Coupled Q. Harmonic Oscillators

[filo85x]

My quantum mechanics teacher give me the following problem:

"Find eigenvalues of the following system: two different particles of mass m in a harmonic oscillator coupled by attractive potential V(x1,x2)=beta*abs(x1-x2)."

Now, I know that standard solving method for this kind of problem is to substitue x1 and x2 with appropriate expressions.

Using reduced mass, i can transform V(x1,x2)=beta*abs(x1-x2) in V(y)=something*abs(y).

This problem, a part the "abs", is similar to a harmonic oscillator in a electric field...but is only similar...i don't know how to manage the "abs". I have thought that the only method for removing abs is to replace y with something always positive (or negative) but my little brain can't find the solution!

Can anyone help me??

Thank you in advance

 

[mark726]

Thank you for sharing your problem with us. The eigenvalues of a system can be found by solving the Schrödinger equation for the system. In this case, we have two particles in a harmonic oscillator coupled by an attractive potential. The Hamiltonian for this system can be written as:

H = H1 + H2 + V(x1,x2)

where H1 and H2 are the single-particle Hamiltonians for each particle and V(x1,x2) is the potential energy due to the interaction between the particles.

To find the eigenvalues of this system, we need to solve the time-independent Schrödinger equation:

Hψ = Eψ

where ψ is the wavefunction and E is the corresponding energy eigenvalue. In this case, we can use the separation of variables method to solve this equation.

First, we can write the wavefunction as a product of two single-particle wavefunctions:

ψ(x1,x2) = ψ1(x1)ψ2(x2)

Substituting this into the Schrödinger equation, we get:

(H1 + H2 + V(x1,x2))ψ1(x1)ψ2(x2) = Eψ1(x1)ψ2(x2)

Next, we can divide both sides by ψ1(x1)ψ2(x2) and rearrange the terms to get:

(H1 + V(x1,x2) - E)ψ1(x1) = (E - H2)ψ2(x2)

This can be further simplified by defining a new variable, y = x1 - x2, and using the reduced mass μ = m/2:

(H1 + V(y) - E)ψ1(x1) = (E - H2)ψ2(x2)

(H1 + β|y| - E)ψ1(x1) = (E - H2)ψ2(x2)

Now we have two single-particle Schrödinger equations, one for each particle, with an additional term β|y| due to the interaction between the particles. These equations can be solved using the standard methods for solving the harmonic oscillator problem.

We can then use the boundary conditions to determine the possible values of E and solve for the eigenvalues. The absolute value term can be handled by considering the different