Question

Problem 2: Infinite Square Well III (7 marks) An electron is confined to an infinite square well, which spans from x = 0 to x- a. Initially, the electron is in an equal linear superposition of the ground and first excited state of the quantum well with zero relative phase. (a) [1 mark] Write down the initial wavefunction Ψ(x, t = 0) of the electron in terms of the energy eigenfunctions. (b) [1 mark] Plot the initial PDF for an electron with this wavefunction trapped in a 10-nm-wide quantum well. Use the appropriate units. It is nicer to have the x-axis in units of nm and the y-axis in units of 1/nm. (c) [1 mark] Write down the time-dependent wavefunction 뾔x,t). (d) [1 mark] Plot the time-evolution of the electron's PDF using a space-time plot. Follow the same units you used in (b). You should observe an oscillation Choose your time range wisely. The energies usually set the time-scale of the system (e) [2 marks] Quantify the oscillation by solving angular frequency of the expec- tation value (x(t) in terms of the energy eigenvalues E1.2. Hint: You don't need to fully solve for 〈x(t), only the oscillatory component. (f [1 mark] What is the oscillation frequency and period (in SI units) of the elec- tron?

Answer 1

Initial wave function psi (x, t = 0) of the electron in term of the energy Eigen function is (after applying suitable boundary conditions) psi (x, t = 0) = { A sin (n pi x/a) from 0 < x < a 0 elsewhere} ---(A) where the normalization constant A equal to squareroot 2/a (as you can easily zee it) PDF of this electron with above wave function is Vertical-bar psi (x, t = 0)^2 Vertical-bar whose plot we can frame with side by side with the above wave function i.e. psi (x, t = 0) time dependent wave function psi (x, t) = psi (x, 0) e^-I e_n t /h where E_n = h^2n^2/8m a^2 \r\n interestingly we can see that PDF of the time dependent wave function i.e equation (B) does not at all depend on time we so we need not to take any botterance about taking time into account in case of calculation are plotting of PDF