Quantal recurrence in the innite square well
a. Classical period:
E = 1 mv 2
2
so
v=
2E/m
and
distance = speed time,
so
period =
distance
=
speed
2L
2E/m
= L 2m/E.
b. Quantal recurrence:
How does the initial wavefunction (x; 0) change with time? Expanded
Mean separation
According to Griths equations [5.19] and [5.21], the mean square separations (xA xB )2 are
for nonidentical particles:
for identical bosons/fermions:
x2 n + x2 m 2 x n x
2
the above 2  mxn 
m
(Surprisingly, none of the integrals on th
Variational estimate for a quartic oscillator
a. The trial wavefunction
2
2
1
(x) = ex /2
4
has

=
=
=
+
2
2
1
ex / dx
+
1
x2
e d
x
1.
b. This trial wavefunction is probably not an energy eigenstate, nevertheless, we can calculate an expected
energ
The Stark Eect
a. Evaluate a matrix element.
200H 210
=
eE 200z210
=
eE 200r cos 210
=
eE
2
r2 dr
d R20 (r)Y00 (, ) r cos R21 (r)Y10 (, )
sin d
0
0
0
Angular part use Griths page 139 for Y m (, )
2
0
d Y00 (, ) cos Y10 (, )
sin d
=
2
sin d
0
0
3
cos
Scaling in the stadium problem
Famed mathematician George Plya coined the term the inventors paradox in his book How to Solve
o
It The more ambitious plan may have more chances of success. I like to phrase this principle as It can
be easier to prove a mor
The WKB Approximation
Griths problem 8.2: Alternative derivation of WKB
(a)
(x) =
d
=
dx
2
d
=
dx2
h
eif (x)/
i
h
f eif /
h
i
1
h
h
f eif / 2 f 2 eif /
h
h
So the Schrdinger equation is
o
i
1
p2 (x)
h
h
h
f eif / 2 f 2 eif / = 2 eif /
h
h
h
or
i f f
h
2
Atoms
Dan Styer, Oberlin College Physics; c 27 February 2014
Griths problem 5.12
term
symbol
S
H
He
Li
Be
B
C
N
(1s)
(1s)2
(1s)2 (2s)
(1s)2 (2s)2
(1s)2 (2s)2 (2p)
(1s)2 (2s)2 (2p)2
(1s)2 (2s)2 (2p)3
L
J
1
2
0
0
0
0
1
0, 1, 2
0, 1, 2, 3
1
2
2
0
1
0
1
2
0
1
Addition of Angular Momenta
Griths problem 4.34
(A) (B)
(a) Remember that S = S + S and that
S s, m = h s(s + 1) m(m 1) s, m 1 .
Thus
S =
h
1 3
2(2)
1
1 ( 2 ) = h ,
2
= h 1(2) 0(1) 1, 1 = h 2 1, 1 .
S 1, 0
So
1, 0
=
S 1, 0
=
=
=
=
1
[ + ]
2
1
(A)
Variational method for nding the ground state energy
Imagine a gymnasium full of fruits
smallest fruit smallest cantaloupe.
Similarly
ground state energy H for any  .
So try out a bunch of states, turn the crank, nd the smallest. Very mechanical.
For e
Quantal recurrence in the Coulomb problem
As in part (b) of the problem Quantal recurrence in the innite square well, we ask how the initial
wavefunction (r; 0) changes with time. In terms of the energy eigenfunctions n (r),
(r; 0) =
cn n (r).
n
This wave
Building basis states
Suppose you had three particles and three building block levels (say the orthonormal levels 1 (x),
3 (x), and 7 (x). Construct and count the possible threeparticle states representing (a) three nonidentical
particles; (b) three ide
Twoelectron ions
Let
Z represent the variational parameter in [7.27].
ZN represent the nuclear charge (1 for H , 3 for Li+ ).
We follow the argument of Griths pages 302303, except:
In equations [7.28] (twice) and [7.29], change (Z 2) to (Z ZN ).
Equati