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Derive energy levels for the case of 2-D potential well by using two approaches: a)
solving Schrodinger equation, b) using de-Broglie equation.

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a) Write down the one-dimensional time-dependent Schro ̈dinger equation for a wavefunction Ψ(t, x) in a potential V (x). b) Write down the one-dimensional time-independent Schro ̈dinger equation for a wavefunc- tion ψ(x) in a potential V (x). c) Assuming that Ψ(t,x) corresponds to an energy eigenstate, write down a mathematical expression that relates the solutions of the one-dimensional time-dependent and time- independentSchro ̈dingerequations,Ψ(t,x)andψ(x).
Consider the electron-hole overlap integral Mnn for a quantum well given by: Mn Pen (2) Pnn (z) dz. %3D - 00 n' and zerd (i) Show that Mon is unity if n otherwise in a quantum well with infinite barriers. (ii) Show that Mon is zero if (n-n') is an odd number in a quantum well with finite barriers.
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4) Consider the one-dimensional wave function given below. (a) Draw a graph of the wave function for the region defined. (b) Determine the value of the normalization constant. (c) What is the probability of finding the particle between x = o and x = a? (d) Show that the wave function is a solution of the non-relativistic wave equation (Schrodinger equation) for a constant potential. (e) What is the energy of the wave function? (x) = A exp(-x/a) for x > o (x) = A exp(+x/a) for x < o