- #1

MMS

- 148

- 4

## Homework Statement

Hello,

I'm asked to show the equivalence of the given Hamiltonian below which describes the interaction between an electron and a nucleus

and the following Hamiltonian

## The Attempt at a Solution

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I've simply written down each Hamiltonian as a sum of four tensor product and calculated it.

The first (given) Hamiltonian gives

And as for the second one, I've also written it the same way (sum of four tensor products) and received

I've went over my calculations a few times now and I can't seem to find a mistake so it got me thinking if it's more than simply a calculation mistake.

For the given Hamiltonian, I'm pretty sure of what I did. I know how to take the tensor product of two matrices and I've checked my answers online for tensor product of Pauli matrices and it seems to be right.

As for the second one, I've written it as (the summation only with the 1/2 taken out the brackets)

(1 ⊗ σ0 + σ0 ⊗ 1)^2+(1 ⊗ σ1 + σ1 ⊗ 1)^2+(1 ⊗ σ2 + σ2 ⊗ 1)^2+(1 ⊗ σ3 + σ3 ⊗ 1)^2

and for each (1 ⊗ σi + σi ⊗ 1)^2 I've calculated the tensor product, added them up and only then taken the square of it (multiplication of the matrix by itself) and the rest with multiplying by the factor and substituting 3I from it is trivial.

Any ideas on where I could've went wrong? Is what I described above right?

Also can I write (1 ⊗ σi + σi ⊗ 1)^2 somehow in the form of a^2+b^2+2ab?

Thanks in advance.