Bilkent University     Department of Physics

Phys 325 Quantum Mechanics I - Assignments

Fall 2019 Homework Assignments

Homework Policy

• Homework 1: (Due Thursday, October 10)
```   1) Consider the wavefunction

{  A (1-|x|/a)  for |x| < a
Ψ(x) =  {
{            0  elsewhere

where A is the normalization constant and a is a constant with units of length.

(a) Normalize the wavefunction.

(b) Find the expectation values <x>, <x2>, <p>, <p2> and the uncertainties Δx and Δp.  What is Δx Δp ?

(c) Find the probability currents at x = ± a/2. Interpret your result.

2)  Repeat problem 1 with the wavefunction

{  A (1-|x|/a) exp(ikox)  for |x| < a
Ψ(x) =  {
{                      0  elsewhere

where ko is a constant wave-number.

```

• Homework 2: (Due Tuesday, October 22)
```  A one-dimensional quantum well has the potential

{  0  for 0 < x < a
V(x) =  {
{  ∞  elsewhere.

At time t=0, the wavefunction of the system is given as

{  A  for 0 < x < a/2
Ψ(x,t=0) =  {
{  0  elsewhere.

(1) Find the expansion coefficients of Ψ(x,t=0) in terms of the eigenfunctions un(x) corresponding to V(x).
(2) Make three plots for the series expansion for Ψ(x,t=0)/A as a function of x/a, keeping 15, 5, and 3 terms in the expansion.
(3) Make three plots for |Ψ(x,t)/A|2 as a function of x/a with 15 terms in the expansion, at times t=0, t=0.1(ℏ/E1), t=ℏ/E1, and t=3(ℏ/E1).
(4) What time interval does ℏ/E1 correspond to for an electron in a 10nm wide well?
```
t=0:  n=15:

• Homework 3: (Due Thursday, November 21)
```
Find the time-dependent expectation values of the following quantities

(1)  x

(2)  xP + Px

when the initial (t=0) state of the system is

(a) A ( un + un+1 )

(b) A exp(a+) uo    with   exp(a+) = 1 + a+ + (a+)2/2! + (a+)3/3! + (a+)4/4! + . . .

where un are the eigenstates of the harmonic oscillator. Use the a+ and a- operators
to solve this problem. (i.e. do NOT use the Hermite polynomials.) Determine the
normalization constant A for each case. Obtain the general expression for the
terms in series in part (b), but do not spend too much time to obtain a functional
form for the sum. (Bonus points if you do!)

```

• Homework 4: (Due Tuesday, December 3)
```
Suppose you were given the opportunity to design your potential barriers for an experiment:

You can produce 1eV high barriers which can have widths which are multiples of 1nm. (i.e., you can fabricate
barriers which are 1eV high and can have a width of 1nm, 2nm, 3nm, etc.) You can then treat these very thin
barriers as delta-function potentials αδ(x-xo) where α can be 1eV-nm, 2eV-nm or 3eV-nm, etc.
xo is the position of the barrier.

Construct a two-barrier potential that will "filter" a certain energy of electrons: Plot the transmission
coefficient of the structure as a function of energy. Try to maximize the ratio of the first maximum of the
transmission coefficient (Tmax) to its value at the first minimum (Tmin1) after the maximum.

Give values for the positions and widths of the barriers (one of the barriers must be at x=0), Tmax, Tmin1 and Tmax/Tmin1.
The student who obtains the largest Tmax/Tmin1 value will be the class champion.
```

• Homework 5: (Due Thursday, December 19)
```Evaluate the following matrix operations:

(1)  Show that [A,B]=0 , find the common eigenvector set.
Expand V in terms of these eigenvectors.

(2)  Show that [A,C]≠0 .
Expand V in terms of eigenvectors of C.

(3)  exp(A) V

(4)  exp(B) V

(5)  exp(A+B) V

(6)  exp(C) V

(7)  exp(A+C) V

[ 1 0 3 ]        [ 2 0 0 ]       [ 1 1 0 ]        [ 1 ]
Use the quantities A = [ 0 2 0 ]    B = [ 0 1 0 ]    C= [ 1 1 0 ]    V = [ 1 ]
[ 3 0 1 ]        [ 0 0 2 ]       [ 0 0 1 ]        [ 1 ]
```

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.