HOMEWORK 2

DUE FRIDAY 7 September

Useful integrals:

$\int_0^a sin^2(\frac{n\pi x}{a})dx = \frac{a}{2}$

$\int_0^a x^2 sin^2(\frac{n\pi x}{a})dx = (\frac{a}{2\pi n})^3 (\frac{4 \pi^3 n^3}{3} - 2 n \pi)$

1. Operators are very important in quantum mechanics

(a) Explain what an operator is.

(b) For the operators $\hat A$ below, and the functions $f$, evaluate $\hat A \cdot f$

$\hat A$	$f(x)$
$\frac{d^3}{dx^3} + x^3$	$e^{-kx}$
$\int_1^3 dx$	$\frac{1}{x} + x^2 + cos(2\pi x)$

2. Determine if the functions

$f_1(x) = cos(\omega x)$ and $f_2(x) = xe^{kx}$ are eigenfunctions of the operator $\hat A = \frac{d^2}{dx^2}$.

If so, what are the eigenvalues?

3. Is $f_1(x)$ from Question (2) an eigenfunction of the operator $\hat O= \frac{d}{dx} + x$?

4. Prove that the set of functions

$$\psi_n(x) = (2a)^{(-1/2) } e^{in\pi x/a}$$ (1)

[where $n=0,\pm1, \pm2 ...$ and $-a \leq x \leq a]$ are ORTHONORMAL.

i.e.

$$\int_{-a}^{a} \psi_n(x) \psi_m(x) dx = \delta_{nm}$$ (2)

$\delta_{nm}$ is a special function that is called the Kronecker delta function:

$$\delta_{nm} = 1 \;\;\;\; if \;\;\;\; n=m$$ (3)

$$\delta_{nm} = 0 \;\;\;\; if \;\;\;\; n \neq m$$ (4)

5. For a particle of mass m confined to a one-dimensional line [a 1-d "box"] with $0 \leq x \leq L$, the wavefunctions for a given quantum number $n$ are given as:

$$\Psi_n(x) = sin(\frac{n \pi x}{L})$$ (5)

Normalize these wavefunctions.

6. Using the normalized wavefunction from the above question, calculate the probability of finding the particle in the box anywhere between $x=0$ and $x=\frac{L}{4}$. Show that the result depends on whether the quantum number $n$ is even or odd.

7. Evaluate $<x^2>$ for the above particle in the box in the excited state where the quantum number $n=2$.

8. Show that $<p>=0$ for the $n=1$ state.