Bilkent University     Department of Physics

Phys 325 Quantum Mechanics I - Assignments

Fall 2016 Homework Assignments

Homework Policy

• Homework 1: (Due Tuesday, October 3)
```Normalize the following wavefunctions (i.e. find the normalization constants A)
and determine the expectation values for <x> and <p> corresponding to them.

ψ(x) = A exp[-(x-xo)2/a2]

ψ(x) = A exp[-(x-xo)2/a2] exp(i x po/ℏ)

The variables a, xo, and po are real constants.  What physical quantities do they correspond to?

```

• Homework 2: (Due Thursday, October 20)
```
A particle is confined to a one dimensional box with zero potential in the region -L/2 < x < L/2 and
infinite potential elsewhere.

1. Find the normalization constants and the expansion coefficients cn for
the series Ψ(x,0) = Σn cn un(x) corresponding to the wavefunctions Ψ(x,0) below:

(All finite only in the interval -L/2 < x < L/2 .)

(a) Ψ(x,0) = A [ exp(-α |x| ) - exp(-α L/2 ) ]

(b) Ψ(x,0) = A x (|x| -L/2 )

2. If Ψ(x,0) = A [ u1(x) + 2 u2(x) ] find <p>t, the expectation value of the momentum as a function of time.

```

• Homework 3: (Due Tuesday, October 25)
```
Consider the Harmonic oscillator raising and lowering operators a+  and  a-.

1. Note that the position and momentum operators are proportional to  (a+ + a-)  and  (a+ - a-).
Express  x  and  P  in terms of  a+  and  a-

2. Expand the quantities  x4  and  P4  in terms of  a+  and  a-.
(Be careful: a+  and  a-  do not commute!)

3. 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-

```

• Homework 4: (Due Thursday, December 29)
```  Using the radial Hydrogenic wavefunctions in Table 4.7 of your textbook, show the orhonormality of the
states |n,l,m> below:

|1 0 0> , |2 0 0> and |3 0 0> .

Note that all three states have l=0 and m=0 so that we know that the integral of the angular part
∫ Y00*(θ,φ) Y00(θ,φ) dΩ    will be equal to 1 in all cases.
```

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.