6.4 Ground State of a System of Bosons

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 $\pp111/{\skew0\vec r}///$ of lowest energy. That makes the system energy eigenfunction for spinless bosons equal to:

\begin{displaymath}
\psi_{\rm gs,\ bosons} =
\pp111/{\skew0\vec r}_1/// \tim...
...r}_2/// \times
\ldots \times
\pp111/{\skew0\vec r}_I/// %
\end{displaymath} (6.8)

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 $I{\vphantom' E}^{\rm p}_{111}$, the number of bosons times the single-particle ground state energy.

Figure 6.2: Ground state of a system of noninteracting bosons in a box.
\begin{figure}
\centering
\setlength{\unitlength}{1pt}
\begin{picture}(...
...35){\makebox(0,0)[r]{${\vphantom' E}^{\rm p}$}}
\end{picture}
\end{figure}

Graphically, the single-particle ground state $\pp111////$ 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 $I$ 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 $\pp111////$ has occupation number $I$, while all other single-particle states have zero.

Note that for a macroscopic system, $I$ will be a humongous number. Even a millimol of particles means well over 10$\POW9,{20}$ particles. Bosons in their ground state are very unfair to the single-particle states: $\pp111////$ gets all of them, the rest gets nothing.


Key Points
$\begin{picture}(15,5.5)(0,-3)
\put(2,0){\makebox(0,0){\scriptsize\bf0}}
\put(12...
...\thicklines \put(3,0){\line(1,0){12}}\put(11.5,-2){\line(1,0){3}}
\end{picture}$
For a system of bosons in the ground state, every boson is in the single particle state of lowest energy.