Laplacian in Polar Cooridinates

[QuantumMech]

I need to take the \nabla^2 of x^2+y^2+z^2. This is how far I got

<br /> \begin{gather*}<br /> \nabla^2 = \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} + \frac{1}{r^2}(\frac{1}{sin^2\theta}\frac{d^2}{d\Phi^2} + \frac{1}{sin\theta}\frac{d}{d\theta} sin\theta\frac{d}{d\theta})\\<br /> \nabla^2(r^2sin^2\theta cos^2\Phi + r^2sin^2\theta sin^2\Phi + r^2cos^2\theta = \frac{1}{sin\theta} \frac{d}{d\theta}(sin\theta \frac{d}{d\theta}) + \frac{1}{sin^2\theta} \frac{d^2}{d\Phi^2})<br /> \end{gather*}<br />


Also, can degeneracy occur with n not in order? Like for a part. in 3D box can degeneracy occur for \Psi_{1,3,5} \Psi_{5,3,1} or do the n have to be next each other like \Psi_{1,2,1} \Psi_{2,1,1}?

 

[dextercioby]

Is that a spherical box?And how does the potential look like...?

Daniel.

 

[QuantumMech]

Im not sure. I just need to use the del operator on x^2+y^2+z^2.

 

[dextercioby]

Del or laplacian...?

Daniel.

 

[QuantumMech]

I mean del squared or laplacian.

Oh, for the 2nd question: a 3D box with V = infinity outside box.

 

[dextercioby]

It's simple.

\nabla^{2}=\Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}

Use it to differentiate what u had to (x^{2}+y^{2}+z^{2}).

Daniel.

 

[QuantumMech]

Oh, but I mean using polar coordinates for x^2+y^2+z^2 = r.

Thanks for the the p chem help dextercioby.

 

[dextercioby]

Nope,i think you mean spherical coordinates and

x^{2}+y^{2}+z^{2}=r^{2}

Daniel.

 

[dextercioby]

And one more thing:please take my advice and compute that in cartesian coordinates...It's easier.Maths should be made as easy as possible,here's an opportunity

Daniel.

 

[Theelectricchild]

That's why I was so confused with the first post, why were we straying away from cartesian when the Laplacian operator is so easily used on the described fct?