Ph135c. Solution set #8, 5/27/10
1.
(a) Proceeding as in last weeks HW, we can write:
=
=
where x0 =
gives:
me ,
GF 2 (gV 2 + 3gA 2 )
2 2
Ee 2 me 2 ( Ee )2
dEe Ee
me
GF 2 (gV 2 + 3gA 2 ) 5
me
2 2
x0
dxx
x2 1(x0 x)2
1
which in this case is about 1.036. Num
Physics 135c
Homework 5
1.) Determine the non-relativistic elastic scattering form factor
F (|q|) =
d3 r(r)eiqr
for the following two densities:
a) (r) = 0 =constant for r R and = 0 for r > R,
b) (r) = 0 er/R
2.) Beginning with the non-relativistic elasti
Physics 135c
Homework 4
1.) The central part of the nucleon-nucleon potential can be written as a sum
of four terms:
V (r) = V0 [W (r) + B (r)P + M (r)Px + H (r)Px P ]
where P is the spin exchange operator and Px is the space coordinate exchange
operator.
Physics 135c
Homework 3
1.) Construct an s-wave scattering length operator a for the n p system such
that the matrix element S M |a|SM for two-nucleon spin states gives as for
spin-singlet states (i.e. S = 0) and at for spin-triplet states (i.e. S = 1) -
Physics 135c
H. W. Assignment 1
1.) Determine the order of energy levels for a very deep (V0 > 2 /ma2 for r > a and
h
V0 = 0 for r a) 3-dimensional potential. Draw an energy level diagram for the rst 10
discrete levels, with the spacing between levels sca
Unification of Nuclear Structure Theory Is Possible
Norman D. Cook
Informatics, Kansai University, Osaka, Japan
Abstract
The impossibility of achieving a unified theory of nuclear structure has been the
conventional wisdom in nuclear physics since the 196
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PHYS 560: Solution 5
George Newman
December 8, 2006
Abstract
The solutions to the fth assignment for Nuclear Physics, PHY560,
Autumn 2006. 1 Question 1: Wong 4.11
Obtain the masses of members of the A = 135 isobar from a table of binding
energies and
PHYS 560: Assignment 6 : SOLUTIONS
Martin J. Savage
December 18, 2009
Abstract
The sixth assignment for Nuclear Physics, PHY560, Autumn
2009.
December 2009
1
Assignment 6 : PHYS 560
Due : Dec 14
1. Wong : 6.1: We wish to construct the allowed values of to
Parity Violation in Beta Decays
The Parity operator is the mirror image and is NOT conserved in Weak decays (is conserved in EM and strong) P( x, y , z ) ( x, y , z )
P ( r , , ) ( r , , + )
non-conservation is on the lepton side, not the nuclear wave f
Physics 135c
H. W. Assignment 6
1.) The semi-empirical mass formula (with coecients in MeV) is
(A 2Z ) 2
Z (Z 1)
23
+
A
A
A 1 /3
with = +25 MeV for even-even nuclei, = 0 MeV for odd-even nuclei and = 25
MeV for odd-odd nuclei. Use this formula to calcula
Physics 135c
H. W. Assignment 2
1.) Show that the deuteron binding energy (Eb = 2.22 MeV) and n p triplet scattering
length (at = +5.4 fm) can be reproduced by a simple square well [V (r ) = V0 , r
r0 , V (r ) = 0, r > r0 ] by determining the parameters
Physics 135c
H. W. Assignment 7
1.) Bertulani Probs. 8.7, 8.8 & 8.9
2.) For this problem you will use Fermis Golden Rule
d =
2
2
|Af i | f
h
to calculate the energy spectrum of electrons emitted in neutron -decay n p + e + e .
Ignoring the proton recoil a
Ph135c. Solution set #7, 5/20/10
1.
8.7
We can write:
d = 2 |Af i |2 (Ee + E )
=
d3 pe d3 p
(2 )3 (2 )3
1
|Af i |2 (Ee + E )pe 2 dpe de p 2 dp d
(2 )5
where = mn mp . To get the electron momentum distribution, we integrate over all variables
except pe . T
Ph135c. Solution set #6, 5/13/10
1.
