What Condition Determines Eigenket of A?

[zhaiyujia]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

From the question we know that A| \alpha > =a|\alpha> , A|\beta> =b|\beta>. And A is a hermitian operator:
<\alpha|A\beta>=<\alpha|b\beta>=b<\alpha|\beta>,
<\alpha|A\beta>=<A\alpha|\beta>=<a\alpha|\beta>=a<\alpha|\beta>,
Therefore a=b? or <\alpha|\beta>=0?
But it's nothing to do with |\alpha>+|\beta>+
It seems no addition is need to constrain on them

 

[Hurkyl]

It seems no addition is need to constrain on them

Of course not; you never applied the condition that |\alpha\rangle + |\beta\rangle is to be an eigenket!

 

[zhaiyujia]

But what is the condition? if my first part is right:
A(|\alpha>+|\beta>)=a(|\alpha>+|\beta>)
is automatic right?

 

[Hurkyl]

Well, no, it's not automatic. Your first part said

"a = b" or "\langle \alpha | \beta \rangle = 0".​

So, you have to consider both cases, not just the "a = b" case.