PHYS212 : Homework

Homework Assignment 9 (due 4/20)

Quantum Review and the Infinite Square Well

  1. Find convenient nuclear energy and time scales. Consider a proton with a wavelength of 1 fm ($10^{-15}$ m). (Nuclear radii are approximately (1.2 fm)$A^{1/3}$ where $A$ is the mass number, so fm are a suitable length scale.)

  2. (a) For a particle trapped in an infinite square well, calculate the probability of finding the particle in the middle half of the well ( $\frac{L}{4} \leq x \leq \frac{3L}{4}$). Express your result in terms of $n$ (look for a pattern in the trig. functions).

    (b) How does your result from part (a) compare with the corresponding probability for a classical particle bouncing back and forth in a region of width $L$? Specifically, does the quantum mechanical result reduce to the classical result in the classical limit ( $n \rightarrow \infty$)?

  3. Investigate an equal superposition of the first two states

\Psi(x,t) = A [\Psi_1(x,t) + \Psi_2(x,t)]

    of an electron in an infinite square well of width 1.0 nm.

    (a) Find the probability density $\varrho(x,t)$ and the overall normalization constant $A$ of the superposition. (Use the normalized forms of $\Psi_1$ and $\Psi_2$.)

    (b) The probability density is time dependent. Use Maple to animate it. You will need the plots package

    [> with(plots):
    Then, if you define the function and limits
    [> rho  := (Your density function of x and t) ;
    [> L    := (well width) ;
    [> tmax := (duration of the animation in fs) ;
    the animation syntax is
    [> animate(rho, x=0..L, t=0..tmax);

    (c) Calculate the expectation value of $x$. (Feel free to have Maple do the integrals.) It is also time dependent. Use Maple to graph it over an appropriate time interval.

    (d) Describe in words what your animation of part (b) and your graph of part (c) tell you, and how they relate to each other.

Copyright © 2003-2009, Lewis A. Riley Updated Tue Apr 14 02:04:45 2009