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# Exercises

1. Use the standard power law expansions,

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

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

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

then

3. A He-Ne laser emits radiation of wavelength . How many photons are emitted per second by a laser with a power of ? 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 . 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 for radiation of wavelength , and for radiation of wavelength . Use this data to calculate Planck's constant (divided by ) 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 is given by

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

7. If the atoms in a regular crystal are separated by demonstrate that an accelerating voltage of about would be required to produce a useful electron diffraction pattern from the crystal. [Modified from Pain 1999.]

8. A particle of mass has a wavefunction

where and are positive real constants. For what potential does satisfy Schrödinger's equation?

9. Show that the wavefunction of a particle of mass trapped in a one-dimensional square potential well of of width , and infinite depth, returns to its original form after a quantum revival time .

10. Show that the normalization constant for the stationary wavefunction

describing an electron trapped in a three-dimensional rectangular potential well of dimensions , , , and infinite depth, is . Here, , , and 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

where , and . Hence, demonstrate that for a shallow well (i.e., ) , whereas for a deep well (i.e., ) (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 and energy , incident on a square potential barrier of height , and width , is

where and . Demonstrate that the coefficient of transmission is unity (i.e., there is no reflection from the barrier) when , where is positive integer.

15. The probability of a particle of mass penetrating a distance into a classically forbidden region is proportional to , where

If and show that is approximately equal to for an electron, and for a proton. [Modified from Pain 1999.]

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