Quantum Mechanics Answers

 Using the classical theory, derive the formula ρ(ν)dν = 8πν2 c 3 kBT dν for the blackbody radiation where ρ(ν) is the energy per unit of volume in the frequency interval ν to ν + dν of the blackbody spectrum of a cavity at temperature T.

A container holds 2 moles of a diatomic gas at temperature T = 420K. At this temperature,

rotational motions (but not vibrational motions) of the gas molecules are allowed.

(a) What is the total internal energy of this gas? (2)

(b) A quantity of energy E = 300 Joules is transferred to the gas. What fraction of this added

energy is responsible for increasing the temperature of the gas? (1)

(c) How much did the internal translational energy increase, and thus how much did the temperature

of the gas increase? (4)

(d) Explain how the predictions from particle statistics fail, when describing the heat capacities

of a diatomic ideal gas.

Show that low-frequency limit of Planck’s Law reduces to the Rayleigh-Jeans Law

and in the high-frequency limit reduces to Wien’s Law

A car starting from rest passes two successive milestones A and B that are 250 m apart. It takes 60 s to pass A and another 60 s to pass from A to B. Find the velocities of the car when it passes milestone A and milestone B. Assume that the acceleration of the car is uniform 

. (a) Find the change in wavelength of 80-pm x-rays that are

scattered 120° by a target. (b) Find the angle between the directions of the recoil electron and the incident photon. (c) Find

the energy of the recoil electron.

The position of a stone dropped from a cliff is described approximately by x = 5t^2 , where x is in meters and t is in seconds. The +x direction is downwards and the origin is at the top of the cliff. Find the velocity of the stone during its fall as a function of time t.

Show that if a generalized coordinate is cyclic then corresponding component of a generalized momenta is a constant of motion.