1. a. Calculate the ground state energy of a proton in an infinite-walled
box in 3 dimensions

with side L= 10 fermi.

E1 for 3-D box = 3h^2/8mL^2=3(hc)^2/(8mc^2L^2)

E1=3*(1240 MeV fm)^2/(8*938 MeV*(10 fm)^2) = 6.147
MeV

b. Calculate the ground state energy of an electron in the same box.

E1=3*(1240 MeV fm)^2/(8*.511 MeV*(10 fm)^2) = 11283.8
MeV

2. For an electron contained in a one-dimensional box of dimension L=.3 nm:

a. Calculate the ground state energy if the walls are infinite.

E1=(hc)^2/(8*mc^2*L^2)=(1240 eV nm)^2/(8*.511 MeV*(.3 nm)^2)=

E1= 4.179 eV

b. Calculate the ground state energy if the walls are 100 eV high. This
involves the calculation

of a new effective dimension L by calculating the distance at which
the wavefunction drops to

1/e times its value inside the box.

Wavefuntion in barrier drops exponentially according to exp(-ax)

a=sqrt((2*mc^2/(hbar*c)^2)*(U-E1))=sqrt((2*.511MeV)/(197 eV nm)^2)*(100-4.179))

a= 50.233 1/nm

add to each side of box the amount 1/a = 0.020
nm

New box width 0.340 nm

New ground state energy = (1240 eV nm)^2/(8*.511 MeV*(.34 nm)^2)=

New E1 = 3.254 eV

3. The calculation of tunneling probability for an alpha particle out
of a nucleus involves

approximating the coulomb barrier with a series of barriers of different
height. An alpha

particle has energy 8 MeV inside the nucleus. Approximate the barrier
with two

rectangular barriers of width 5 fm, the first at height 20 MeV and
the second at height

10 MeV. Calculate the tunneling probabability for the alpha particle.
The mass energy

of an alpha particle is 3727 MeV.

The wavefunction drops off exponentially in the barrier.

exp(-sqrt((2*mc^2)/(hbar*c)^2*(U-E))*x)

For the first barrier:

exp(-sqrt((2*3727 MeV)/(197 MeV fm)^2*(20-8)MeV)*x)=
5.05E-04

The probability is the square of the ratio of wavefunctions, so prob
= 2.55E-07

For the second barrier:

exp(-sqrt((2*3727 MeV)/(197 MeV fm)^2*(10-8)MeV)*x)=
4.51E-02

The probability is the square of the ratio of wavefunctions, so prob
= 2.03E-03

The product of the tunneling probabilities is 5.19E-10