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

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


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?