Eigenfunctions and dirac notation for a quantum mechanical system.

[Paintjunkie]

QUESTION
A quantum mechanical system has a complete orthonormal set of energy eigenfunctions,
|n> with associate eigenvalues, E_n. 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.

 

[Dick]

QUESTION
A quantum mechanical system has a complete orthonormal set of energy eigenfunctions,
|n> with associate eigenvalues, E_n. 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.

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

 

[Paintjunkie]

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

 

[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 ∞

 

[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

 

[Dick]

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

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^?

 

[Paintjunkie]

that they are linear and Hermitian

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

 

[Dick]

that they are linear and Hermitian

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

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

 

[Paintjunkie]

maybe that E_n is 1 and 2 ?

 

[Dick]

maybe that E_n is 1 and 2 ?

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

 

[Paintjunkie]

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

 

[Paintjunkie]

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

I guess that means that α and β are 1?

 

[Dick]

I guess that means that α and β are 1?

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?

 

[Paintjunkie]

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

 

[Dick]

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

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.