Question

I have the following Hamiltonian describing two spin 1/2 systems, represented by the pauli matrices σ1

and σ2:
H=Dσ1z+J(σ1⋅σ2)
The two spins are coupled by J
but at the same time the first spin σ1
is also under the influence of D
that influences it's z-component. My questions now are:
Is this Hamiltonian even allowed in a physical way? I.e. can I just divide the spin σ1
like this and claim that on the one hand it is interacting with another spin but on the other hand it is also influenced by something in which I do not take the second spin into account anymore and just add the Hamiltonians up. Somehow I feel that a perturbation is more appropiate.
Now of course I want to find the eigenstates and eigenvalues of this Hamiltonian, for this I couple the spins and try the following 4x4 Hamiltonian with the new 4x4 spin operators:
H=D(σ1z⊗1(2x2))+J(σ1x⊗σ2x+σ1y⊗σ2y+σ1z⊗σ2z)
I then naively diagonalize this Hamiltonian with Matlab and get some eigenvalues. The eigenvalues seem to agree with the expected energies. But my problem now is that I'm not sure if I this is correct based on my first question and I have no idea what the eigenstates are supposed to mean, I feel like I need to do a unitary transformation or a base change to make sense out of the eigenstates that I get. I also can't find operators that now commute with this Hamiltonian, i.e. σ2
does not commute anymore. Where I have used new operator for the total spin in the following form (here only for the z-component):
Sz=σ1z⊗1(2x2)+1(2x2)⊗σ2z.
Is there a way to find good quantum numbers again that are conserved since I'm really confused now how I am supposed to label the eigenstates, i.e. which m_z value and total spin they have.
Options:
A) The eigenvalues and eigenstates from the diagonalization in MATLAB are valid and correctly represent the system without requiring any further transformations.
B) A unitary transformation is needed to interpret the eigenstates properly due to the combined spin operators used in the Hamiltonian.
C) The eigenvalues seem correct, but a non-unitary transformation is required for accurate interpretation of the eigenstates.
D) The eigenvalues and eigenstates obtained from the diagonalization in MATLAB are incorrect, and the Hamiltonian needs reconsideration in terms of proper spin interactions.

Answer 1

The Hamiltonian you provided is allowed in a physical way, and you correctly construct a 4x4 Hamiltonian to find the eigenstates and eigenvalues. No further transformations are needed to interpret the eigenstates.

Step-by-step explanation:

The Hamiltonian you provided is indeed allowed in a physical way. It describes the interaction between two spin 1/2 systems, where one spin is coupled to the other spin and also influenced by an external field.

The addition of the Hamiltonians is valid because the spins are treated independently in this case.

To find the eigenstates and eigenvalues of this Hamiltonian, you correctly construct a 4x4 Hamiltonian using the new 4x4 spin operators. Diagonalizing this Hamiltonian will give you the eigenvalues, which seem to agree with the expected energies.

No further unitary or non-unitary transformations are needed to interpret the eigenstates. The eigenstates obtained from the diagonalization in MATLAB are valid and correctly represent the system without requiring any additional transformations.