Center of mass system of two interacting identical Fermions

[Dustgil]

Homework Statement

Consider the center of mass system of two interacting fermions with spin 1/2.

a) What is the consequence of the Pauli exclusion principle on the two-particle wave function?

b)Let S1 and S2 be the spin operators of the two individual fermions. Show that the operators

$$P_{+/-}=(\frac{1}{2}) +/- (\frac{1}{4}) +/- S_1 \cdot S_2/\hbar^2$$

are the projection operators of the triplet states and the singlet states of the spin wave functions, respectively.

c) Using the Pauli exclusion principle and the symmetry properties of the spin and relative orbital angular momentum L, find the allowed values of L for any bound triplet state of the two-particle system.

d) Again using the Pauli exclusion principle and the symmetry properties of the space coordinate, show that the particles in a triplet state can never scatter through an angle of 90 degrees in their center of mass system.

Homework Equations

The Attempt at a Solution

I understand part a. Since the particles are fermions, the total state must be antisymmetric under the exchange operator. Since the singlet configuration is antisymmetric, the corresponding space wavefunction must be symmetric, and vice versa for the triplet states since they are symmetric.

part b)

The first thing I needed was the dot product of S1 and S2.

$$S^2=(S_1 + S_2) \dot (S_1 + S_2) = S_{1}^{2}+S_{2}^{2}+2S_{1}\cdot S_{2}$$
$$S_{1}\cdot S_{2}=\frac{1}{2}(S^{2}-S_{1}^{2}-S_{2}^{2})$$

I also know the P needs to be applied to the states 1,-1 | 1,0 | 0,0 | 1,1.

I can just plug the found dot product into P and apply P to each state to see what I get, and this is where I'm stuck...I believe I can break P up and apply each of the terms to the state individually, but I don't know what this is, for example:

$$\frac{1}{2}|1,1>$$

I feel like I'm not understanding something important here.

 

[Orodruin]

What you need to show is that the $S^2$ eigenvalue of $P_\pm |\psi\rangle$ is $s(s+1)$ with $s = 1$ and $0$, respectively (i.e., eigenvalues 2 and 0), for any state $|\psi\rangle$, that $P_\pm^2 = P_\pm$, and that $P_+ + P_- = 1$. (The latter should be trivial)

Also, note that the LaTeX command \pm gives $\pm$.

 

[Dustgil]

Could you elaborate further on what you mean in the first sentence? I can apply the found dot product into P to get

$$P_{\pm}=((\frac{1}{2}) \pm (\frac{1}{4}) \pm (\frac{(S^{2}-S_{1}^{2}-S_{2}^{2})}{\hbar^{2}}))|\psi>$$

but this gives me the eigenvalues of P. To find the eigenvalues of S^2, do I only look at the term containing S^2 within P?

Thanks for the heads up about \pm.

 

[Dustgil]

Also, a question on why we need to prove those facts. I understand $$P_{\pm}^{2}=P_{\pm}$$ as that's the definition of the projection operator. I'm unsure how the first and third requirements tie into things though.

 

[Orodruin]

The first requirement is that the operators project onto the correct eigenstates. The last requirement is that the spaces they project on together span the entire space. For example, you could imagine a projection operator on a single $s = 1$ state.

Could you elaborate further on what you mean in the first sentence? I can apply the found dot product into P to get

$$P_{\pm}=((\frac{1}{2}) \pm (\frac{1}{4}) \pm (\frac{(S^{2}-S_{1}^{2}-S_{2}^{2})}{\hbar^{2}}))|\psi>$$

but this gives me the eigenvalues of P. To find the eigenvalues of S^2, do I only look at the term containing S^2 within P?

Thanks for the heads up about \pm.

The equation you wrote here is not complete, you seem to be missing some state somewhere. If $P_+$ is a projector onto the states with $s = 1$, then $P_+|\psi\rangle$ must be a state with $s = 1$. You can check this by applying $S^2$ to the state. If it is indeed an eigenstate of $S^2$ with $s = 1$, then you should find that
$$
S^2 P_+ |\psi\rangle = s(s+1) P_+ |\psi\rangle = 2 P_+ |\psi\rangle.
$$

Also, next LaTeX lesson: > is a greater than sign $>$ and is typeset as a relation, adding additional space on the sides. \rangle is a delimeter that can be used for writing kets $\rangle$.

 

[Dustgil]

Well, I can only think of two ways to do the dot product, and they must both be wrong. The way I thought was correct is

$$S^{2} = (S_{1}+S_{2})\cdot (S_{1}+S_{2})=S_{1}^{2}+S_{2}^{2}+2S_{1}\cdot S_{2}$$
$$2S_{1}\cdot S_{2}=(S^{2}-S_{1}^{2}-S_{2}^{2})\frac{1}{2}$$

The other way would be $$S_{1}\cdot S_{2}=S_{1x}S_{2x}+S_{1y}S_{2y}+S_{1z}S_{2z}$$

But this just returns S^2 when I plug the spin matrices in...