Help with Eigenfunction Questions

[adi_butler]

I am a little stuck understanding and answering the following questions. Can anyone help me with them?



"A system has four eigenstates of an observable, with corresponding eigenvalues 3/2, 1/2, -1/2 and -3/2, and normalized eigenfunctions
Psi_{3/2}, Psi_{1/2}, Psi_{-1/2} and Psi_{-3/2} respectively. (cant get tex to work properly)

Measurements of the observable are made on three systems that are all in the same superposition state, and yield the values 3/2, -1/2 and -3/2"

1)What can you say about the state of the system after each measurement?

2)What can you say about the original superposition state?

3)If many measurements on systems that are all in the same superposition state never yield the result -1/2, and give the result 3/2 twice as often as the other two possible results, deduce the normalized form of the superposition state

4)What is the expectation value of the observable in this state?

 

[jcsd]

Show what you've done so far and where your stuck

 

[adi_butler]

Well I am new to quantum and i missed the lecture explaining what eigenvalues and eigenstates are, I've heard of them before but have no idea what they are

 

[slyboy]

Questions

Well, you should probably have a read of your textbook because eigenvalues and eigenvectors are pretty important for understanding QM. It's probably not appropriate to answer your homework for you, but here is a summary of what you need to know:

- Observables in QM are represented by hermitian operators, H.

- The eigenstates of H are the solutions to the following equation:

H Psi = h Psi

where Psi is a vector, called an eigenvector of H, and h is just a number, called an eigenvalue of H.

- Let's label the distinct solutions to this equation by an index j, i.e. as Psi_j and h_j

- QM says that when H is measured, the answer is always one of the eigenvalues h_j. After the measurement, the state will become the corresponding eigenstate Psi_j.

- Further, is the initial state is Phi, then the probability of obtaining the value h_j is: |<Psi_j|Phi>|^2

- The Psi_j's usually form a complete orthonormal basis (although one has to be a bit careful when H is degenerate, i.e. when there is more than one Psi_j corresponding to a particular value of h_j). Therfore, the initial state can be written in terms of the eigenvectors:

Phi = \sum_j a_j Psi_j

and then

|<Psi_j|Phi>|^2 = |a_j|^2


That should be enough to answer your questions

 

[adi_butler]

Thank you slyboy, it wasnt my homework its a question from a past paper that i didnt have a clue about. I didnt realize how easy the question was, doesn't actually require a lot of work at all! Thanks for the help