Two particles in a potential (wave equation and harmonic oscillators)

[Psycopathak]

Homework Statement

Please bear with me, I'm not that good with LaTeX.

Consider the harmonic oscillator problem. Define $$\Phi$$n(x) as the n-th wave function for one particle, with coordinate x and energy (n+1/2) $$\overline{h}$$$$\omega$$, where n=0, 1,… Now, let’s consider a system consisting of two particles which have the same mass; each particle experiences the same potential energy function and therefore each has the same angular frequency $$\omega$$. Consider a situation where the total energy of this two-particle system is E_total = 3 $$\overline{h}$$$$\omega$$. We write the wave functions $$\Psi$$(x_A,x_B) for the system of two particles, where x_A and x_B are the positions of the two particles.
You will be asked below whether the following state is (or is not) allowed; it is a product of two n=1 wave functions (one for each particle):
$$\Psi$$_maybe(x_A,x_B) =$$\Phi$$₁(x_A)$$\Phi$$₁(x_B)

(a) Suppose first that the particles are of different species, called A and B. Is the state $$\Psi$$_maybe(x_A,x_B) an allowed state of the two-particle system, or not?

(b) For this “different species case”, what is the degeneracy g of the 2-particle system with total energy E_total=3 $$\overline{h}$$$$\omega$$?

(c) Now, consider instead the case when both particles are identical bosons. Is the state labeled $$\Psi$$_maybe(x_A,x_B) possible, or not?

Homework Equations

From what I know, the time independent schrodinger wave equation.

H$$\Psi$$ = (-h/2m)($$\partial$$²/$$\partial$$x²) + V$$\Psi$$ = E$$\Psi$$

Right Moving particle solution
$$\Psi$$(x) = Aexp(ipx)/$$\overline{h}$$)

E = p²/2m + C

The Attempt at a Solution

I am pretty confused with what this problem is asking, so I don't have an attempt at a solution to offer. I figured that if you have $$\Psi$$_maybe(x_a,x_b) could be looked at as:
$$\Psi$$(x_a) * $$\Psi$$(x_b)

Unfortunately from here, I am really not sure where to go on. I've been reading this problem for a week with no such luck.

 

[MathematicalPhysicist]

For b), you should use the fact that E=3hw=(n+1/2)hw+(m+1/2)hw=(m+n+1)hw, so if I am not mistaken degenrate state is one which for two different eigenvalues we have the same energy, i.e check for values of m and n which satisfy n+m+1=3 and partition the ways to choose them, for example, n=0 and m=2 and n=2 and m=0 are two non degenrate states, the degenrate state is n=m=1, and thus g=?