Exercises

1. Use the standard power law expansions,

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

   $$\sin x = x - \frac{x^{\,3}}{3!} + \frac{x^{\,5}}{5!}-\frac{x^{\,7}}{7!}+\cdots,$$

   $$\cos x =1-\frac{x^{\,2}}{2!} + \frac{x^{\,4}}{4!}-\frac{x^{\,6}}{6!}+\cdots,$$

   which are valid for complex $x$, to prove Euler's theorem,
   $${\rm e}^{\,{\rm i}\,\theta} = \cos\theta + {\rm i}\,\sin\theta,$$
   where $\theta$ is real.
2. Equations (8.27) and (8.28) can be combined with Euler's theorem to give
   $$\delta(k) = \frac{1}{2\pi}\int_{-\infty}^\infty {\rm e}^{\,{\rm i}\,k\,x}\,dx,$$
   where $\delta (k)$ is a Dirac delta function. Use this result to prove Fourier's theorem; that is, if
   $$f(x) = \int_{-\infty}^{\infty} \bar{f}(k)\,{\rm e}^{\,{\rm i}\,k\,x}\,dk,$$
   then
   $$\bar{f}(k) = \frac{1}{2\pi}\int_{-\infty}^\infty f(x)\,{\rm e}^{-{\rm i}\,k\,x}\,dx.$$

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 aluminum 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 aluminum. [Adapted from Gasiorowicz 1996.]

6. Show that the de Broglie wavelength of an electron accelerated across a potential difference $V$ is given by
   $$\lambda = 1.23\times 10^{-9}\,V^{\,-1/2}\,{\rm m},$$
   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
   $$\psi(x,t)= A\,\exp\left[-a\,(m\,x^{\,2}/\hbar+ {\rm i}\,t)\right],$$
   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 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
    $$\psi(x,y,z) = A\,\sin\left(l_x\,\pi\,\frac{x}{a}\right)\sin\left(l_y\,\pi\,\frac{y}{b}\right)\sin\left(l_z\,\pi\,\frac{z}{c}\right)$$
    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, $l_x$, $l_y$, and $l_z$ are positive integers. [From Pain 1999.]

11. Derive Equation (11.89).

12. Consider a particle trapped in the finite potential well whose potential is given by Equation (11.75). 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
    $$\frac{P_{\rm out}}{P_{\rm in}}= \frac{\cos^3 y}{\sin y\,(y + \sin y\,\cos y)},$$
    where $(\lambda-y^{\,2})^{1/2} = y\,\tan y$, 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 (11.105) from Equations (11.101)–(11.104).

14. Derive expression (11.114) from Equations (11.110)–(11.113).

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

16. A stream of particles of mass $m$ and energy $E>0$ encounter a potential step of height $W (<E)$; that is, $U(x)=0$ for $x<0$, and $U(x)=W$ for $x>0$, with the particles incident from $-\infty$. Show that the fraction reflected is
    $$R = \left(\frac{k-q}{k+q}\right)^2,$$
    where $k^{\,2}= (2\,m/\hbar^{\,2})\,E$ and $q^{\,2}= (2\,m/\hbar^{\,2})\,(E-W)$.

17. Consider the half-infinite potential well
    $$U(x) = \left\{\begin{array}{lll} \infty&\mbox{\hspace{1cm}}&x\leq 0\\ -V_0&&0<x<L\\ 0 &&x\geq L \end{array}\right.,$$
    where $V_0> 0$. Demonstrate that the bound-states of a particle of mass $m$ and energy $-V_0<E<0$ satisfy
    $$\tan\left(\sqrt{2\,m\,(V_0+E)}\,\,L/\hbar\right) = - \sqrt{(V_0+E)/(-E)}.$$

18. Given that the number density of free electrons in copper is $8.5\times 10^{28}\,{\rm m}^{-3}$, calculate the Fermi energy in electron volts, and the velocity of an electron whose kinetic energy is equal to the Fermi energy.

19. Obtain an expression for the Fermi energy (in eV) of an electron in a white-dwarf star as a function of the stellar mass (in solar masses). At what mass does the Fermi energy equal the rest mass energy?