Fall 2020 Homework Assignments
The wavefunction of a particle at time t=0 is given as Ψ(x) = A exp(-|x-a|/b). Where A is the normalization constant and a and b are constants with units of length. (1a) Find A, assuming it is real and positive. (1b) Find the expectation values <x>, <x2>, and <p>. (1c) Find the expectation value of <p2> in two ways: once by using the equation <p2> = ∫ Ψ (ℏ/i ∂/∂x)2 Ψ dx and second by using <p2> = ℏ2 ∫ |∂/∂x Ψ|2 dx. (Be careful when differentiating discontinious functions!) (1d) Find Δx and Δp. For the second part of this problem, use the function Ψ(x) = A exp(-|x-a|/b) exp(iqx). (2a) Find A, assuming it is real and positive. (2b) Find the expectation values <x>, <x2>, and <p>. (2c) Find the expectation value of <p2> using your favorite method. (2d) Find Δx and Δp.
At time t=0, the wavefunction of a particle inside an infinite one-dimensional quantum well (extending from x=-a/2 to x=a/2) is given by { 0 for x<0 and x>a/2 Ψ(x,0) = { { A x(x-a/2) for 0<x<a/2. (a) Find the series expansion for the wavefunction Ψ(x,t) at later times t>0. (b) After calculating part (a) analytically, sum the first 10 terms of the series on a computer and obtain plots of |Ψ(x,t)|2 as a function of x/a for the time values t=0, τ/3, τ/2 and τ where τ=ħ/E1 where E1 is the ground state energy. Submit your analytical work, computer code, and the plots as homework.
At time t=0, a particle in a harmonic oscillator potential is known to be in the state Ψ(x,0) = ψ3(x)/2 + A ψ4(x) where ψn(x) are the normalized stationary states of the harmonic oscillator. (a) What is the value of A so that the wavefunction Ψ is normalized? Find the following expectation values as functions of time: (b) <x> (c) <x2> (d) <p> (e) <p2> (f) Δx and Δp
1) Calculate the quantities <0|A|0> , <0|A|2> and <2|A|0> where |n> is the n'th eigenvector of the harmonic oscillator and A is the operator given below: (a) xp3 (b) xp3+ p3x (c) xp2x (d) px2p
Consider a 3 dimensional harmonic oscillator with Hamiltonian H = ( px2 + py2 + pz2 )/2m + k( x2 + y2 + z2 )/2 . (a) Separate the Schrodinger Equation in variables x, y, and z and obtain three independent oscillators in these directions. (b) Using our knowledge of the one dimensional harmonic oscillator, list the states for the lowest three energy levels of the 3 dimensional harmonic oscillator. How many fold degenerate are these levels? (Use the notation |nx,ny,nz> to represent the states.) (c) Consider the operator Lz = x py - y px. Show that the states below are eigenstates of this operator. Find the corresponding eigenvalues. (i) |0 0 0> (ii) (|1 0 0> + i|0 1 0>)/√2
One of your classmates asks the following question: "Due to the commutation relation [Lx,Ly] = iħLz and the generalized uncertainty principle, we should have σLx σLy ≥ |<[Lx,Ly]>| /2 = ħ/2 |<Lz>| . Then we are going to have σLx σLy ≥ 0 if we have <Lz>=0. Doesn't that correspond to zero uncertainty?" The answer is that σLx σLy ≥ 0 is still an inequality, and you should be able to find a state for which |<ψ|Lz|ψ>|=0 such that the corresponding uncertainties for Lx or Ly are small. One state which will make |<ψ|Lz|ψ>|=0 is an angular momentum state of the form |ψ> = |ℓ,0>. Use this state in determining the expectation values of Lx, Lx2, Ly and Ly2 and hence determine σLx σLy for this state. Discuss your result, especially the ℓ=0 case. Hint: Use Lx = (L+ + L-)/2 and Ly = -i(L+ - L-)/2. Remember also that L± L∓ = L2 - Lz2 ± ħ Lz .
An electron is known to be in the angular momentum state |Ψ> = A|0,0> + ( |1,1> + |1,0> )/2 (a) Find the constant A through normalization. (b) What is the expectation value of Lz for this state? (c) What is the expectation value of Lx for this state? The following measurements of the angular momenta are to be made on the electron (all on |Ψ>): (d) What are the possibilities and their probabilities for a measurement of Lz? (e) What are the possibilities and their probabilities for a measurement of L2? (f) What are the possibilities and their probabilities for a measurement of L2 carried out on the electron after a measurement of its Lz results in zero? (g) What are the possibilities and their probabilities for a measurement of L2 carried out on the electron after a measurement of its Lz results in ħ?
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
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