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2. Quantum Mechanics and Vector Spaces. 2.1 Physics of Quantum mechanics Principle of superposition Measurements 2.2 Redundant mathematical structure 2.3 Time evolution The Schr ödinger equation Time evolution operator Example: Electron Spin Precession.
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2. Quantum Mechanics and Vector Spaces 2.1 Physics of Quantum mechanics • Principle of superposition • Measurements 2.2 Redundant mathematical structure 2.3 Time evolution • The Schrödinger equation • Time evolution operator • Example: Electron Spin Precession
We can produce interference between different components of a quantum state, e.g. Two-slit experiment photons, electrons, buckyballs (C60)… Bragg diffraction: interference between particles reflected from different planes in a crystal Photons, electrons, neutrons, H2 molecules Superconducting Quantum Interference Devices (SQUIDS): interference between electric currents travelling around loop in opposite directions. “The most beautiful experiment in physics” …according to Physics World readers (2002) 2.1.1 Principle of Superposition Credit: Tonomura et al, Hitachi Corp.
SG−x SG x SG x Interference experiments | | SG−x | SG z φ | | Feynman thought experiment • Destructive interference genuine wave-like superposition, not just addition of probabilities. • Interference pattern depends on both relative amplitude and ‘phase difference’ between components represent as complex amplitude. • Interference always seen whenever theory predicts it should be detectable. • Physical states can be added and multiplied by complex numbers, i.e. they have the structure of a vector space.
Why not stick with wave functions? • Don’t take ‘vector’ too seriously • …it’s a metaphor • Really a general theory of “superposables” • So you can always think of waves instead if that helps. • Often we’re interested in quantum numbers, not the wave pattern: vector approach avoids calculating wave functions when not needed. • Wave function picture incomplete: • If you know ψ(r) you know everything about: • position, momentum, KE, orbital angular momentum • …but nothing about spin (+ other more obscure quantities) Vector space allows us to easily include spin.
Only certain results found in quantum measurement: some quantities quantized (ang. mom., atomic energy levels) some continuous (position, momentum of a free particle). We can prepare quantum states that will definitely give any allowed result for a quantized observable an arbitrarily small spread for continuous observables. There is ‘something there’ to measure. SG z Ag SG z SG z Ag 2.1.2 Measurements
SG−z SG z SG z Measurement (continued) | SG−z SG z Ag | • If we superpose definite states of a given observable, & measure the same observable, we randomly get one of the superposed values—never an ‘intermediate’ result. • Probability of result a, Prob(a) |amplitude|2 in superposition. • We always get some result: Probs = 1.
Represent states of definite results (eigenstates) as a set of orthonormal basis vectors. Represent physical states as normalised vectors. Probability amplitude for result ai from state ψ: ci = ai |ψ. zero amplitude to get anything but ai in “definite ai” state. Use projectors instead, if degenerate. General state can always be decomposed into a superposition: Sum of probabilities = 1 is Pythagoras rule in N-D vector space! 1 cz|z |ψ cx|x cy|y Mathematical model
2.2 Redundant Mathematical Structure • A mathematical model for a physical process may contain things that don’t have any physical meaning. • e.g. in electromagnetism, vector potential is undetermined up to a gauge change: A A + • Bad thing? May make the maths much easier! • In QM, physical states are represented by normalised vectors: • Ambiguous up to factor of eiθ, i.e. |ψ and eiθ|ψ represent the same state. • Normalised vectors do not make a vector space—maths requires vectors of all lengths. • Really, physical state equivalent to a ‘ray’ through the origin: normalisation is a convention as we could write: • Vectors of a particular length & phase needed when analysing a vector into a superposition.
Redundancy (continued) • Vector space may include unphysical vectors: • all those with infinite energy, i.e. outside the domain of the energy operator, Ĥ, (e.g. discontinuous wave functions). • Should other operators (x ? p ?) have finite expected values? • Do all possible self-adjoint operators represent physical observables? • In practice, no: we only need a few dozen. • In theory, no: some self-adjoint ops represent things disallowed by ‘superselection’ — e.g. real particles are either bosons or fermions, not some mixture.
