PHYS851 Quantum Mechanics I, Fall 2009
Semester Outline
1. Dirac Notation
(a) Bras and Kets
i. Hermitian conjugation
(b) Operators
i. Hermitian conjugation of operators
ii. Properties of Hermitian operators
iii. Properties of Unitary operators
(c) c-numbe
Momentum vs. Wavevector
Instead of momentum, it is often convenient
to use wavevector states:
Wavevector definition:
Important commutators:
K=
P
h
[X , K ]= i
Eigenvalue equation:
K k =k k
k =c p
Relation to momentum eigenstates:
k k = (k k )
[P, K ]= 0
Lecture 13: The classical limit
Phy851/fall 2009
?
Wavepacket Evolution
For a wavepacket in free space, we have
already seen that
x = x0 +
p0
t
M
p = p0
So that the center of the wavepacket obeys
Newtons Second Law (with no force):
p
d
x=
dt
m
d
p =0
d
Lecture 14: Motion in 1D
Phy851/fall 2009
Simple Problems in 1D
To Describe the motion of a particle in 1D, we
need the following four QM elements:
ih
d
(t ) = H (t )
dt
Schrdinger's
equation
P2
H=
+V (X )
2m
Energy of a
particle
x (t ) = ( x, t )
Defin
Lecture 15: Simple problems in 1D
and Probability Current I
Phy851 Fall 2009
Continuity Theorem
From previous Lecture:
Theorem:
the wavefunction and its first derivative
must be everywhere continuous.
Exception: where there is a (x-x0) or
(x-x0) in the
Lecture 16:
Probability Current II and 1D
Scattering
Phy851 Fall 2009
Continuity Equation
dP( x, t )
j( x ) j( x + ) =
dt
d ( x, t )2
j( x ) j( x + ) =
dt
j ( x ) j ( x + ) d ( x, t )
=
2
dt
d
d
j ( x, t ) = ( x, t )
dx
dt
This is the standard continuity
Lecture 17:
Scattering in One Dimension
Part 2
Phy851 Fall 2009
Transfer Matrix
For the purpose of propagating a wave
through multiple elements, it is more
convenient to relate the amplitudes on the
right-side of the boundary to those on the
left-side vi
Lecture 19:
Quantization of the simple harmonic
oscillator
Phy851 Fall 2009
Systems near equilibrium
The harmonic oscillator Hamiltonian is:
P2 1 2
H=
+ kX
2m 2
Or alternatively, using
k
=
m
P2 1
H=
+ m 2 X 2
2m 2
Why is the SHO so important?
Answer:
Lecture 20:
Quantum SHO: Part 2
Phy851 Fall 2009
Recap
Introduced dimensionless variables:
X
X=
H
H=
h
P= P
h
=
h
m
1212
H= P + X
2
2
Introduce normal variables:
A=
1
(X + iP ) A = 1 (X iP )
2
2
A, A ] = 1
[
1
H = A A+
2
Energy eigenvalues:
H n = (n +
Lecture 21:
The Parity Operator
Phy851 Fall 2009
Parity inversion
Symmetry under parity inversion is
known as mirror symmetry
x x
P : y a y
z z
Parity inversion cannot
be generated by
rotations
Formally, we say that f(x) is symmetric
under parity i
Lecture 30:
The Hydrogen Atom
Phy851 Fall 2009
Example 2: Hydrogen Atom
The Hamiltonian for a system consisting of an
electron and a proton is:
Pp2
Pe2
e2
H=
+
rr
2me 2m p 4 0 Re R p
In COM and relative coordinates, the
Hamiltonian is separable:
H = H C
Lecture 34:
The `Density Operator
Phy851 Fall 2009
The QM `density operator
HAS NOTHING TO DO WITH MASS PER
UNIT VOLUME
The density operator formalism is a
generalization of the Pure State QM we
have used so far.
New concept: Mixed state
Used for:
De
Lecture 11:
X and P, part II
PHY851/fall 2009
Quick review:
Normalization
The constant C is found from normalization:
This leads to the results:
1
xp=
eipx / h
2h
1
px= xp =
e ipx / h
2h
The momentum operator
But we have really learned even more.
By t
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 9: SOLUTIONS
1. The Parity Operator: [20 pts] Determine the matrix element x|x and use it to simplify the
identity = dx dx |x x|x x |, then use this identity to compute 2 , 3 , and n .
From these
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 10: Solutions
Topics Covered: Tensor product spaces, change of coordinate system, general theory of angular momentum
Some Key Concepts: Angular momentum: commutation relations, raising and lowerin
PHYS851 Quantum Mechanics I, Fall 2009
HOMEWORK ASSIGNMENT 12
Topics Covered: Motion in a central potential, spherical harmonic oscillator, hydrogen atom, orbital
electric and magnetic dipole moments
1. [20 pts] A particle of mass M and charge q is constr
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
Lecture 1: Demystifying h and i
We are often told that the presence of h
distinguishes quantum from classical theories.
Q: Is h necessary at all?
By changing units we can of course make h
disappear from QM
But if it is truly fundamental, shouldnt this
sam
Lecture I: Dirac Notation
To describe a physical system, QM assigns a
complex number (`amplitude) to each distinct
available physical state.
(Or alternately: two real numbers)
What is a `distinct physical state?
Various common vector notations:
Just a
Notation:
Operators
In QM, an operator is an object that acts on a ket,
transforming it into another ket
Let A represent a generic operator
An operator is a linear map
A:H!H
A|"#= |"#
Operators are linear:
A(a |"1#+b |"2#) = aA |"1#+bA|"2#
a and b are
Remark: Commutation of Operators
Since operators are matrices, they do not
necessarily commute
AB ! BA
We define the commutator of the operators A
and B as
[A, B]= AB ! BA
The properties of most physically important
operators (e.g. X, P, L, S,) can genera
Lecture 6: Time Propagation
Ordinary Functions of Operators
Outline:
Ordinary functions of operators
Let us define an `ordinary function, f(x), as a
function that can be expressed as a power
series in x, with scalar coefficients:
Powers
Functions of di
General Study of Two-Level Systems
Goals:
Study Quantum Resonance behavior
Discuss Avoided Crossings and Adiabatic Passage
Study connection between Spin-1/2 and general twolevel systems
Examples of two level systems:
A Spin 1/2 particle
A two-level a
PHY-851: QUANTUM MECHANICS I
Final Exam /Total: 40 points/
December 11, 2001
NAME.
A. MULTIPLE CHOICE (encircle the correct answers) /12/.
1. Thermal neutrons (energy 0.025 eV) have their de Broglie wavelength
equal to
a. 2 105 cm;
b. 2 108 cm;
c. 2 1013