zero. This means that the only non-zero entries are along the main diagonal.

## 2. How is a diagonal matrix used in quantum mechanics?

In quantum mechanics, diagonal matrices are used to represent quantum states and operators. The diagonal elements of a matrix correspond to the eigenvalues of the operator, and the non-diagonal elements represent the probability amplitudes for transitions between different eigenstates.

## 3. What is perturbation theory in quantum mechanics?

Perturbation theory is a mathematical technique used to approximate the solutions of a quantum mechanical problem by adding a small perturbation, or disturbance, to a known solution. This is often used when the exact solution is difficult or impossible to calculate.

## 4. How is perturbation theory applied to diagonal matrices?

In quantum mechanics, perturbation theory is used to approximate the eigenvalues and eigenvectors of a diagonal matrix by adding a small perturbation to the diagonal elements. This allows us to calculate the effects of small changes to the system and make predictions about its behavior.

## 5. What are the limitations of perturbation theory in quantum mechanics?

While perturbation theory is a useful tool in quantum mechanics, it is not always accurate. It is only valid for small perturbations and breaks down for larger perturbations. Additionally, it may not be applicable to systems with degenerate eigenvalues or when the perturbation affects multiple eigenstates. In these cases, other techniques such as variational methods may be necessary.

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