91Ó°ÊÓ

The wave equation describing the transverse vibrations of a stretched membrane under tension \(T\) and having a uniform surface density \(\rho\) is $$ T\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)=\rho \frac{\partial^{2} u}{\partial t^{2}} $$ Find a separable solution appropriate to a membrane stretched on a frame of length \(a\) and width \(b\), showing that the natural angular frequencies of such a membrane are $$ \omega^{2}=\frac{\pi^{2} T}{\rho}\left(\frac{n^{2}}{a^{2}}+\frac{m^{2}}{b^{2}}\right) $$ where \(n\) and \(m\) are any positive integers.

Schrödinger's equation for a non-relativistic particle in a constant potential region can be taken as $$ -\frac{\hbar^{2}}{2 m}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}\right)=i \hbar \frac{\partial u}{\partial t} $$ (a) Find a solution, separable in the four independent variables, that can be written in the form of a plane wave, $$ \psi(x, y, z, t)=A \exp [i(\mathbf{k} \cdot \mathbf{r}-\omega t)] $$ Using the relationships associated with de Broglie \((\mathbf{p}=\hbar \mathbf{k})\) and Einstein \((E=\hbar \omega)\), show that the separation constants must be such that $$ p_{x}^{2}+p_{y}^{2}+p_{z}^{2}=2 m E $$ (b) Obtain a different separable solution describing a particle confined to a box of side \(a(\psi\) must vanish at the walls of the box). Show that the energy of the particle can only take the quantised values $$ E=\frac{\hbar^{2} \pi^{2}}{2 m a^{2}}\left(n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\right) $$ where \(n_{x}, n_{y}, n_{z}\) are integers.