Yukawa potential sacttering in Born approx.
Department of
Physics and Astronomy
Differential
elastic scattering
cross section in Born
approximation: Yukawa
potential
Below we show a plot of the differential
elastic scattering cross section in Born approxi
Intermediate Quantum Mechanics
Spring 2017
Solutions to Homework 8
1) Calculate the following commutators:
x, L
y]
[L
+, L
z]
[L
z]
[L2 , L
, L
z]
[L
+, L
]
[L
Solution
x = ypz zpy
L
y = zpx xpz
L
[
xi , xj ] = 0
+ = L
x + iL
y
L
[
pi , pj ] = 0
Intermediate Quantum Mechanics
Spring 2017
Solutions to Homework 3
1) Show that the vector specifying the state of an electron that is spin up in the n
direction is given by:
!
cos 2
|n i =
sin 2
where
nx = sin
ny = 0
nz = cos .
Hint: you might want to u
Intermediate Quantum Mechanics
Solutions to Homework 4
1) A particle is in the state |i with probability distribution in x given by the Gaussian:
1/2
1
2
2
(x)(x) =
ex /2x
2
2x
where (x) is the representation of |i in the position, x, basis.
a) Find an e
Intermediate Quantum Mechanics
Spring 2017
Homework 6
1) If the initial wave packet has a small enough , then we can have:
1
(t)
=
4
h
m
> c
This would seem to violate relativity since initially the probability of finding the particle say
a lightyear away
Intermediate Quantum Mechanics
Spring 2016
Solutions Homework 8
1) In class we discussed expressions for the ground state energy and the Bohr radius of the
hydrogen atom in terms of the constants: h
, c, and m. We can form muonic hydrogen by
binding a mu
Intermediate Quantum Mechanics
Spring 2017
Solutions to Homework 1
1) For the following objects at room temperature, T 300 K, determine the approximate
distance scale at which effects due to quantum mechanics would become manifest.
a soccer ball: m 0.1 k
Intermediate Quantum Mechanics
Spring 2016
Solutions to Homework 2
1) Prove that the trace of an operator, the sum of the diagonal elements of the matrix,
is independent of the basis chosen to represent the operator. In other words, for the
unprimed basis
Intermediate Quantum Mechanics
Spring 2017
Solutions Homework 10
1) For the spin system of two electrons we have for the total angular momentum:
Jz = S1z + S2z
J2 =
J+ = S1+ + S2+
J = S1 + S2
S1 + S2 S1 + S2 = Jz2 + h
Jz + J+ J
h
h
i
and
S2i = i
S1i =
2
Intermediate Quantum Mechanics
Spring 2017
Solutions Homework 7
a
a
1) Evaluate the commutation relations [H,
] and [H,
].
Solution
a
[H,
] = [(
h
a a
+h
/2), a
] = h
[
a a
, a
] = h
(
a a
a
a
a
a
)
h
h
i
i
= h
a
a
a
[
a, a
] + a
a
a
= h
Phase shift analysis: elastic N-N scattering
Department of
Physics and Astronomy
Phase shift analysis of measured elastic proton and neutron scattering cross
sections
Below we show measured data (Nijmegen 1993) for proton-proton and neutron-proton
elastic
Intermediate Quantum Mechanics
Spring 2017
Homework 5
1) An electron in the state of spin-up in the x-direction | i is placed in a region with a
uniform magnetic field, B, in the z-direction. After a period of time equal to T /8 = /4
where = eB/m:
a) what