Spring 2015 Homework Assignments
1) Consider the periodic function { 10 for |x| < 5 f(x) = { { 0 otherwise with f(x+100) = f(x). a) Find the expression for the Fourier series coefficients c_{n} . b) Sum the series Σ c_{n} exp(i 2πnx/100) for values of n such that i) |n| < 10 ii) |n| < 50 iii) |n| < 100 and plot the result for the three cases. c) Repeat the above for the function { 5 for |x| < 10 g(x) = { { 0 otherwise with g(x+100) = g(x). 2) Find the energies for the photons corresponding to the electromagnetic radiation associated with the following sources of radiation in electron-Volts. For each case state what value of wavelength you are using. (Make a guess if you cannot find a precise number.) - a red LED - your GSM telephone - Bilkent Radio - X-ray machine at a health center - Full body x-ray scanner (used in US airports)
An electron is known to be at the lowest energy state of a one dimensional box which extends from x=0 to x=L/2. (a) What is the wavefunction ψ_{o}(x) of the particle? At time t=0 the size of the box is suddenly doubled so that it now extends from x=-L/2 to x=L/2. At time t=0, the particle is still in its original state. (b) What are the stationary states of the particle in the extended box? (c) Expand ψ_{o}(x) in terms of these stationary states. (i.e. find the expression for the expansion coefficients.) (d) Write down the form of this expansion when t > 0. Plot |ψ_{o}|^{2}/L for values of t = 0 , 0.5 πħ/E_{1} , πħ/E_{1} , 1.5 πħ/E_{1}, and 2πħ/E_{1} keeping 10 terms in the expansion. E_{1} is the energy of the lowest state. (Note that when displaying physical quantities in plots, it is best to construct the plots in terms of unitless quantities. For example, you should plot the magnitude-squared of the wavefunction |ψ_{o}|^{2}/L as a function of x/L. I have already given you a time variable τ = 2πħ/E_{1} so that your plots will correspond to the unitless values t/τ = 0 , 1/4 , 1/2 , 3/4 and 1 .) (e) Give the values of E_{1} (in eV), and τ for a box of size L=10nm.
Consider the following wavefunction for an electron: ψ(x) = A exp(-|x|/x_{o}) where x_{o} is a length parametrizing its width. (a) Find the normalization constant A assuming that it is real and positive. (b) Find the "momentum space wavefunction", i.e. the Fourier transform g(k). (c) Find the expectation values <x>, <x^{2}>, <p>, <p^{2}>. (Hint, it will be easier to work with ψ(x) rather that g(k) when determining the momentum expectation values. Be careful when differentiating discontinious functions!) (d) What are the uncertainties Δx and Δp. (e) What is the expectation value for the kinetic energy of this particle?
Consider the wavefunction { A(1 - 2|x|/a) for |x| < a/2 ψ(x) = { { 0 otherwise. Find its Fourier transform g(k).
For a particle with mass m and potential energy { ∞ for |x| > L { V(x) = { V_{o} for |x| < a { { 0 otherwise construct the forms of the solutions to the time-independent Schrodinger equation. (Write down the forms of the solutions, and the relevant boundary conditions, but do not attempt to solve these equations.) The parameters L, a, and V_{o} are constant quantities, with a < L. Note that since the potential (and therefore the Schrodinger equation) has the symmetry V(-x) = V(x), your solutions will be either even ( ψ(-x)=ψ(x) ) or odd ( ψ(-x)=-ψ(x) ). Discuss the solutions for (a) when E < V_{o} (b) when E > V_{o}.
Consider the angular momentum operator (vector) L = r x p where the vector quantities are defined through their components: L = iL_{x} + jL_{y} + kL_{z} with i, j, and k the unit vectors in the x, y, and z directions. Let us also define the operators "total angular momentum" L^{2} = L_{x}^{2} + L_{y}^{2} + L_{z}^{2} "raising operator" L_{+} = L_{x} + i L_{y} "lowering operator" L_{-} = L_{x} - i L_{y} (the i in these equations is √-1 ) (a) Calculate the following commutators: [L_{z},L_{y}] [L_{z},L_{x}] [L_{z},L^{2}] [L_{z},L_{+}] [L_{z},L_{-}] (b) Suppose that I have found an eigenfunction Ψ_{a} of the L_{z} operator so that L_{z} Ψ_{a} = a Ψ_{a}. Consider the wavefunction Ψ_{b} = L_{+} Ψ_{a} and show that L_{z} Ψ_{b} = (a+ħ) Ψ_{b}. (c) Repeat part (b) using the wavefunction Ψ_{c} = L_{-} Ψ_{a}. What do you get for the new eigenvalue?
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