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esorolla
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Homework Statement
Hello everybody:
I have a problem with the Schrödinger equation in 3D in spherical coordinates, since I'm trying to calculate the discrete set of possible energies of a particle inside a spherical box of radius "a" where inside the sphere the potential energy is zero and out the sphere is infinite.
First problem was when I was looking at wikipedia webpage for Schrödinger equation and I have to confess that I don't understand why when we deal with spherical coordinates the first thing that appears is the spherical harmonics. I tell you this because in my case I don't see any dependency on the angular variables, theta and phi, since there is not any privilegiated direction in the free particle inside the spherical box. Therefore I should recover the same result that the one showed in the wikipedia webpage where the Schrödinger's equation in spherical coordinates is dealt with when there is rotational symmetry (I'd say!)
What I was trying to solve was the Schrödinger equation in spherical coordinates assuming the function psi doesn't depend on the angular variables, theta and phi, since the problem has spherical symmetry (by definition of the problem). Then I got only derivatives in the radial coordinate:
Homework Equations
View attachment eq1.bmp
Making a change of variable
[tex]\Psi=U(r)/r[/tex]
I found the energy of the free particle inside the spherical box:
View attachment eq3.bmp
The problem arises when I compare this result to that one of a free particle inside a cubic of side l in cartesian coordinates, since I expected something similar, but the problem is that I found:
View attachment eq2.bmp
Then I see a difference, because in the cartesian case for the cube I have one result and in the sphere I have another one. It can be said that it is not shocking because the systems are different, but I'm wondering something...
I was studying statistical physics (perhaps I should include this now in other thread, I don't know...) and the problem arose when I saw that in the calculation of the partition function for the "particle-in-a-box" problem the partition function reads:
[tex]\ q_{trans}=\sum_{n_{x}}\sum_{n_{y}}\sum_{n_{z}}exp(-\beta\frac{h^2}{8m l^2}(n_{x}^2+n_{y}^2+n_{z}^2))=(\sum_{n}exp(-\beta\frac{h^2 n^2}{8m l^2}))^3 [/tex]
and if you compare it to the partition function for the case that I've solved for the particle in the spherical box, I don't find the same, since the factor 3 in the exponent is missed.
Now my question is, why? Is there any energy that I didn't take into account and it should be taken into account? The energy related to the angular momentum of the particle? I don't see it, even in this case that energy doesn't recover the case of the "particle-in-a-box" in cartesian coordinates.
So, either I have made a mistake all along the steps described in the previous discussion which recovers almost the good solution without the factor 3, or there isn't anything wrong but something missed.
The Attempt at a Solution
In my opinion, as the partition function has to take into account all the states, even if there is degeneracy, I have the intuition that there is some kind of degeneracy that gives me the same result in both cases, but I really don't see how. Because the free particle in the sphere doesn't have (I'd say) any degeneracy, since the eigensolutions of the free particle in the spherical box is (inside the sphere)
[tex]\Psi(r)=\frac{2i sin(\alpha r)}{\sqrt{8\pi a}r}[/tex]
and the boundary conditions establish that [tex]\alpha a=n\pi[/tex] for n integer. And [tex]\alpha[/tex] is related to the energy through
[tex]\alpha^2=\frac{2mE}{\hbar^2}[/tex]
Thank you very much and sorry for the, perhaps, too technical thread.
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