The ground state for a system of noninteracting spinless bosons is
simple. The ground state is defined as the state of lowest energy, so
every boson has to be in the single-particle state of
lowest energy. That makes the system energy eigenfunction for
spinless bosons equal to:
If the bosons have spin, this is additionally multiplied by an arbitrary combination of spin states. That does not change the system energy. The system energy either way is , the number of bosons times the single-particle ground state energy.
Graphically, the single-particle ground state is the point closest to the origin in wave number space. It is shown as a fat blue dot in figure 6.2 to indicate that all bosons are bunched together in that state.
Physicists like to talk about “occupation numbers.” The occupation number of a single-particle state is simply the number of particles in that state. In particular, for the ground state of the system of noninteracting spinless bosons above, the single-particle state has occupation number , while all other single-particle states have zero.
Note that for a macroscopic system, will be a humongous number. Even a millimol of particles means well over 10 particles. Bosons in their ground state are very unfair to the single-particle states: gets all of them, the rest gets nothing.
Key Points
- For a system of bosons in the ground state, every boson is in the single particle state of lowest energy.