# Prove Hermitian with two different wave functions

• pfollansbee
In summary, the condition that Â is Hermitian leads to the equation ∫ψ_m (r)^* Âψ_n (r)dr = ∫Â^* ψ_m (r)^* ψ_n (r)dr. To solve this, we need to expand out the expressions <m|\hat A|n> and <n|\hat A|m>^* and show that they are equal. One approach is to set \phi _m = (\psi -c_n \phi _n)/c_m and then use this to simplify the equations. However, if you are still struggling, it may be helpful to ask your professor for further clarification and guidance.
pfollansbee

## Homework Statement

Let $$ψ(r)= c_n ϕ_n (r) + c_m ϕ_m (r)$$ where $$ϕ_n(r)$$ and $$ϕ_m (r)$$ are independent functions.
Show that the condition that Â is Hermitian leads to
$$∫ψ_m (r)^* Âψ_n (r)dr = ∫Â^* ψ_m (r)^* ψ_n (r)dr$$

## Homework Equations

$$∫ψ(r)^* Â ψ(r)dr = ∫Â^* ψ(r)^* ψ(r)dr$$

## The Attempt at a Solution

It is obvious to me that if
$$<m|\hat A|n> = <\hat A m|n>$$
then
$$<m|\hat A|n> = <n|\hat A|m>^*$$

My professor gave me a hint and said that I need to expand these out and show that they are equal. This is where my problem lies. I have no idea how to expand these out. I have tried a few ways, like setting
$$\phi _m = (\psi -c_n \phi _n)/c_m$$
This certainly did not seem like the correct approach to me.

Maybe someone here can give me another hint as to how this goes. I have asked my professor three times to talk to me about it, but he seems content in misunderstanding me and talking about other problems that we have already solved.

Hmm no replies... oh well. Here is the solution that I came up with. Just in case anyone else happens to happen upon a similar problem, this may help.

#### Attachments

• solution.png
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## What does it mean for a matrix to be Hermitian?

A matrix is Hermitian if it is equal to its own conjugate transpose. In other words, the matrix is symmetric about its main diagonal and the elements in the upper triangle are the complex conjugates of the elements in the lower triangle.

## How do you prove that a matrix is Hermitian?

To prove that a matrix is Hermitian, you need to show that it is equal to its own conjugate transpose. This can be done by taking the transpose of the matrix and then taking the complex conjugate of each element. If the resulting matrix is equal to the original matrix, then it is Hermitian.

## Can a matrix be Hermitian with only one wave function?

No, a matrix can only be Hermitian if it is equal to its own conjugate transpose. This means that there must be at least two wave functions involved in order for a matrix to be Hermitian.

## Why is it important for a matrix to be Hermitian?

Hermitian matrices have many important properties and are widely used in quantum mechanics and other areas of physics. They have real eigenvalues and orthogonal eigenvectors, making them useful for solving equations and representing physical systems.

## Can two different wave functions result in the same Hermitian matrix?

Yes, it is possible for two different wave functions to result in the same Hermitian matrix. This is because the Hermitian matrix is determined by the complex conjugate of the wave function, so different wave functions with the same complex conjugate will result in the same Hermitian matrix.

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