130_notes.dvi

from the electrons grows faster than the pressure of gravity, the star will stay at about
earth size even when it cools.

If the star is more massive, the Fermi energy goes up and it becomes possible to absorb the electrons
into the nucleons, converting protons into neutrons. The Fermi energy needs to be above 1 MeV.
If the electrons disappear this way, the star collapses suddenly down to a size for which the Fermi
pressure of the neutrons stops the collapse (with quite a shock). Actually some white dwarfs stay
at earth size for a long time as they suck in mass from their surroundings. When they have just
enough, they collapse forming a neutron star and making a supernova. The supernovae are all nearly
identical since the dwarfs are gaining mass very slowly. The brightness of this type of supernova has
been used tomeasure the accelerating expansion of the universe.

We can estimate theneutron star radius.

R→R

MN

me

N

(^13)
2 −
(^53)
= 10
Its about 10 kilometers. If the pressure at the center of a neutron star becomes too great, it collapses
to become a black hole. This collapse is probably brought about by general relativistic effects, aided
by strange quarks.

13.2 The 3D Harmonic Oscillator

The 3D harmonic oscillator can also be separated in Cartesian coordinates. For the case of a
central potential,V=^12 mω^2 r^2 , this problem can also be solved nicely in spherical coordinates using
rotational symmetry. The cartesian solution is easier and better for counting states though.

Letsassume the central potentialso we can compare to our later solution. We could have three
different spring constants and the solution would be as simple. The Hamiltonian is

H =

p^2

2 m

+

1

2

mω^2 r^2

H =

p^2 x

2 m

+

1

2

mω^2 x^2 +

p^2 y

2 m

+

1

2

mω^2 y^2 +

p^2 z

2 m

+

1

2

mω^2 z^2

H = Hx+Hy+Hz

The problem separates nicely, giving usthree independent harmonic oscillators.

E=

(

nx+ny+nz+

3

2

)

̄hω

ψnx,ny,nz(x,y,z) =unx(x)uny(y)unz(z)

This was really easy.

This problem has a different Fermi surface inn-space than did the particle in a box. The boundary
between filled and unfilled energy levels is a plane defined by

nx+ny+nz=

EF

̄hω

−

3

2