Hamiltonian and spin states

[kylie14]

Sorry if this question is very general/vague, but I would prefer a general answer rather than a specific solution... I'll put more detail in if necessary though.

So, say we have a Hamiltonian for a system (of fermions, spin 1/2); then we find its eigenvalues and hence eigenstates. These are then energy eigenstates, yes? What I really need is the spin states; how do I get them?

The only infomation I have, other than the Hamiltonian, is that the spin 1/2 particles are described by the Dirac equation (2D).

I'm thinking pauli spin matrices might be useful here?

Obviously, you get (0,1) and (1,0) {column not row vectors there obviously) for spin up and spin down; but I think I need some kind or linear combination of these?

Sorry if it's not clear, I'm a bit out of my depth here!

 

[kuruman]

Here is a general answer.

If your Hamiltonian does not have spin operators in it, then any two orthonormal linear combinations of spin-up and spin-down states will do. That's because the energy degeneracy is 2 as far as spin is concerned.

If your Hamiltonian does have spin operators in it, then the procedure for finding the energy eigenstates will give the appropriate two linear combinations of spin-up and spin-down states.

 

[kylie14]

Thanks for your reply.
The hamiltonian can be written as a linear combination of spin operators if that's what you mean?
The problem I have is that I don't remember considering spin when I formed the hamiltonian (although I have confirmation that the hamiltonian is correct).

 

[kuruman]

It's not a matter of "can be written", it's a matter of "is it written" so that spin operators appear explicitly in the Hamiltonian. In other words, does your Hamiltonian contain terms that would lift the spin degeneracy or does it not?
If it does, then the procedure for finding the eigenstates (usually a diagonalization) will provide the correct spin eigenstates.
If it does not, then any linear combination of spin states will do, or you can ignore spin. If you don't remember whether it does or does not, I cannot help you.

 

[paranormal]

Dear scientist,

Thank you for your question. The Hamiltonian is a fundamental concept in quantum mechanics that describes the total energy of a system. In your case, the Hamiltonian is for a system of fermions with spin 1/2. The eigenvalues and eigenstates of the Hamiltonian represent the possible energy levels and corresponding states of the system.

To obtain the spin states, we need to consider the spin operators. In this case, the Pauli spin matrices are indeed useful. These operators act on the spin states and give the spin values of the particles. As you mentioned, the spin up and spin down states can be represented as (0,1) and (1,0) respectively. However, for a system of multiple particles, we need to consider the spin states of each particle separately.

For example, if we have two particles with spin 1/2, the total spin states can be represented as a linear combination of the individual spin states. This can be written as (spin of particle 1) x (spin of particle 2). For instance, if particle 1 has spin up and particle 2 has spin down, the total spin state would be (1/2) x (-1/2) = -1/4. This is known as the singlet state. Similarly, there are other possible combinations such as (1/2) x (1/2) = 1 (triplet state) and (1/2) x (-1/2) + (-1/2) x (1/2) = 0 (mixed state).

In summary, to obtain the spin states of a system described by the Hamiltonian, we need to consider the spin operators and take into account the spin states of each individual particle. I hope this helps clarify your question. Please let me know if you need any further information.

Best regards,