Alan0354
Full Member level 4
For \[\nabla^2 u(r,\theta, \phi)=0,\;u(r,\theta, \phi)=r^{n}Y_{nm}(\theta,\phi) \]
But I have issue with this, for spherical coordinates:
\[\nabla^2u=\frac{\partial^2{u}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial{u}}{\partial {r}}+\frac {1}{r^{2}}\left(\frac{\partial^2{u}}{\partial {\theta}^2}+\cot\theta\frac{\partial{u}}{\partial {\theta}}+\csc\theta\frac{\partial^2{u}}{\partial {\phi}^2}\right)\]
Let \[u=R(r)Y(\theta,\phi) \] where \[Y(\theta,\phi) \] is the spherical harmonics.
\[\Rightarrow\; r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-k R=0\]
and
\[\frac{\partial^2{Y}}{\partial {\theta}^2}+\cot\theta\frac{\partial{Y}}{\partial {\theta}}+\csc^2\theta\frac{\partial^2{Y}}{\partial {\phi}^2}+k Y=0\]
For Euler equation: \[r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-k R=0\] where \[k=n^2\]. and the solution is \[R=r^n\].
Here, because of the condition, only ##k=n(n+1)## is used for bounded solution.
\[ r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-k R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-n(n+1) R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+\;r\frac{\partial{R}}{\partial {r}}-(n+1/2)^2R\]
Which gives
\[R=r^{(n+1/2)}\]
What have I done wrong?
But I have issue with this, for spherical coordinates:
\[\nabla^2u=\frac{\partial^2{u}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial{u}}{\partial {r}}+\frac {1}{r^{2}}\left(\frac{\partial^2{u}}{\partial {\theta}^2}+\cot\theta\frac{\partial{u}}{\partial {\theta}}+\csc\theta\frac{\partial^2{u}}{\partial {\phi}^2}\right)\]
Let \[u=R(r)Y(\theta,\phi) \] where \[Y(\theta,\phi) \] is the spherical harmonics.
\[\Rightarrow\; r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-k R=0\]
and
\[\frac{\partial^2{Y}}{\partial {\theta}^2}+\cot\theta\frac{\partial{Y}}{\partial {\theta}}+\csc^2\theta\frac{\partial^2{Y}}{\partial {\phi}^2}+k Y=0\]
For Euler equation: \[r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-k R=0\] where \[k=n^2\]. and the solution is \[R=r^n\].
Here, because of the condition, only ##k=n(n+1)## is used for bounded solution.
\[ r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-k R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-n(n+1) R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+\;r\frac{\partial{R}}{\partial {r}}-(n+1/2)^2R\]
Which gives
\[R=r^{(n+1/2)}\]
What have I done wrong?