Foundations of Quantum Mechanics

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Foundations of Quantum Mechanics

• PHYS 20602

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Aim of course

• To introduce the idea of Hilbert space and to use
  it as a framework to solve problems in quantum
  mechanics.
• Wikipedia definition
• A Hilbert space is a real or complex inner
  product space that is complete under the norm
  defined by the inner product ??,?? by

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Overview

• Vector spaces (7 lectures)
• mathematical introduction
• Hilbert spaces (4 lectures)
• what that definition means
• What quantum mechanics has to do with Hilbert
  space and related ideas (2 lectures)
• Angular momentum (5 lectures)
• a case where Hilbert space methods become very
  easy (much easier than using wave mechanics)
• Perturbation theory (4 lectures)
• a case where Hilbert space notation makes things
  look much easier (even though it really is wave
  mechanics)

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Why this course?

• Many mysteries in QM
• Wave-particle duality
• Non-local interactions
• Quantum gravity
• What happens to Schrödingers Cat?
• But QM is also 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.

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Books

• Shankar (US postgraduate text)
• Very clear
• This course is based on a drastically
  trimmed-down version of Shankars approach.
• Shankars coverage of ang. mom. relies on parts
  of his book we will skip.
• Chapter 1 recommended!
• 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.

• Other US PG texts
• (all cover much more than this course)
• Merzbacher old school
• Sakurai more condensed than Shankar.
• Ballentine similar to Shankar but more condensed
  and rigorous.
• Undergrad QM texts
• Isham excellent on formal part of course but
  does not do examples (ang. mom., perturbation
  theory)
• Townsend covers examples but skips formal maths.
• Feynman vol III brilliant on concepts but rather
  qualitative.

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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

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Definitions Groups

• A group is a system G, ? of a set, G, and an
  operation, ?, such that
• The set is closed under ?, i.e. a?b ? G for any
  a,b ? G
• The operation is associative, i.e. a?(b?c)
  (a?b)?c
• There is an identity element e ? G, such that a?e
  e?aa
• 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.

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Definitions Fields

• A field is a system F,, ? such that
• F, is an abelian group with
• identity element written 0
• () inverse of a written -a.
• For convenience, a (-b) can be written a - b.
• Let F x x ? F and x ? 0, i.e. F excluding
  0.
• F, ? is an abelian group with
• identity element written 1
• (?) inverse of a written a-1.
• For convenience a ? b-1 can be written a / b.
• The operation ? is distributive with respect to
  the operation , i.e. a ? (b c) a ?b a ? c

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Definitions Vector Space

• A vector space over a field F, is a set, written
  V(F), of elements called vectors, such that
• There is an operation, , such that V(F), is
  an abelian group with
• identity element written 0 (the zero vector).
• inverse of vector x written -x
• For every ?, ? ? F and x, y ? V(F), products such
  as ?x are vectors in V(F) and
• ? ( ? x ) (? ? ? ) x
• 1 x x
• ? (x y ) ? x ? y
• (? ? ) x ? x ? x