Next: Exercises (N.B. Neglect spin
Up: MultiParticle Systems
Previous: TwoParticle Systems
Identical Particles
Consider a system consisting of two identical particles of mass .
As before, the instantaneous state of the system is specified by the
complex wavefunction
. However, the only thing
which this wavefunction tells us is that the probability of finding the
first particle between and , and the second
between and , at time is
. However, since the particles are
identical, this must be the same as the probability of finding the
first particle between and , and the second
between and , at time (since, in both
cases, the physical outcome of the measurement is exactly the same).
Hence, we conclude that

(451) 
or

(452) 
where is a real constant. However, if we swap the labels on
particles 1 and 2 (which are, after all, arbitrary for identical particles), and repeat the argument, we also conclude that

(453) 
Hence,

(454) 
The only solutions to the above equation are and .
Thus, we infer that for a system consisting of two identical particles, the wavefunction
must be either symmetric or antisymmetric under interchange
of particle labels: i.e., either

(455) 
or

(456) 
The above argument can easily be extended to systems containing more
than two identical particles.
It turns out that whether the wavefunction of a
system containing many identical particles is symmetric or antisymmetric
under interchange of the labels on any two particles is determined by the nature
of the particles themselves. Particles with wavefunctions which are symmetric under
label interchange
are said to obey BoseEinstein statistics, and
are called bosonsfor instance, photons are bosons. Particles with
wavefunctions which are antisymmetric
under label interchange are said to obey FermiDirac
statistics, and are called fermionsfor instance, electrons,
protons, and neutrons are fermions.
Consider a system containing two identical and noninteracting bosons.
Let be a properly normalized, singleparticle, stationary wavefunction corresponding to a state of definite energy .
The stationary wavefunction of the
whole system is written

(457) 
when the energies of the two particles are and . This
expression automatically satisfies the symmetry requirement on the
wavefunction. Incidentally, since the particles are
identical, we cannot be sure which particle
has energy , and which has energy only that one particle
has energy , and the other .
For a system consisting of two identical and noninteracting fermions,
the stationary wavefunction of the whole system takes
the form

(458) 
Again, this expression automatically satisfies the symmetry requirement on
the wavefunction. Note that if then the total wavefunction
becomes zero everywhere. Now, in quantum mechanics, a null wavefunction
corresponds to the absence of a state. We thus conclude that it
is impossible for the two fermions in our system to occupy the
same singleparticle stationary state.
Finally, if the two particles are somehow distinguishable then the stationary
wavefunction of the system is simply

(459) 
Let us evaluate the variance of the distance, , between the
two particles, using the above three wavefunctions. It is easily
demonstrated that if the particles are distinguishable then

(460) 
where

(461) 
For the case of two identical bosons, we find

(462) 
where

(463) 
Here, we have assumed that , so that

(464) 
Finally, for the case of two identical fermions, we obtain

(465) 
Equation (462) shows that
the symmetry requirement on the total wavefunction of two identical bosons forces the
particles to be, on average, closer together than two similar distinguishable
particles. Conversely, Eq. (465) shows that the symmetry requirement on the total wavefunction of two identical fermions forces the
particles to be, on average, further apart than two similar distinguishable
particles. However, the strength of this effect depends on square of the magnitude of
, which measures the overlap between the
wavefunctions and . It is evident, then, that
if these two wavefunctions do not overlap to any great extent then identical
bosons or fermions will act very much like distinguishable particles.
For a system containing identical and noninteracting fermions,
the antisymmetric stationary wavefunction of the system
is written
This expression is known as the Slater determinant, and automatically
satisfies the symmetry requirements on the wavefunction.
Here, the energies of the particles are
.
Note, again, that if any two particles in the system have the same energy
(i.e., if for some )
then the total wavefunction is null. We conclude that it is impossible
for any two identical fermions in a multiparticle system to
occupy the same singleparticle stationary state. This important result is known as
the Pauli exclusion principle.
Subsections
Next: Exercises (N.B. Neglect spin
Up: MultiParticle Systems
Previous: TwoParticle Systems
Richard Fitzpatrick
20100720