Quantum Real Estate

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 NN particles with energy EE is associated with a particular set of states of the gas molecules as represented in the joint 6N6N 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. SlnΓS\sim\ln \Gamma where Γ\Gamma is the "volume" of the microstates in the phase space.

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 LL. The Hamiltonian (total energy) of the particle is given by its kinetic energy, H^=P^22m\hat{H} = \frac{\hat{P}^2}{2m}, where P^\hat{P} is the momentum operator. The allowed energies for the particle are given by En=n22π22mL2E_n = \frac{n^2\hbar^2\pi^2}{2mL^2} where \hbar is Planck's constant divided by 2π2\pi, mm is the mass of the quantum particle, and nn is a positive integer.

Pretend that P^\hat{P} is a continuous variable and assume that the position xx of the particle in the well varies from 00 to LL. Calculate the volume of the phase space (the space of xx and pp) associated with particles of energy level nn or lower.

i.e. calculate Vol(En)=EEndxdp\mbox{Vol}\left(E_n\right) = \displaystyle\int_{E\leq E_n}dxdp

This gives a classical notion of the volume of the phase space occupied by particles of energy level nn or below. Next, calculate the number of states of energy EnE_n or lower.

Divide this classical "volume" of phase space by the number of states of energy less than or equal to EnE_n to find the "volume" of phase space that is occupied by each eigenstate of the particle. Express your answer in multiples of \hbar.

This gives a value to the smallest division of phase space.

Assumptions

  • You definitely do not need calculus or any deep knowledge of quantum mechanics for this problem.
  • Because this system is one dimensional, the phase space "volume" is two dimensional (xx and pp).
  • Make sure you account for the entire range of values for pp that contribute to the volume.
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