Problem about identical particles

[Lebnm]

Can someone help me with this excercise?: Consider two electrons constrained to move in one dimension. They interact through the potential
$$
V(x) = \begin{cases}
0, & \text{ if } |x| > a \\
-V_{0}, & \text{ if } |x| \leq a
\end{cases}
$$
where $x$ is the relative coordinate. The total spin is $S = 1$, and the objective is to determine the state of the system.
I know I can write the state as $|\Psi \rangle = |P \rangle \otimes |\psi \rangle \otimes |SM \rangle$, where $|P \rangle$ is an eigenstate of the momentum of CM, and the wave function $\psi (x) = \langle x | \psi \rangle$ is the solution of $$-\frac{\hbar^{2}}{2\mu}\frac{\mathrm{d^{2}}\psi }{\mathrm{d} x^{2}} + V(x)\psi(x) = E \psi(x).$$ Now, I know that the states $|P \rangle$ and $|SM \rangle$ with $S=1$ are symmetric over permutation of the electrons, but I don't know how to determine if $|\psi \rangle$ is either symmetric or antisymmetric. In this case, the permutation of the electrons change $x \rightarrow -x$, so it has the same effect of parity. The hamiltnian above is obviously invariant over parity, and there is a theorem that says $|\psi \rangle$ is an eigenstate of parity (suposing that the spectrum of the hamiltonian is non-degenerated), so it have to be symmetic or antisymmetic. Do I need to determine the wave function $\psi(x)$ to it? I tried to do this, but I get four constants that I can't determine.

 

[DrClaude]

Now, I know that the states $|P \rangle$ and $|SM \rangle$ with $S=1$ are symmetric over permutation of the electrons, but I don't know how to determine if $|\psi \rangle$ is either symmetric or antisymmetric.

What does the Pauli principle tell you?

 

[Lebnm]

Ok, I understood. The total state have to be antisymmetric, so $| \psi \rangle$ is antisymmetric, since $|P\rangle$ and $|S,M\rangle$ are symmetric. Solving the problem for $\psi(x)$, I find

$$
\psi(x) =
\begin{cases}
Ae^{ikx} + Be^{-ikx}, & \text{ if } |x| > a \\
Ce^{ilx} + De^{-ilx}, & \text{ if } |x| \leq a
\end{cases}
$$

where $k = \sqrt{2\mu E}/\hbar$, $l = \sqrt{2\mu(E+V_{0})}/\hbar$ and $E = E_{TOT} - E_{CM}$. Imposing that $\psi(-x) = - \psi(x)$, we have $A = -B$ and $C = - D$, so

$$
\psi(x) =
\begin{cases}
Asin(kx), & \text{ if } |x| > a \\
Csin(lx), & \text{ if } |x| \leq a
\end{cases}
$$

But $\psi$ also need to be continuous, what implies that $C = A sin(ka)/sin(la)$, and $A$ have to be chosen such that $\langle \Psi | \Psi \rangle = 1$. The orbital part of the wave function will be

$$
\Psi(x, X) \propto e^{iPX/\hbar}
\begin{cases}
sin(kx), & \text{ if } |x| > a \\
\frac{sin(ka)}{sin(la)}sin(lx), & \text{ if } |x| \leq a
\end{cases}
$$

where $X$ is the coordinate of C.M. Now I have to multiply this by the spin state. But I have three of them: $|1,1 \rangle$, $|1,0 \rangle$ and $|1,-1 \rangle$. Do I need to take a linear combination of them? In this case, my wave function would depend of three constants, is it correct?

 

[DrClaude]

Can you post the full text of the problem?

 

[Lebnm]

Actually, this is a problem that I saw in the internet and I don't remember the exact text. But the potential is that $V$ above and the total spin is $S = 1$. Is this enough to find the soluction?

 

[DrClaude]

Actually, this is a problem that I saw in the internet and I don't remember the exact text. But the potential is that $V$ above and the total spin is $S = 1$. Is this enough to find the soluction?

No, far from it. At best, this is an exercise to understand the constraints on the spatial wave function.