Find the possible total energies (Quantum Physics)

[BookWei]

Homework Statement

I'm doing problems for practice in quantum physics.
Consider two particles of the mass m in one dimension with coordinates being denoted by x and they are
connected by a spring with spring constant k. Suppose that the total momentum of the system is p.
Find all possible total energies for the following cases :
(1)two particles are different (2)two particles are identical fermions.

2. The attempt at a solution
(1) I try to guess the answer...
Total energy is the sum of potential and kinetic energy. Now our particles have the same mass and they are one-dimensional. Moreover, they are non-identical. Now potential energy is based on spring constant K therefore V=1/2*K*x^2 . Now considering the harmonic oscillator in classic sense total energy E= T +V
Therefore E = P^2/2m + 1/2*mω^2*x^2.

considering energy from quantum mechanical point of view, we know P= -iℏ d/dx =p' and x=x' hamiltonian becomes, H= 1/2 p'^2/2m + 1/2*mω'^2*x^2
now considering the particles time independent
H'ψ(x) = Eψ(x)
the eigenvalues of this Hamiltonian is based on En = (n+ 1/2)ℏω where ground state has non-zero energy.

(2) I have no idea how to start this problem.

Thank you for your help.

 

[PeroK]

Thank you for your help.

I'm not sure at all how to interpret this problem. My first thought was that you have two particles in a harmonic oscillator, but I suspect that is not what is intended.

I'm not sure how you can have two fermions connected by a spring. How could you attach a fermion to a spring?

Perhaps what is meant is simply to calculate the potential for a classical system of this type and then translate that to a quantum potential - based on particle repulsion and attraction, rather than a spring! - and solve the resulting Schrodinger equation.

In any case, your solution to 1), which appears to be simply a solution to a single particle SHO, cannot be correct.

 

[BookWei]

Thanks a lot.
I upload the original problem file.

I will try to solve the problem (1).

 

[PeroK]

Thanks a lot.
I upload the original problem file.

I will try to solve the problem (1).

Okay, so the question setter does believe you can attach fermions and bosons to a spring. On the face of it, the natural length of the spring should be relevant, but I think you'll just have to do the maths and see what happens.

 

[TSny]

H'ψ(x) = Eψ(x)

You have a two-particle system. So, you can consider the wavefunction to be a function of two position coordinates: ψ(x₁, x₂).

Do you know how to do a coordinate transformation which separates out the center-of-mass motion and the motion relative to the center of mass (with a reduced mass μ)?

 

[BookWei]

I tried to solve this problem for two days.
But I still do not know how to solve it...

 

[TSny]

I suggest that you change variables from $x_1$ and $x_2$ to $X_c$ and $x$, where $X_c$ is the coordinate of the center of mass and $x = x_2 - x_1$. This is a standard method for reducing the two-body problem to two independent one-body problems.

Hopefully you've seen this before. See the first 3 or 4 pages here: http://physics.oregonstate.edu/~corinne/COURSES/ph426/notes2.pdf

 

[BookWei]

I suggest that you change variables from $x_1$ and $x_2$ to $X_c$ and $x$, where $X_c$ is the coordinate of the center of mass and $x = x_2 - x_1$. This is a standard method for reducing the two-body problem to two independent one-body problems.

Hopefully you've seen this before. See the first 3 or 4 pages here: http://physics.oregonstate.edu/~corinne/COURSES/ph426/notes2.pdf

Thank you so much !