Question

1. For what kinetic energy is the de Broglie wavelength of an electron equal to its Compton wavelength? Express your answer in units of mc2 in doing the calculation, and then use mc2 = 0.5 MeV.(Answer: 0.2 MeV)

2. A beam of electrons with energy 1.0 eV approaches a potential barrier with U = 2.0 eV, whose width is 0.10 nm (see a figure below). Estimate the fraction of electrons that tunnel through the barrier. (Hint: use the relation of probability and the wave function, and the expression for wave function for barrier penetration)

3. Calculate the Fermi energy of sodium (n= 2.65·10^28 m^-3). This metal has a single valence electron per atom.(Answer: 3.25 eV.)

Answer 1

1. Compton wavelength has the value 0.00243 nm = 2.43 x 10^-12 m. For a electron of mass m, kinetic energy E, the momentum of the electron will be

De Broglie hypothesis says that all matter has both particle and wave nature. The wave nature of a particle is quantified by de Broglie wavelength defined as λ=h/p, where p is the momentum of the particle. For this problem

2. We can express the rate of escape as: (In the semiclassical limit (finely spaced energy levels compared to the barrier height)

3. The free electron density of Sodium is n= 2.65 x 10²⁸ m^-3. Therefore the Fermi energy will be