130_notes.dvi

1.17 Identical Particles

Identical particles present us with another symmetry in nature. Electrons, for example, are indis-
tinguishable from each other so we must have a symmetry of the Hamiltonian under interchange
(See section 12.4) of any pair of electrons. Lets call the operatorthat interchanges electron-1 and
electron-2P 12.

[H,P 12 ] = 0

So we can make our energy eigenstates also eigenstates ofP 12. Its easy to see (by operating on
an eigenstate twice withP 12 ), that the possible eigenvalues are±1. It is a law of physics that
spin^12 particles calledfermions(like electrons) always areantisymmetric under interchange,
while particles withinteger spin called bosons(like photons) always aresymmetric under
interchange. Antisymmetry under interchange leads to the Pauli exclusion principle that no two
electrons (for example) can be in the same state.

1.18 Some 3D Problems Separable in Cartesian Coordinates

We begin our study of Quantum Mechanics in 3 dimensions with a few simple cases of problems that
can be separated in Cartesian coordinates (See section 13). This ispossible when the Hamiltonian
can be written

H=Hx+Hy+Hz.

One nice example of separation of variable in Cartesian coordinates isthe3D harmonic oscillator

V(r) =

1

2

mω^2 r^2

which has energies which depend on three quantum numbers.

Enxnynz=

(

nx+ny+nz+

3

2

)

̄hω

It really behaves like 3 independent one dimensional harmonic oscillators.

Another problem that separates is theparticle in a 3D box. Again, energies depend on three
quantum numbers

Enxnynz=

π^2 ̄h^2

2 mL^2

(

n^2 x+n^2 y+n^2 z

)

for a cubic box of sideL. We investigate the effect of the Pauli exclusion principle by filling our 3D
box with identical fermions which must all be in different states. We can use this to model White
Dwarfs or Neutron Stars.

In classical physics, it takes three coordinates to give the locationof a particle in 3D. In quantum
mechanics, we are finding that it takesthree quantum numbersto label and energy eigenstate
(not including spin).