Vector Spaces for Vector Spaces for Quantum MechanicsQuantum Mechanics

PHYS 20602PHYS 20602

Aim of courseAim of course

►To introduce the idea of vector spaces To introduce the idea of vector spaces and to use it as a framework to solve and to use it as a framework to solve problems in quantum mechanics.problems in quantum mechanics. More general than wave mechanics, e.g. More general than wave mechanics, e.g.

natural way of treating spinnatural way of treating spin Unifies original wave mechanics and matrix Unifies original wave mechanics and matrix

mechanics approaches to quantum mechanics approaches to quantum mechanicsmechanics

Neat notation makes complicated algebra Neat notation makes complicated algebra easier (once you understand it!)easier (once you understand it!)

OverviewOverview

1.1. Vector spaces (9 lectures)Vector spaces (9 lectures)……mathematical introductionmathematical introduction

2.2. Quantum mechanics and vector spaces (3 lectures)Quantum mechanics and vector spaces (3 lectures)……applying the maths to physics applying the maths to physics

3.3. Angular momentum (4 lectures)Angular momentum (4 lectures)……a case where vector space methods become very easy (much a case where vector space methods become very easy (much

easier than using wave mechanics)easier than using wave mechanics)

4.4. Function spaces (3 lectures)Function spaces (3 lectures)……the connection to wave mechanicsthe connection to wave mechanics

5.5. The simple harmonic oscillator (2 lectures)The simple harmonic oscillator (2 lectures)……using vector space notation to make operator algebra easy…using vector space notation to make operator algebra easy…

and solving the basic problem for quantum field theory.and solving the basic problem for quantum field theory.

6.6. Entanglement (1 lecture)Entanglement (1 lecture)……weird quantum properties of multi-particle systemsweird quantum properties of multi-particle systems

Why this course?Why this course?

►QM is QM is mathematically mathematically hard to pin down…hard to pin down… Quantum rules Quantum rules

(Planck/Einstein/Bohr(Planck/Einstein/Bohr: 1900-1916): 1900-1916)

Wave mechanics Wave mechanics (Schrödinger 1926)(Schrödinger 1926)

Matrix mechanics Matrix mechanics (Heisenberg 1925)(Heisenberg 1925)

Path integrals Path integrals (Feynman 1948)(Feynman 1948)

► This course gives This course gives you the most you the most general formulation, general formulation, linking all the others linking all the others (von Neumann, (von Neumann, Dirac, 1926, + help Dirac, 1926, + help from later from later mathematicians).mathematicians).

► Should help you tell Should help you tell what is physics from what is physics from what is maths in QM.what is maths in QM.

BooksBooks

► Shankar (US postgraduate Shankar (US postgraduate text):text): Very clearVery clear This course is based on a This course is based on a

drastically trimmed-down drastically trimmed-down version of Shankar’s version of Shankar’s approach.approach.

Shankar’s coverage of ang. Shankar’s coverage of ang. mom. relies on parts of his mom. relies on parts of his book we will skip.book we will skip.

Chapter 1 recommended!Chapter 1 recommended!► Townsend (US undergrad Townsend (US undergrad

tex):tex): Intuitive approachIntuitive approach covers examples but skips covers examples but skips

formal maths.formal maths.

► Undergrad QM texts:Undergrad QM texts: Isham: excellent on formal Isham: excellent on formal

part of course but does not part of course but does not do examples (angular do examples (angular momentum, harmonic momentum, harmonic oscillator)oscillator)

Feynman vol III: brilliant on Feynman vol III: brilliant on concepts but rather concepts but rather qualitative.qualitative.

► Maths texts:Maths texts: Byron & Fuller (US PG text): Byron & Fuller (US PG text):

fairly rigorous, but very fairly rigorous, but very clear.clear.

Boas; Riley Hobson & Bence Boas; Riley Hobson & Bence (Standard UK undergrad (Standard UK undergrad references): basic coverage references): basic coverage of most relevant maths. of most relevant maths.

1. Vector Spaces1. Vector Spaces

Mathematicians are like a Mathematicians are like a certain type of certain type of Frenchman: when you Frenchman: when you talk to them they talk to them they translate it into their translate it into their own language, and own language, and then it soon turns into then it soon turns into something completely something completely different. different. — — Johann Wolfgang von Johann Wolfgang von

Goethe, Goethe, Maxims and Maxims and ReflectionsReflections

Definitions: GroupsDefinitions: Groups

A A groupgroup is a system [G, is a system [G, ] of a set, G, and an ] of a set, G, and an operation, operation, , such that, such that

1.1. The set is The set is closedclosed under under , i.e. , i.e. aab b G for any G for any a,ba,b G G

2.2. The operation is The operation is associativeassociative, i.e. , i.e. aa(b(bc)c) = = (a(ab)b)cc

3.3. There is an There is an identity elementidentity element ee G, such that G, such that aae e = e= ea=aa=a

4.4. Every Every aa G has an G has an inverse elementinverse element aa−1−1 such that such that aa−1−1a a = = aaaa−1−1 = e = e

If the operation is If the operation is commutativecommutative, i.e. , i.e. aab = bb = baa, , then the group is said to be then the group is said to be abelianabelian..

Definitions: Vector SpaceDefinitions: Vector Space

A complex vector space, is a set, written VA complex vector space, is a set, written V(CC), of of elements called elements called vectorsvectors, such that:, such that:

1.1. There is an operation, There is an operation, ++, such that [V(, such that [V(CC), ), ++] is an abelian ] is an abelian group withgroup with

identity element written 0 (“the zero vector”).identity element written 0 (“the zero vector”). inverse of vector inverse of vector xx written written −−xx

2.2. For any complex numbers For any complex numbers , , CC and vectors and vectors x, yx, y V(V(CC), products such as ), products such as xx are vectors in V( are vectors in V(CC) and ) and

a)a) ( ( x x ) = () = ( ) ) xxb)b) 1 1 x x = = xxc)c) ((xx ++ y y ) = ) = x x ++ y yd)d) (( + + ) ) xx = = x x ++ x x

We can also have real vector spaces, where We can also have real vector spaces, where , , RR i.e. real numbers (includes “ordinary” vectors).i.e. real numbers (includes “ordinary” vectors).