(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have a problem in which I have a two-atomic molecule, and I'm supposed to find the energy and wave function in theground state, given the particles' masses [tex]m_1,m_2[/tex] and the potential [tex]V(r)=kr^2[/tex], where [tex]r[/tex] is the distance between the particles.

I don't necessarily need this problem solved (you can change V to whatever you want). I just want to see what a solution looks like.

2. Relevant equations

The time-independent schrÃ¶dinger equation for a two-particle system is

[tex]-\frac{\hbar^2}{2m}\nabla_1^2\psi-\frac{\hbar^2}{2m}\nabla_2^2\psi+V\psi=E\psi.[/tex]

If the potential is dependent only on [tex]\mathbf r=\mathbf r_1-\mathbf r_2[/tex], then the above equation can be separated into the variables [tex]\mathbf r[/tex] and [tex]\mathbf R=(m_1\mathbf r_1+m_2\mathbf r_2)/(m_1+m_2)[/tex] so that

[tex]-\frac{\hbar^2}{2(m_1+m_2)}\nabla^2\psi_R=E_R\psi_R[/tex]

[tex]-\frac{\hbar^2}{2\mu}\nabla^2\psi_r+V\psi_r=E_r\psi_r[/tex], [tex]\mu[/tex] is the reduced mass

[tex]E=E_r+E_R[/tex]

3. The attempt at a solution

My book mentions (Introduction to Quantum Mechanics by Griffiths, in problem 5.1) that if the potential for a two-particle system is dependent only on the separation between the particles, then we can separate the time-independent schrÃ¶dinger equation into two single-particle equations (see the above section), one that looks like a free particle equation (its potential is 0), and the other with the same potential as the original problem.

The second equation is just an harmonic oscillator, but I don't know what to do with the first. The problem is that a free particle doesn't have a ground state (right?), so I don't know how the whole thing can have one.

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# Homework Help: What's the wave function for a two-particle system given distance-dependent potential

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