Assignment 5 - PHYS 4010
due: October 19 , 2012
1. Consider the harmonic oscillator Hamiltonian:
2
1
+ m 2 x2
2
2m x
2
It is always a good idea, especially in computational physics, to convert
a problem into dimensionless variables. Dene a new dimensionle

PHYS 4010
QUANTUM MECHANICS
Assignment 7
due November 5, 2012
1. Construct
100 (r), 200 (r), 211 (r), 210 (r), 211 (r)
where the
nlm (r) Rnl (r)Ylm (, )
are the eigenfunctions of H, the hydrogen atom Hamiltonian.
2. Show
100 (r), 200 (r), 211 (r), 210 (r)

PHYS 4010
QUANTUM MECHANICS
Assignment 6
due October 26, 2012
1. Consider a potential barrier of height V0 and width a with its left edge
located at the origin. Assume particles of energy E, are arriving from the
left. Set up and solve the appropriate bou

PHYS 4010 3.0 Addition of angular momentum
In quantum mechanics we sometimes have to add angular momenta. In
the hydrogen atom the electron can have orbital angular momentum L in
addition to its intrinsic spin S. We also have systems of many electrons
whe

PHYS 4010 3.0
Angular momentum
Classically the angular momentum vector L is dened by
L=rp
In quantum mechanics the angular momentum operator is given by
L = i r
Explicitly we can write the components of this operator as
Lx = i
y z z y
Ly = i
z x x z
Lz =

1
QUANTUM MECHANICS
(PHYS4010) LECTURE NOTES
Lecture notes based on a course given by Roman Koniuk.
The course begins with a formal introduction into quantum mechanics and then
moves to solving different quantum systems and entanglement
York University, 2

Assignment 4 - PHYS 4010
due: October 3, 2012
1. A rectangular pulse is described by
1 if |x| < 1
0 if |x| > 1
f (x) =
Find the fourier transform gc () of the pulse. Plot the pulse and its
transform. Increase the pulse width and recalculate and replot the

Assignment 1 - PHYS 4010
due: September 14, 2012
1. Demonstrate by direct substitution that the rst 5 eigenfunctions
of the one-dimensional square-well are indeed eigenfunctions of the
hamiltonian.
2. Plot the rst 5 eigenfunctions n (x)
3. Plot the rst 5

PHYS 4010 3.0
Spin
Spin is the intrinsic angular momentum of a particle. It is not related to
the spatial coordinates of a particle as is orbital angular momentum. There
is no coordinate representation for spin. We will construct a matrix representation.

PHYS 4010 3.0
Tunneling
Tunneling is a quantum mechanical eect, where particles can emerge on
the other side of a potential barrier from which they are classically forbidden.
Consider the potential barrier of height V0 and width 2a pictured below.
This po

Assignment 3 - PHYS 4010
due: September 28, 2012
1. Construct the fourier series expansion for the sawtooth function.
f (x) =
x
x 2
if 0 x <
if < x 2
Plot the sum of the rst N terms, where N = 1, 5, 10, 100, 1000
2. Construct the fourier series expansion

PHYS 4010 3.0
Matrix mechanics
Let the space of wavefunctions be the Hilbert space H . Let a basis set
of functions be B.
B = cfw_1 , 2 , 3 ,
For example
particle in a box : B = cfw_
2/a sin(nx/a)
2
harmonic oscillator : B = cfw_An Hn ()e /2
hydrogen :

Assignment 2 - PHYS 4010
due: September 26, 2012
Particle in a box
Consider the state
1
(x, t) = 1 (x)ei1 t + 2 (x)ei2 t
2
where En = n are the eigenvalues and n (x) are the eigenfunctions.
For the state (x, t):
1. Calculate x .
2. Calculate p .
3. And th

PHYS 4010 3.0
Hydrogen atom
The momentum operator in three dimensions is
p=
i
Therefore the momentum-squared operator is
p2 =
2
2
=
2
1 2
1
1
2
r
+ 2
sin
+ 2 2
r2 r
r
r sin
r sin 2
We notice that this can be written as
p2 = p2 +
r
L2
r2
where
1 2
r
r2

PHYS 4010 3.0
Entanglement
An entangled multi-particle state is by denition is a state that cannot
be represented by a simple product of single-particle states.
We will review some of the properties of a two-particle entangled spinstate. Consider the stat