# Hermitian Operators: Identifying & Solving Examples

• Axiom17
In summary, the conversation is about determining whether certain operators are Hermitian or not. The definition of a Hermitian operator is given as A_{ab} \equiv A_{ba}^{*}, and the method to determine Hermiticity is to check if \langle \psi | A | \psi \rangle= \langle \psi |A| \psi \rangle^*. The example operator \hat{A} \psi(x) \equiv exp(ix) \psi(x) is shown to not be Hermitian through this method.
Axiom17

## Homework Statement

I have some operators, and need to figure out which ones are Hermitian (or not).

For example:

1. $$\hat{A} \psi(x) \equiv exp(ix) \psi(x)$$

## Homework Equations

I have defined the Hermitian Operator:

$$A_{ab} \equiv A_{ba}^{*}$$

## The Attempt at a Solution

I just don't know where to start with this

Last edited:
One good way is to see if something is hermitian is if

$$\langle \psi | A | \psi \rangle= \langle \psi |A| \psi \rangle^*$$

if A is hermitian then the equality will hold.

Ok but I don't understand the calculations to do

The way you expressed the opeator A_{ab}=A*_{ba} is a matrix notation, useful when acting on a set of vectors like v_{b}. But in your problem, how does the operator act on the wavefuntion Psi(x)? Can you re-express your definition above for Hermiticity in this specific case?

Umm I have this:

$$\langle a|\hat{A}|b \rangle = \int dV \psi_{a}^{*}(r)r^{2}\psi_{b}(r)=[\langle b|\hat{A}|a \rangle]^{*}$$

But I don't know if that's any use

Still don't get this

So, for example

$$\hat{A} \psi(x) \equiv exp(ix) \psi(x)$$

$$\langle \psi | A | \psi \rangle= \int \psi^*(x) exp(ix) \psi(x)$$

$$\langle \psi | A | \psi \rangle^*= \int \psi^*(x) exp(ix)^* \psi(x)=\int \psi^*(x) exp(-ix) \psi(x) \neq \int \psi^*(x) exp(ix) \psi(x)$$

The operator is not hermitian.

## 1. What is a Hermitian operator?

A Hermitian operator is a mathematical operator that has a set of properties, including being self-adjoint and having real eigenvalues. It is commonly used in quantum mechanics to represent physical observables, such as position and momentum.

## 2. How do you identify if an operator is Hermitian?

An operator is Hermitian if it is self-adjoint, meaning it is equal to its own adjoint, and if its eigenvalues are all real numbers. This can be determined by calculating the adjoint of the operator and checking if it is equal to the original operator.

## 3. What is the significance of Hermitian operators in quantum mechanics?

Hermitian operators are used to represent physical observables in quantum mechanics, such as position, momentum, and energy. They have real eigenvalues which correspond to the measurable values of these observables.

## 4. Can you give an example of a Hermitian operator?

One example of a Hermitian operator is the momentum operator, which is represented by the mathematical expression p = -iħ(d/dx). It is self-adjoint and has real eigenvalues, making it a valid Hermitian operator.

## 5. How do you solve examples involving Hermitian operators?

To solve examples involving Hermitian operators, you first need to identify the operator and its properties, such as being self-adjoint and having real eigenvalues. Then, you can use mathematical techniques, such as eigenvalue equations and eigenvectors, to find the eigenvalues and corresponding eigenvectors of the operator. These values can then be used to solve for the observable being represented by the operator.

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