Quantum Mechanics I: Phys. 486
Discussion, Week 13
based on Griffiths, Prob. 6.21
Consider the 8 n=2 states in the hydrogen atom. We would like to calculate the weak-field
Zeeman effect for these states. As you recall, in this case, the energy of the atom
Quantum Mechanics I: Phys. 486
Discussion, Week 4
Griffiths, Prob. 2.29
Analyze the odd bound state wave functions for the finite square well.
a) Derive the transcendental equation for the allowed energies, and solve it graphically and then
with your calc
Let us consider a quantum particle trapped in the one-dimensional infinite quantum well described by the
potential:
= cfw_ 0,
oo,
0 < x <a,
otherwise
(1)
where a is a real positive constant.
(a) (2pts) From the Schrodinger equation, -'ilifJ'IJ(x, t)jat
in
-~-~-~-
A hydrogen atom has the initial state at t
1
=0
v'7 [12, 1' 0) -
213, 2, 1) + i 13, 1' 1) + 14, 2, 0) J
(a) (5pts) At t = 0, L 2 is measured. Find the possible outcomes and their associated probabilities.
L)./ n, xJVI)
~11/~
.J-
(o.fv~.:.
-1
f)
4
Phys. 486 Discussion, Week 13
Di Zhou
April 10, 2015
(a) The unperturbed n = 2 states energy is E2 = 13.6 eV/22 = 3.4eV.
(b) The weak-field Zeeman effect is the case Bext Bint (where fine structure
dominates); the good quantum numbers are n, l, j, mj .
(c
QtSG3.
A particle of mass m moving in a harmonic oscillator potential (V = !mw 2 x 2 ) has the initial wave function
w(x ,O) =
where~=
Jmw/nx.
mw)
( -7rft
1 4
2
1 -ee-~
2
;z,
v'3
Note that w(.T,O) is properly normalized.
(a) (4pts) Let <I>(.T) b e a solut
Quantum Mechanics I: Phys. 486
Discussion, Week 12
based on Griffiths, Prob. 6.29
In this problem, we will investigate so-called finite size corrections for the hydrogen atom, i.e.,
the general effect of the non-zero size of the nucleus on atomic energy l
For parts (a) - (d), consider two observables A and B, represented as
(
)
(
)
4 3
2 0
A=
, B=
.
3 4
0 2
( )
( )
1
1
3
1
with eigenvalue +5, and | i =
with eigenvalue 5.
Eigenvectors of A are |+ i =
1
3
( 10
)
( ) 10
1
0
Eigenvectors of B are |+ i =
with
td\
The time-independent wave functio of a quantum particle is given in Fig. 1.
Im 7/;(x)
h
X
a
X
-a
a
Figure 1: The real (left) and imaginary (right) part of the wave function 'lj!(x). Here, a and hare a positive
constant.
(a) (4pts) The wave function in
Quantum Mechanics I: Phys. 486
Discussion, Week 6
based on Griffiths, Prob. 3.27
An operator A, representing observable A, has two eigenstates,
and a 2 , respectively. Similarly, operator
eigenstates,
(A
a) Show that
lj/1
and lf/2 , with eigenstates a 1
B
4%
Note Title 10/24/2015
1. For quantum mechanics problems with spherically symmetric potentials, the time independent
Schrodinger equation expression includes the operator I.2
2
L2- :12 .1 abwa 1 6
5111663 63 sin 669352
a] Show explicitly that the
Quantum Mechanics I: Phys. 486
Discussion, Week 1
based on Griffiths, Prob. 6.1
Suppose we put a delta-function bump in the center of an infinite square well
a
H x
2
where is a constant.
a) Find the first order perturbation to the allowed energies. Expla
A particle in a 3D harmonic potential V =
~mw 2 r 2
has the wavefunction
1/J(f') = Axy e-a
where r 2 = x 2
+ y 2 + z2
2
2
r ,
and n is some constant.
(a) (2pts) Write the wave function in spherical coordinates (x
You may find these useful:
cos(2x) = cos 2
P470
Chapter 3 Building Hadrons from Quarks
Mesons in SU(2) We are now ready to consider mesons and baryons constructed from quarks. As we recall, mesons are made of quark-antiquark pair and baryons are made of three quarks. Consider mesons made of u and