(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that |[tex]\alpha[/tex]> and |[tex]\beta[/tex]> are eigenkets(eigenfunctions) of a hermitian operator A. Under what condition can we conclude that |[tex]\alpha[/tex]> + |[tex]\beta[/tex]> is also an eigenket of A?

2. Relevant equations

It's quite basic, I don't think any addtional equations are needed except the definations.

3. The attempt at a solution

From the question we know that A| [tex]\alpha[/tex] > =a|[tex]\alpha[/tex]> , A|[tex]\beta[/tex]> =b|[tex]\beta[/tex]>. And A is a hermitian operator:

<[tex]\alpha[/tex]|A[tex]\beta[/tex]>=<[tex]\alpha[/tex]|b[tex]\beta[/tex]>=b<[tex]\alpha[/tex]|[tex]\beta[/tex]>,

<[tex]\alpha[/tex]|A[tex]\beta[/tex]>=<A[tex]\alpha[/tex]|[tex]\beta[/tex]>=<a[tex]\alpha[/tex]|[tex]\beta[/tex]>=a<[tex]\alpha[/tex]|[tex]\beta[/tex]>,

Therefore a=b? or <[tex]\alpha[/tex]|[tex]\beta[/tex]>=0?

But it's nothing to do with |[tex]\alpha[/tex]>+|[tex]\beta[/tex]>+

It seems no addition is need to constrain on them

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# Homework Help: QM eigenket

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