# Eigenfunctions and dirac notation for a quantum mechanical system.

1. Dec 2, 2013

### Paintjunkie

QUESTION
A quantum mechanical system has a complete orthonormal set of energy eigenfunctions,
|n> with associate eigenvalues, En. The operator $\widehat{A}$ corresponds to an observable such that
Aˆ|1> = |2>
Aˆ|2> = |1>
Aˆ|n> = |0>, n ≥ 3
where |0> is the null ket. Find a complete orthonormal set of eigenfunctions for
$\widehat{A}$. The observable is measured and found to have the value +1. The system is unperturbed and then after a time t is remeasured . Calculate the probability
that +1 is measured again.

I would really appreciate any guidance on this. I cannot find this in my book at all. I know my teacher has spoken about it bra-ket notation and such. but it never really made sense. where can I find examples of problems like this but not this.

2. Dec 2, 2013

### Dick

You could start out by telling me what ${\widehat A}(|1>+|2>)$ and ${\widehat A}(|1>-|2>)$ are.

3. Dec 3, 2013

### Paintjunkie

Aˆ(|1>+|2>) = |2> + |1> ?
Aˆ(|1>−|2>) = |2> - |1> ? I really don't know

4. Dec 3, 2013

### Paintjunkie

I found this I feel like it should help me but I don't know...

For every observable A, there is an operator $\hat{A}$ which acts upon the
wavefunction so that, if a system is in a state described by |ψ>, the
expectation value of A is
<A>= <ψ|$\hat{A}$|ψ>= ∫ dx ψ*(x) $\hat{A}$ ψ(x)

that integral is from -∞ to ∞

5. Dec 3, 2013

### Paintjunkie

ok so I am reading a little more and.

Aˆ(|1>+|2>) = Aˆ|1>+Aˆ|2> ==>α=1, β=1
Aˆ(|1>−|2>) = Aˆ|1>-Aˆ|2> ==>α=1,β=-1

<1|2> = <-2|1>

idk this does not really make sense to me

6. Dec 3, 2013

### Dick

Yes, A^(|1>+|2>)=|1>+|2> and A^(|1>-|2>)=|2>-|1>=(-1)*(|1>-|2>). What does that tell about eigenvalues and eigenvectors of A^?

7. Dec 3, 2013

### Paintjunkie

that they are linear and Hermitian

or I guess that's for A^.....

8. Dec 3, 2013

### Dick

Be more specific. Looking at those two equations I see two eigenvectors of A^ and their corresponding eigenvalues.

9. Dec 3, 2013

### Paintjunkie

maybe that En is 1 and 2 ?

10. Dec 3, 2013

### Dick

Noo. x is an eigenvector of A^ if A^(x)=cx for some constant c. What does A^(|1>+|2>)=|1>+|2> tell you?

11. Dec 3, 2013

### Paintjunkie

is the probability that |1> will give 2 at t=0 |En|2 and that equals 1 ?

12. Dec 3, 2013

### Paintjunkie

I guess that means that α and β are 1?

13. Dec 3, 2013

### Dick

That's still not saying anything about eigenvalues or eigenvectors. Look, A^(|1>+|2>)=|1>+|2> tells me that |1>+|2> is an eigenvector of A^. Why do I say that and what's the corresponding eigenvalue?

14. Dec 3, 2013

### Paintjunkie

I don't know I give up. thanks for trying man.

15. Dec 3, 2013

### Dick

Yeah, something isn't clicking here. But in case you decide to take another crack at it, I'm trying to get you to see that |1>+|2> is an eigenvector of A^ with an eigenvalue of +1 and |1>-|2> is an eigenvector of A^ with an eigenvalue of -1. Review the definition of eigenvector, there's nothing very hard about this part. You are just getting confused by other stuff.

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