# Spectrum of two particles system hamiltonian

Consider the following hamiltonian describing a system of two identical spin 1/2 particles in one dimension: $$H = H_1 +H_2 - \lambda \vec {s_1} . \vec {s_2}$$ Where $$H_i$$ is the Hamiltonian of an harmonic oscillator.

I want to determine the spectrum of the first 3 energy levels. I don't know if it is correct to define: $$\vec{s} = \vec{s_1} + \vec{s_2}$$ Such that: $$\vec{s_1}.\vec {s_2} =\frac{1}{2} (s^2 - s_1^2 - s_2^2)$$

And therefore: $$H_{spin} = - \lambda \hbar^2 (s (s+1)-\frac{1}{2})$$

What are the implications of the fact that the two particles are identical?

• Welcome to physics stack exchange! This looks like a homework problem. Homework is not in general dealt with. Nevertheless, two hints. There is an error in your last expression and for two identical fermions you need an antisymmetric two particle state. – my2cts Jan 7 at 12:34
• Thank you! I forgot a 1/2 that multiplies the last expression. Sorry for the homework style question, I thought an example could use. – Cecilia Bignotti Jan 7 at 13:27