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## Homework Statement

I am having too many troubles finding the eigenfunctions of a given Hamiltonian. I just never seem to know what exactly to do. My idea here is not for you to help me solve each problem below, but I would like to just set the equations. I know you guys don't like it when somebody posts 325235 different questions in the same topic, but please understand that I really don't want to open a new thread for each of the basic problems below:

a) Particle with spin ##S=1## is in magnetic field ##\vec B=(0,0,B)##. At time ##t=0## the particle is in ground state. Find eigenfunctions and energies if ##H=-\lambda \vec S \vec B## where ##\lambda >0##.

b) Two particles with spin 1/2 interact. This describes Hamiltonian ##H=\lambda(S_{1x}S_{2y}-S_{1y}S_{2x})##. Find eigenfunctions and energies for ##\lambda >0##.

c) Two particles with spin 1/2 have Heisenberg interaction ##H=J\vec S_1 \vec S_2## where ##J>0##. Find eigenfunctions and energies.

d) Or the same as c) only with ##S_1=1## and ##S_2=1/2##.

## Homework Equations

## The Attempt at a Solution

For most of them I really don't know what to do. I know that the idea is to solve Schrodinger equation. But...

a) Ok, because ##S=1## the basis is ##\left \{ |1,1>,|1,0>,|1,-1> \right \}##. Right?

What is the ground state of a particle with spin 1?

So I would start like this: ##-\lambda B_zS_z(\alpha |1,1>+ \beta|1,0>+\gamma |1,-1>)=E(\alpha |1,1>+ \beta|1,0>+\gamma |1,-1>)##. but How do I determine the values of ##\alpha, \beta, \gamma ##?

b) I don't know what the basis is? Is it ##\left \{ |\uparrow>,|\downarrow> \right \}## or is it ##\left \{ |\uparrow>|\uparrow>,|\uparrow>|\downarrow>,|\downarrow>|\downarrow>,|\downarrow>|\uparrow> \right \}##. And still, how would I determine the values of the coefficients before the basis vectors?

... As you can see, my questions are in all the cases more or less the same, therefore I published them in the same topic. :/ I hope you don't mind.