Quantum

energy per oscillator

U=k_BT

Boltzmann distribution

a probability distribution that gives the probability of a certain state as a function of that state’s energy and temp of the system to which distribution is applied

oscillation N_j

Total energy is then

average energy is then

if atom is considered a crystal-3 directions of motion- 3 oscillators

total energy per mole

plancks distribution and einstein heat capacity frequency oscillator population

wave particle duality

• electrons through a double slit at significant rates.

• constructive interference giving maxima and destructive interference giving minima.

• electrons behave like waves

• single slit-electrons act like particles