Up to this point, this book has presented energy levels in the form of an energy spectrum. In these spectra, each single-particle energy was shown as a tick mark along the energy axis. The single-particle states with that energy were usually listed next to the tick marks. One example was the energy spectrum of the electron in a hydrogen atom as shown in figure 4.8.
However, the number of states involved in a typical macroscopic system
can easily be of the order of 1
For almost all practical purposes, the energy levels of a macroscopic system of noninteracting particles in a box form a continuum. That is schematically indicated by the hatching in the energy spectrum to the right in figure 6.1. The spacing between energy levels is however very many orders of magnitude tighter than the hatching can indicate.
It can also normally be assumed that the lowest energy is zero for
noninteracting particles in a box. While the lowest single particle
energy is strictly speaking somewhat greater than zero, it is
extremely small. That is numerically illustrated by the values for a
ground state, may have
a relatively more significant energy.
The spacing between the lowest and second lowest energy is comparable to the lowest energy, and similarly negligible. It should be noted, however, that in Bose-Einstein condensation, which is discussed later, there is a macroscopic effect of the finite spacing between the lowest and second-lowest energy states, miniscule as it might be.
The next question is why quantum mechanics is needed here at all. Classical nonquantum physics too would predict a continuum of energies for the particles. And it too would predict the energy to start from zero. The energy of a noninteracting particle is all kinetic energy; classical physics has that zero if the particle is at rest and positive otherwise.
Still, the (anti) symmetrization requirements cannot be accommodated using classical physics. And there is at least one other important quantum effect. Quantum mechanics predicts that there are more single-particle states in a given energy range at high energy than at low energy.
To express that more precisely, physicists define the “density
of states” as the number of single-particle states per unit
energy range. For particles in a box, the density of states is not
that hard to find. First, the number
(It should be noted that for the above expression for
To get the density of states on an energy basis, eliminate
Note that the density of states grows like
The given expression for the density of states is not valid if the
particle speed becomes comparable to the speed of light. In
particular for photons the Planck-Einstein expression for the energy
must be used,
It is conventional to express the density of states for photons on a
frequency basis instead of an energy basis. Replacing
- The spectrum of a macroscopic number of noninteracting particles in a box is practically speaking continuous.
- The lowest single-particle energy can almost always be taken to be zero.
- The density of states
is the number of single-particle states per unit energy range and unit volume.
- More precisely, the number of states in an energy range
- To use this expression, the energy range
should be small. However, should still be large enough that there are a lot of states in the range.
- For photons, use the density of modes.