We have a particle moving in a 3-D potential well V=1/2*m*(omega^2)*(r^2). we use separation of variables in cartesian coords to show that the energy levels are:(adsbygoogle = window.adsbygoogle || []).push({});

E(Nx,Ny,Nz)=hbar*omega(3/2 + Nx + Ny + Nz)

where Nx,Ny,Nz are integers greater than or equal to 1.

Therefore we can say that the ground state is unique, the first excited state has 3 fold degeneracy, and the second excited state has 6 fold degeneracy etc.

We now have to rewrite the wavefunctions of the 1st and 2nd excited states in terms of the 1d oscillator wavefunctions. We then have to rewrite the wavefunctions again using spherical harmonics and functions of r. We then find the L^2 and Lz eigenvalues of our wavefunctions.

For the 1st excited state this is easy, but I'm struggling with the 2nd one.

I know that we have 6 different wavefunctions. N.b the wavefunctions for the 1-d oscillator are as follows:

phi0(x)=A0*exp(-(x^2)/2(a^2))

phi1(x)=A1*x*exp(-(x^2)/2(a^2))

phi2(x)=A2*(2(x^2)-(a^2))*exp(-(x^2)/2(a^2))

n.b A0,A1,A2 are constants and a is a constant involving w, hbar etc.

Therefore our six wavefuntions for the 3-D case in the second excited state will be

phi(r)=k*(2(x^2)-(a^2))*exp(-(r^2)/2(a^2))

phi(r)=k*(2(y^2)-(a^2))*exp(-(r^2)/2(a^2))

phi(r)=k*(2(z^2)-(a^2))*exp(-(r^2)/2(a^2))

phi(r)=k*xy*exp(-(r^2)/2(a^2))

phi(r)=k*yz*exp(-(r^2)/2(a^2))

phi(r)=k*xz*exp(-(r^2)/2(a^2))

where k is a constant.

I'm finding it really difficult to express these functions as linear combinations of the spherical harmonics. thanks very much for your help.

Sachi

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# Homework Help: QM- 3d SHO potential well

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