particles with energy is associated with a particular set of states of the gas molecules as represented in the joint dimensional space of the positions and momenta of the gas molecules. In general, the entropy of a macrostate is proportional to the natural logarithm of how many microstates can give rise to the given macrostate, i.e. where is the "volume" of the microstates in the phase space.In classical statistical mechanics, states of a system are distinguished by the values of their extensive variables, i.e. energy, volume, etc. Different values of the extensive variables carry over into different arrangements of the microscopic states of the system. For example, an ideal gas of
However, without a natural unit with which to measure the "volume", the value of the entropy is arbitrary up to an additive constant. We can fix this by exploring quantum analogues to classical systems.
Consider the particle in a well, a quantum particle that is trapped in a 1d line of length . The Hamiltonian (total energy) of the particle is given by its kinetic energy, , where is the momentum operator. The allowed energies for the particle are given by where is Planck's constant divided by , is the mass of the quantum particle, and is a positive integer.
Pretend that is a continuous variable and assume that the position of the particle in the well varies from to . Calculate the volume of the phase space (the space of and ) associated with particles of energy level or lower.
This gives a classical notion of the volume of the phase space occupied by particles of energy level or below. Next, calculate the number of states of energy or lower.
Divide this classical "volume" of phase space by the number of states of energy less than or equal to to find the "volume" of phase space that is occupied by each eigenstate of the particle. Express your answer in multiples of .
This gives a value to the smallest division of phase space.