Download Vector Spaces for Quantum Mechanics

Vector Spaces for
Quantum Mechanics
PHYS 20602
Aim of course
► To
introduce the idea of vector spaces and
to use it as a framework to solve problems
in quantum mechanics.
 More general than wave mechanics, e.g. natural
way of treating spin
 Unifies original wave mechanics and matrix
mechanics approaches to quantum mechanics
 Neat notation makes complicated algebra easier
(once you understand it!)
Overview
1.
Vector spaces (9 lectures)
…mathematical introduction
2.
Quantum mechanics and vector spaces (3 lectures)
…applying the maths to physics
3.
Angular momentum (4 lectures)
…a case where vector space methods become very easy (much easier
than using wave mechanics)
4.
Function spaces (3 lectures)
…the connection to wave mechanics
5.
The simple harmonic oscillator (2 lectures)
…using vector space notation to make operator algebra easy…and
solving the basic problem for quantum field theory.
6.
Entanglement (1 lecture)
…weird quantum properties of multi-particle systems
Why this course?
► QM
is mathematically
hard to pin down…
 Quantum rules
(Planck/Einstein/Bohr:
1900-1916)
 Wave mechanics
(Schrödinger 1926)
 Matrix mechanics
(Heisenberg 1925)
 Path integrals
(Feynman 1948)
► This
course gives you
the most general
formulation, linking all
the others (von
Neumann, Dirac, 1926,
+ help from later
mathematicians).
► Should help you tell
what is physics from
what is maths in QM.
Books
►
Shankar (US postgraduate text):
 Very clear
 This course is based on a
drastically trimmed-down
version of Shankar’s approach.
 Shankar’s coverage of ang.
mom. relies on parts of his book
we will skip.
 Chapter 1 recommended!
►
Townsend (US undergrad tex):
 Intuitive approach
 covers examples but skips
formal maths.
►
Undergrad QM texts:
 Isham: excellent on formal part
of course but does not do
examples (angular momentum,
harmonic oscillator)
 Feynman vol III: brilliant on
concepts but rather qualitative.
►
Maths texts:
 Byron & Fuller (US PG text):
fairly rigorous, but very clear.
 Boas; Riley Hobson & Bence
(Standard UK undergrad
references): basic coverage of
most relevant maths.
1. Vector Spaces
Mathematicians are like a
certain type of Frenchman:
when you talk to them
they translate it into their
own language, and then it
soon turns into something
completely different.
— Johann Wolfgang von
Goethe, Maxims and
Reflections
Definitions: Groups
A group is a system [G, ] of a set, G, and an
operation, , such that
1. The set is closed under , i.e. ab  G for any a,b  G
2. The operation is associative, i.e. a(bc) = (ab)c
3. There is an identity element e  G, such that ae =
ea=a
4. Every a  G has an inverse element a−1 such that
a−1a = aa−1 = e
If the operation is commutative, i.e. ab = ba, then
the group is said to be abelian.
Definitions: Vector Space
A complex vector space, is a set, written V(C), of elements
called vectors, such that:
1.
There is an operation, +, such that [V(C), +] is an abelian group
with


2.
identity element written 0 (“the zero vector”).
inverse of vector x written −x
For any complex numbers ,   C and vectors x, y  V(C),
products such as x are vectors in V(C) and
a)
b)
c)
d)
 (  x ) = (   ) x
1x=x
 (x + y ) =  x +  y
( +  ) x =  x +  x
We can also have real vector spaces, where ,   R i.e.
real numbers (includes “ordinary” vectors).