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Homework # 8, Theoretical Mechanics and Quantum Mechanics: 5010
Due Wed Oct 31
Problem 1. What is the wavelength of a hydrogen atom moving with a velocity corresponding to the mean kinetic energy at thermal equilibrium at 20 degrees Centigrade? [2
Problem 2. If you rewrite the uncertainty relation for a free particle in terms of the
uncertainty of its wavelength ∆λ rather than ∆p how will the uncertainty relation look
like? [2 points]
Problem 3. Consider a virus of size 10 nm. Suppose that its density is roughly equal to
that of water and that the virus is located in a region that is approximately equal to its
size. What is the minimum speed of the virus? [2 points]
Problem 4. Consider a monochromatic beam of light of intensity I (units of J · s−1 · m−2 )
striking a perfectly reflecting surface at 90 degrees to it. Using the concept of photons
calculate the radiation pressure. [2 points]
Problem 5. Photons in a waveguide have a relation between the wavelength λ and frequency
ν different from photons in free space,
λ= q
ν 2 − ν02
where ν0 is some threshold frequency below which propagation of waves in the waveguide is
impossible. Still, Einstein’s relations between energy and frequency (E = hν) and momentum and the wavelength (p = h/λ) are valid.
a) Find the group velocity of the waveguide photons vg (p) and express it via phonon momentum. [1 point]
b) Find the expression of the group velocity via the photon’s phase velocity vp [1 point].