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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

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Vector Spaces for Quantum Mechanics

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  1. Vector Spaces for Quantum Mechanics PHYS 20602

  2. 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!)

  3. Overview • Vector spaces (9 lectures) …mathematical introduction • Quantum mechanics and vector spaces (3 lectures) …applying the maths to physics • Angular momentum (4 lectures) …a case where vector space methods become very easy (much easier than using wave mechanics) • Function spaces (3 lectures) …the connection to wave mechanics • The simple harmonic oscillator (2 lectures) …using vector space notation to make operator algebra easy…and solving the basic problem for quantum field theory. • Entanglement (1 lecture) …weird quantum properties of multi-particle systems

  4. 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. Why this course?

  5. 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. Books

  6. 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 1. Vector Spaces

  7. 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 elemente  G, such that ae = ea=a • Every a  G has an inverse elementa−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.

  8. Definitions: Vector Space A complex vector space, is a set, written V(C), of elements called vectors, such that: • 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 • For any complex numbers ,   C and vectors x, y  V(C), products such as x are vectors in V(C) and •  (  x ) = (   ) x • 1 x = x •  (x+y ) =  x+ y • ( +  ) x =  x + x We can also have real vector spaces, where ,   R i.e. real numbers (includes “ordinary” vectors).

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