- #1
eoghan
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- 7
Homework Statement
Let there be 3 particles with mass m moving in the 1D potential:
[tex]\frac{k}{2}[(x_1-x_2)^2 + (x_2-x_3)^2 + (x_1-x_3)^2][/tex]
where [tex]x_i[/tex] is the coordinate of the particle i.
1)Show that with the following coordinat change the Schroedinger equation is easy to solve:
[tex]y_1=x_1-x_2[/tex]
[tex]y_2=\frac{1}{2}(x_1+x_2)-x_3[/tex]
[tex]y_3=\frac{1}{3}(x_1+x_2+x_3)[/tex]2) Find the eigenstates and the energies of the equation you got in point 1)
Homework Equations
The Attempt at a Solution
[tex]x_1-x_2=y_1[/tex]
[tex]x_2-x_3=y_2-\frac{1}{2}y_1[/tex]
[tex]x_1-x_3=y_2+\frac{1}{2}y_1[/tex]
[tex]V=\frac{k}{2}\left[\frac{3}{2}y_1^2+2y_2^2\right][/tex]
[tex]H=\frac{P_1^2}{2m}+\frac{P_2^2}{2m}+\frac{P_3^2}{2m}+V[/tex]
So I have 2 independent harmonic oscillators with angular frequencies [tex]\sqrt{\frac{3k}{2m}}[/tex] and [tex]\sqrt{\frac{2k}{m}}[/tex]
and a free particle whose eigenfunction is [tex]exp\left[\frac{i}{\hbar}\vec P\vec r\right][/tex]
So the eigenstates are the tensor product of the eigenstates of two harmonic oscillators and an exponential.
The energies are [tex](a+\frac{1}{2})\hbar w_1+(b+\frac{1}{2})\hbar w_2 + E[/tex]
where w1 and w2 are the two frequencies of the two harmonic oscillators and E is the energy of the free particle.