Fall 2021 Homework Assignments
This first homework is to test your knowledge of basic calculus which will be needed throughout this course. If you are experiencing major problems in carrying out these operations, then you have defficiencies in the knowledge prerequisite to this course. Chances are you will have trouble in following this course, and will not be able to perform well in the exams. I suggest you do a quick review of the relevant material.
Find the expectation values <x>, <x2>, <p>, <p2> and <K>, where x, p and K are the position, momentum and the kinetic energy of a particle with the following wavefunctions: (a) A exp(-|x|/xo) (b) A exp(-|x|/xo + ikx) (c) A exp(-|x-a|/xo) where A, xo, k and a are constants with appropriate units. Normalize all wavefunctions using the normalization constant A. Warning: Be careful when differentiating discontinious functions!
A one dimensional structure is constructed using two quantum wells as shown in the figure. The depth of the quantum wells is Vo=2eV, and their widths are 1nm. (Since the wells are so narrow, they may be treated as delta-function potentials.) The wells are a=10nm apart. Use computer calculations to find the following: (a) Find the energies of the ground state and the first excited state. (b) Plot the transmission coefficient of the structure as a function of energy. Your plot should contain the "interesting" range of energies which shows the variation of the transmission coefficient.
A particle is known to be in the state Ψ(x,0) = ( un + un+1 ) /√2 at time t=0, where un is the n'th eigenstate of a harmonic oscillator. Find the following quantities as a function of time: (a) Average value of x: <x> (b) Deviation in x: σx Repeat (a) and (b) for the case Ψ(x,0) = ( un + un+2 ) /√2
Consider the Harmonic oscillator raising and lowering operators a+ and a-. 1. Expand the quantities x4 and P4 in terms of a+ and a-. (Be careful: a+ and a- do not commute!) 2. Using the commutation relation for a+ and a-, rearrange the above quantities so that in each term all the a+ are on the left and all the a- are on the right. For example, a-a-a+a+ = a- ( 1 + a+a- ) a+ = a-a+ + a-a+a-a+ = ( 1 + a+a-) + ( 1 + a+a-) ( 1 + a+a-) = 2 + 3a+a- + a+a-a+a- = 2 + 3a+a- + a+ ( 1 + a+a- ) a- = 2 + 4a+a- + a+a+a-a-
(a) Find the operator function sin(A) where [0 1 0] A = [1 0 1] [0 1 0] (b) In a three-state quantum system, the Hamiltonian has the matrix form [0 1 0] H = [1 0 1] ε [0 1 0] where ε is a constant with units of energy. The initial state vector in the three-dimensional Hilbert space is given as [1] Ψ(0) = [0] [0] Find the time dependent state vector given by Ψ(t) = exp(-itH/ℏ) Ψ(0)
Evaluate the following commutators: (a) [x , Lx] (b) [x , Ly] (c) [x , Lz] (d) [x2 , Lx] (e) [y2 , Ly] (f) [z2 , Lz] (g) [r2 , Lz] (h) [r2 , L2] You can use (carefully explained) symmetry arguments to simplify your algebra.
A particle is confined to move inside a two-dimensional square well with the length of its sides equal to a. We will use the vectors |n,m> to represent the eigenstates of the particle, corresponding to the wave function Ψnm(x,y) = 2 sin(nπx/a) sin(mπy/a) / a. Find the expectation value of the angular momentum operator Lz if the particle is in the state (a) ( |1,2> + |1,3> ) / √2 (b) ( |1,2> + i |2,1> ) / √2
Rotate the spin state |↑> around the direction (i + j + k)/√3 (where i, j and k are the unit vectors in the x, y and z directions) by the angles 2π/3 and 4π/3. Interpret the resulting rotated state. (Hint: Draw a coordinate system, place a vector representing |↑> on it, identify the rotation axis (i + j + k)/√3, and discuss what a rotation by 120° would to this vector.)
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
Please name the file in the format:
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.