Problems

1. A He-Ne laser emits radiation of wavelength $\lambda = 633$ nm. How many photons are emitted per second by a laser with a power of 1 mW? What force does such laser exert on a body which completely absorbs its radiation? [modified from Squires]

2. The ionization energy of the hydrogen atom in its ground state is $E_{ion} = 13.60$ eV (1 eV is the energy acquired by an electron accelerated through a potential difference of 1 V). Calculate the frequency, wavelength, and wave-number of the electromagnetic radiation which will just ionize the atom. [from Squires]

3. The maximum energy of photoelectrons from aluminium is 2.3 eV for radiation of wavelength $2000\,\AA$, and 0.90 eV for radiation of wavelength $2580\,\AA$. Use this data to calculate Planck's constant, and the work function of aluminium. [from Gasiorowicz]

4. The relationship between wavelength and frequency for electromagnetic waves in a waveguide is
   
   $$\lambda = \frac{c}{\sqrt{\nu^2 - \nu_0^{\,2}}}.$$
   
   
   What are the group and phase velocities of such waves as functions of $\nu_0$ and $\lambda$? [modified from Gasiorowicz]

5. Nuclei, typically of size $10^{-14}$ m, frequently emit electrons with energies of 1-10 MeV. Use the uncertainty principle to show that electrons of energy 1 MeV could not be contained in the nucleus before the decay. [from Gasiorowicz]

6. A particle of mass $m$ has a wavefunction
   
   $$\psi(x,t) = A\,\exp[-a\,(m\,x^2/\hbar + {\rm i}\, t)],$$
   
   
   where $A$ and $a$ are positive real constants. For what potential function $V(x)$ does $\psi$ satisfy the Schrödinger equation? [from Griffiths]