# Finding Matrix Elements of Operators in Quantum Mechanics

• aaaa202
In summary, "matrix element" is a term used in quantum theory to refer to numbers that represent an operator in a specific basis. It can also refer to the integral of a function that resembles a matrix multiplication. The expectation value of an operator in a specific state can also be considered a matrix element in that basis, but is more commonly referred to as an average or expectation value.
aaaa202
If you have an operator a represented in some basis l1>, l2> you find its matrix elements by doing

Aij = <ilAlj>

But more oftenly you are interested in the expectation value of A. So you take:

<ψlAlψ>. My teacher tends to call these numbers matrix elements too. But which matrix element in A is this and how do I realize that? :)

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The term "matrix element" is used by people in quantum theory to name more different, but similar things. The basic meaning is that the matrix element is one number from a set of numbers that serve as numeric representation of some operator ##\hat{A}## in some basis. In your example,

$$A_{ij} = \langle i | \hat{A} | j\rangle,$$

is indeed a matrix element of a finite or countably infinite matrix A, in the basis consisting of vectors |i>. This matrix is numerical representation of the operator ##\hat{A}##.

But the term is used also for the numbers that do not belong to these two categories, if the collection of them at least somehow reminds of a matrix. For example, in some papers/books, when you encounter the integral of the form

$$I(q) = \int K(q,q') f(q') dq',$$

the authors sometimes refer to the function ##K(q,q')## as to a ##matrix## (continuously indexed), because it reminds of the matrix multiplication, as in

$$I_i = \sum_j K_{ij} f_j.$$

In calculating the matrix elements, some basis always has to be chosen, and it so happens that the resulting matrix elements are generally dependent on this basis.

Special elements are the diagonal elements:

$$A_{ii} = \langle i | \hat{A} |i\rangle.$$

The expression

$$\langle \psi |\hat{A}|\psi\rangle$$

looks similar, but it is not usually called matrix element. It is usually called average or expectation value of the quantity A in "state" ##\psi##. The reason is that it is not frequent to use symbol ##\psi## for a member of a basis; the letter ##\Phi_i## or ket ##|i>## is much better for this purpose.

Nevertheless, ##\langle \psi |\hat{A}|\psi\rangle## can be called matrix element of ##\hat{A}## if really needed, provided ##\psi## is an element of some basis. It is then one of the diagonal matrix elements in that basis.

This isn't in response to Jano, just adding in,
Jano L. said:
$$I(q) = \int K(q,q') f(q') dq',$$

the authors sometimes refer to the function ##K(q,q')## as to a ##matrix## (continuously indexed), because it reminds of the matrix multiplication, as in

$$I_i = \sum_j K_{ij} f_j.$$

$K(q,q')$ is also sometimes called (maybe even more often) the ##kernel##. Building the matrix elements of an operator is as easy as multiplying by 1, twice
$$\hat{A} = \sum_{m}\sum_{n}|n\rangle \langle n|\hat{A}|m\rangle \langle m|$$
since $1=\sum_m |m\rangle \langle m|$. This is the matrix representation of the operator.

## 1. What is a "matrix element" in quantum mechanics?

A matrix element in quantum mechanics refers to the numerical value associated with the transition between two quantum states. It is calculated by taking the inner product of the two states, which involves multiplying the wavefunctions and integrating over all space.

## 2. How do you find the matrix elements of operators in quantum mechanics?

To find the matrix elements of operators in quantum mechanics, you first need to choose a basis set of wavefunctions to represent the quantum states. Then, you apply the operator to these wavefunctions and calculate the inner product to find the matrix elements. This process can be repeated for all possible combinations of states.

## 3. What is the significance of finding matrix elements in quantum mechanics?

Finding matrix elements is crucial in understanding the behavior of quantum systems and predicting the outcomes of quantum measurements. It allows us to calculate probabilities of transitions between states and make predictions about the behavior of quantum particles.

## 4. What is the difference between diagonal and off-diagonal matrix elements?

Diagonal matrix elements refer to the values on the main diagonal of a matrix, while off-diagonal elements refer to the values outside of the main diagonal. In quantum mechanics, diagonal elements represent the energy of a particular state, while off-diagonal elements represent the probability amplitude for a transition between states.

## 5. How do matrix elements change when the basis set is changed?

When the basis set is changed, the matrix elements will also change. This is because the new basis set will have a different set of wavefunctions, resulting in different inner products and therefore different matrix elements. However, the overall behavior and predictions of the quantum system will remain the same as long as the basis set is complete and orthogonal.

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