- #1

- 19

- 0

Was wondering if anyone could point me in the right direction for this one?

Show that the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal?

Thanks

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- Thread starter leila
- Start date

- #1

- 19

- 0

Was wondering if anyone could point me in the right direction for this one?

Show that the eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal?

Thanks

- #2

LeonhardEuler

Gold Member

- 859

- 1

Alright, say you have an hermitian operator, O with eigenfunctions |a_{1}> and |a_{2}>, with eigenvalues of a_{1} and a_{2} respectively. Then:

O|a_{1}>=a_{1}|a_{1}> (1)

<a_{1}|O=a_{1}*<a_{1}| (2)

O|a_{2}>=a_{2}|a_{2}> (3)

and

<a_{2}|O=a_{2}*<a_{2}| (4)

Now right multiply |a_{1}> in equation (4) and left multiply by <a_{2}| in equation (1) to get two expressions for <a_{2}|O|a_{1}>. Subtract the two equations and observe.

-edit: keep in mind that the eigenvalues of hermitian operators are real. You can prove this by letting a_{1}=a_{2}

O|a

<a

O|a

and

<a

Now right multiply |a

-edit: keep in mind that the eigenvalues of hermitian operators are real. You can prove this by letting a

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