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In a system of two identical non-interacting spinless bosons, one particle is in a stationary state $\psi_{1}(\textbf{r})$ with positive parity and another is in stationary state $\psi_{2}(\textbf{r})$ with negative parity. Determine:

a) the coordinate probability distribution for one particle if the position of another is arbitrary.

b) the probability of finding one particle at $z\geq 0$

c) the probabilty of finding both particles at $z\geq 0$

d) Answer a, b, and c for two fermions in the same spin states

e) Answer a, b, and c for two distinguishable particles

f) Two identical spinless bosons of mass m form a molecule with interaction potential $U = \frac{k}{2}\left ( \mathbf{r}_{1}- \mathbf{r}_{2} \right )^{2}$. Determine the spectrum of stationary states of the molecule.

My attempts:

a) The bosons don't interact so the probability distribution is just $\left | \psi_{1,2}\left ( \textbf{r} \right ) \right |^{2}$.

b,c) With that in mind, $\left | \psi_{1,2}\left ( \textbf{r} \right ) \right |^{2}$ must be even under parity so the probability of finding either particle at $z\geq 0$ is 0.5 and since they are non-interacting, the probability of finding both at $z\geq 0$ is 0.25 .

d,e) The same is true for the distinguishable particles and for the fermions, for the same reasons. (The fermions are already in different states so the Pauli exclusion principle is satisfied.)

f) My guess is this involves solving a two-particle Schoedinger equation

$$\frac{-\hbar^{2}}{2m}\left ( \bigtriangledown_{1}^{2} +\bigtriangledown_{2}^{2} \right )\Psi + U\Psi =E\Psi$$

where $\Psi \left ( \textbf{r}_{1},\textbf{r}_{2} \right ) =\frac{1}{\sqrt{2}}\left ( \psi _{1}\left ( \textbf{r}_{1} \right )\psi _{2}\left ( \textbf{r}_{2} \right )+\psi _{1}\left ( \textbf{r}_{2} \right )\psi _{2}\left ( \textbf{r}_{1} \right ) \right )$

with some propagator $$G=G_{0}-iG_{0}UG$$ which has the form $$G = -i\hbar\sum \frac{ \psi _{n}\left ( \textbf{r}_{1} \right )\psi _{n}^{*}\left ( \textbf{r}_{2} \right )}{E-E_{n}+i\eta }$$ $$\eta \rightarrow +0$$ but I don't actually know how I would go about calculating this when I don't know $\psi _{n}$.

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You can do the calculations in centre of mass frame of reference. When you do so your Hamiltonian decouples into a free particle at centre of mass coordinates and a harmonic oscillator for the relative coordinates. You then don’t need to solve for the spectrum because you know the spectrum of both of the decoupled Hamiltonians and the total will be a direct product of the two.

I shall leave the math of the coordinate transformation to you.

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