We write the semi-empirical mass formula as:
Eb = aA bA2/3 c
(A 2Z )2
Z (Z 1)
d
+
A
A
A1/3
First, we maximize this with respect to Z :
0=
Eb
4(A 2Z )
2Z 1
= c 1/3 + d
Z
A
A
= Z
2c
8d
+
1/3
A
A
Z=
+
c
A1 / 3
+ 4d
cA2/3 +
Ph135c. Solution set #4, 4/29/10
1.
This problem amounts to determining the action of P and Px on the various angular momentum
states. The spin exchange operator acts on the singlet as 1 and on the triplet as +1, since these states
are anti-symmetric and
Ph135c. Solution set #3, 4/22/10
1.
First recall that:
n p =
ss
12
(J 2 2 )
sn sp
2
3
Since 2 = 2 = s(s + 1) = 4 (since the proton and neutron are spin- 1 ), and since J 2 = j (j + 1)
sn sp
2
is zero for the singlet and 2 for the triplet, we nd:
3
4
n p =
Ph135c. Solution set #2, 4/15/10
1.
First consider the bound state. As shown in the book, its energy is given by solving the following
transcendental equation:
K cot(Kro ) = k
where:
2(Vo EB )
K=
k=
2EB
here is the reduced mass for the proton-neutron syst
Ph135c. Solution set #1, 4/9/10
1.
We can approximate this potential by an innite radial square well (taking care to shift the energies
by V0 at the end):
0 r<a
r>a
V (r) =
We look for a solution of Schrodingers equation in the form:
k
m (r, , )
=
uk (r)
Physics 135c
H. W. Assignment 9
1.) Bertulani Probs. 12.2, 12.3, 12.4 & 12.8
2.) In class we derived two-component neutrino oscillations. Perform a more rigorous
derivation by using a phase dierence between the two components of ( = c = 1):
h
= (E2 E1 )t
Physics 135c
H. W. Assignment 8
1.) (a.) Use the procedure from HW#7 to calculate the mean lifetime of tritium for
-decay: 3 H 3 He + e + e (i.e. assume massless neutrinos and ignore nuclear recoil
and Coulomb distortion). The energy released in the deca
Ph135c. Solution set #9, 6/3/10
1.
12.2 In the iron core, the number of protons is:
Np =
Z 1.4Msun
7.7 1056
Z +N
mp
Each of these is converted into a neutron and neutrino by a process which consumes an energy of
E mn mp me 0.78 M eV , for a total energy
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 13: Solutions
1. In this problem you will derive the 22 matrix representations of the three spin observables from
rst principles:
(a) In the basis cfw_| z , | z , the matrix representation of Sz i
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 13
Topics Covered: Spin
Please note that the physics of spin-1/2 particles will gure heavily in both the nal exam for 851, as well
as the QM subject exam.
Spin-1/2: The Hilbert space of a spin-1/2
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 5
1. In problem 4.3, we used a change of variables to map the equations of motion for a sinusoidally driven
two-level system onto the time-independent Rabi model. Here we will investigate how this
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 5
1. In problem 4.3, we used a change of variables to map the equations of motion for a sinusoidally driven
two-level system onto the time-independent Rabi model. Here we will investigate how this
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 4: Solutions
1. The 2-Level Rabi Model: The standard Rabi Model consists of a bare Hamiltonian H0 =
and a coupling term V = |1 2| + |2 1|.
2
2
2
(|2 2| |1 1|)
(a) What is the energy, degeneracy, a
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 4
1. The 2-Level Rabi Model: The standard Rabi Model consists of a bare Hamiltonian H0 =
and a coupling term V = |1 2| + |2 1|.
2
2
2
(|2 2| |1 1|)
(a) What is the energy, degeneracy, and state ve
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 3:
Fundamentals of Quantum Mechanics
1. [10pts] The trace of an operator is dened as T r cfw_A =
set.
m
m|A|m , where cfw_|m is a suitable basis
(a) Prove that the trace is independent of the cho