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: )  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 VC of elements called vectors, such that: 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  identity element written 0 (“the zero vector”).  inverse of vector x written −x 2.For any complex numbers ,   C and vectors x, y  V(C), products such as  x are vectors in V(C) and a)  (  x ) = (    ) x b) 1 x = x c)  (x + y ) =  x +  y d) (  +  ) x =  x +  x We can also have real vector spaces, where ,   R i.e. real numbers (includes “ordinary” vectors).