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## Homework Statement

Two identical spin-1/2 particles of mass

*m*moving in one dimension have the Hamiltonian $$H=\frac{p_1^2}{2m} + \frac{p_2^2}{2m} + \frac{\lambda}{m}\delta(\mathbf r_1-\mathbf r_2)\mathbf s_1\cdot\mathbf s_2,$$ where (

*p*

_{i},

**r**

_{i},

**s**

_{i}) are the momentum, position, and spin operators for the

*i*th particle.

(a) What operators, besides the Hamiltonian, are constants of motion and provide good quantum numbers for the stationary states?

(b) What are the symmetry requirements for the spin and spatial wavefunctions?

(c) If

*λ*> 0, find the energy and quantum numbers for the bound state.

## Homework Equations

An operator

*A*is a constant of motion if [

*H*,

*A*] = 0.

##(\mathbf s_1 + \mathbf s_2)^2 = s_1^2 + s_2^2 + 2\mathbf s_1\cdot\mathbf s_2##.

Total mass ##M = m_1+m_2##. Reduced mass ##\mu = \dfrac{m_1m_2}{M}##.

Center of mass ##R = \dfrac{m_1r_1 + m_2r_2}{M}##. Separation ##r=r_1-r_2##.

## The Attempt at a Solution

First off, if it's a 1-D problem, I have no idea why the position operators are vectors. But I guess I can get over that.

(a) Operators which commute with

*H*. I think

*p*

_{1}should be safe, since

*p*

_{1}commutes with itself, with

*p*

_{2}, and with that whole interaction term. Likewise

*p*

_{2}. But I don't know about the positions and spins. Do

**s**

_{1}and

**s**

_{1}·

**s**

_{2}commute? What about

*r*_{1}and

*δ*(

*r*

_{1}–

*r*

_{2})?

My inclination is to rewrite the Hamiltonian in terms of better variables. I can introduce better spatial variables for the problem by considering the center of mass

*R*and the separation

*r*, as defined above. And I can introduce the total spin

**S**, also defined above, to decompose the dot-product.

$$H=\frac{p_R^2}{2M} + \frac{p_r^2}{2\mu} + \frac{\lambda}{m}\delta(r)\frac12(S^2-s_1^2-s_2^2).$$

Looking at it this way, it seems like

*p*

_{R},

*p*

_{r},

*s*

_{1},

*s*

_{2},

*S*, and

*R*all commute with

*H*. Still not sure about

*r*

_{1},

*r*

_{2}, and

*r*.

(b) I think I'm good on this one. Spin-1/2 particles means fermions. Thus the states must be overall anti-symmetric. So if the spatial wavefunction is symmetric, the spin wavefunction must be antisymmetric (the "para" state); and vice versa (the "ortho" state).

(c) ?