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Exercises

  1. Use the standard power law expansions,

    $\displaystyle {\rm e}^x$ $\displaystyle = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots,$    
    $\displaystyle \sin x$ $\displaystyle = x - \frac{x^3}{3!} + \frac{x^5}{5!}-\frac{x^7}{7!}+\cdots,$    
    $\displaystyle \cos x$ $\displaystyle =1-\frac{x^2}{2!} + \frac{x^4}{4!}-\frac{x^6}{6!}+\cdots,$    

    which are valid for complex $ x$ , to prove Euler's theorem,

    $\displaystyle {\rm e}^{ {\rm i} \theta} = \cos\theta + {\rm i} \sin\theta,
$

    where $ \theta$ is real.
  2. Equations (724) and (725) can be combined with Euler's theorem to give

    $\displaystyle \delta(k) = \frac{1}{2\pi}\int_{-\infty}^\infty {\rm e}^{ {\rm i} k x} dx,
$

    where $ \delta (k)$ is a Dirac delta function. Use this result to prove Fourier's theorem: that is, if

    $\displaystyle f(x) = \int_{-\infty}^{\infty} \bar{f}(k) {\rm e}^{ {\rm i} k x} dk,$

    then

    $\displaystyle \bar{f}(k) = \frac{1}{2\pi}\int_{-\infty}^\infty f(x) {\rm e}^{-{\rm i} k x} dx.$

  3. A He-Ne laser emits radiation of wavelength $ \lambda =633 {\rm nm}$ . How many photons are emitted per second by a laser with a power of $ 1\,{\rm mW}$ ? What force does such a laser exert on a body that completely absorbs its radiation?

  4. The ionization energy of a hydrogen atom in its ground state is $ E_{\rm ion}= 13.6 {\rm eV}$ . Calculate the frequency (in hertz), wavelength, and wavenumber of the electromagnetic radiation that will just ionize the atom.

  5. The maximum energy of photoelectrons from Aluminium is $ 2.3\,{\rm eV}$ for radiation of wavelength $ 200\,{\rm nm}$ , and $ 0.90\,{\rm eV}$ for radiation of wavelength $ 258 {\rm nm}$ . Use this data to calculate Planck's constant (divided by $ 2\pi$ ) and the work function of Aluminium. [Adapted from Gasiorowicz 1996.]

  6. Show that the de Broglie wavelength of an electron accelerated across a potential difference $ V$ is given by

    $\displaystyle \lambda = 1.23\times 10^{-9} V^{ -1/2} {\rm m},
$

    where $ V$ is measured in volts. [From Pain 1999.]

  7. If the atoms in a regular crystal are separated by $ 3\times 10^{-10}\,{\rm m}$ demonstrate that an accelerating voltage of about $ 2 {\rm kV}$ would be required to produce a useful electron diffraction pattern from the crystal. [Modified from Pain 1999.]

  8. A particle of mass $ m$ has a wavefunction

    $\displaystyle \psi(x,t)= A \exp\left[-a (m x^2/\hbar+ {\rm i} t)\right],
$

    where $ A$ and $ a$ are positive real constants. For what potential $ U(x)$ does $ \psi(x,t)$ satisfy Schrödinger's equation?

  9. Show that the wavefunction of a particle of mass $ m$ trapped in a one-dimensional square potential well of of width $ a$ , and infinite depth, returns to its original form after a quantum revival time $ T=4\,m\,a^2/\pi\,\hbar$ .

  10. Show that the normalization constant for the stationary wavefunction

    $\displaystyle \psi(x,y,z) = A\,\sin\left(n_x\,\pi\,\frac{x}{a}\right)\sin\left(n_y\,\pi\,\frac{y}{b}\right)\sin\left(n_z\,\pi\,\frac{z}{c}\right)
$

    describing an electron trapped in a three-dimensional rectangular potential well of dimensions $ a$ , $ b$ , $ c$ , and infinite depth, is $ A=(8/a b c)^{1/2}$ . Here, $ n_x$ , $ n_y$ , and $ n_z$ are positive integers. [From Pain 1999.]

  11. Derive Equation (1193).

  12. Consider a particle trapped in the finite potential well whose potential is given by Equation (1179). Demonstrate that for a totally symmetric state the ratio of the probability of finding the particle outside to the probability of finding the particle inside the well is

    $\displaystyle \frac{P_{\rm out}}{P_{\rm in}}= \frac{\cos^3 y}{\sin y\,(y + \sin y\,\cos y)},
$

    where $ \sqrt{\lambda-y^2} = y \tan y$ , and $ \lambda = V/E_0$ . Hence, demonstrate that for a shallow well (i.e., $ \lambda\ll 1$ ) $ P_{\rm out}\simeq 1 - 2\,\lambda$ , whereas for a deep well (i.e., $ \lambda\gg 1$ ) $ P_{\rm out}\simeq (\pi^2/4) / \lambda^{3/2}$ (assuming that the particle is in the ground state).

  13. Derive expression (1209) from Equations (1205)-(1208).

  14. Show that the coefficient of transmission of a particle of mass $ m$ and energy $ E$ , incident on a square potential barrier of height $ V<E$ , and width $ a$ , is

    $\displaystyle \vert T\vert^{\,2} = \frac{4\,k^{\,2}\,q^2}{4\,k^{\,2}\,q^2 + (k^{\,2}-q^2)^{\,2}\,\sin^2(q\,a)},
$

    where $ k=\sqrt{2 m E/\hbar^2}$ and $ q=\sqrt{2\,m\,(E-V)/\hbar^2}$ . Demonstrate that the coefficient of transmission is unity (i.e., there is no reflection from the barrier) when $ q\,a=n\,\pi$ , where $ n$ is positive integer.

  15. The probability of a particle of mass $ m$ penetrating a distance $ x$ into a classically forbidden region is proportional to $ {\rm e}^{-2\,\alpha\,x}$ , where

    $\displaystyle \alpha^2 = 2\,m\,(V-E)/\hbar^2.
$

    If $ x=2\times 10^{-10}\,{\rm m}$ and $ V-E = 1\,{\rm eV}$ show that $ {\rm e}^{-2\,\alpha\,x}$ is approximately equal to $ 10^{-1}$ for an electron, and $ 10^{-38}$ for a proton. [Modified from Pain 1999.]

next up previous
Next: Physical Constants Up: Wave Mechanics Previous: Square Potential Barrier
Richard Fitzpatrick 2013-04-08