1. 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

2. 2.1.1 Principle of Superposition We can produce interference between different components of a quantum state, e.g. ► Two-slit experiment  photons, electrons, buckyballs (C 60 )… ► Bragg diffraction: interference between particles reflected from different planes in a crystal  Photons, electrons, neutrons, H 2 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) Credit: Tonomura et al, Hitachi Corp.

3. ► 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. Interference experiments SG x SG−x SG z SG−x φ Feynman thought experiment |  |  |  |  | 

4. 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.

5. 2.1.2 Measurements ► 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. Ag SG z Ag

6. Measurement (continued) ► 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. Ag SG z SG−z SG−z |  | 

7. Mathematical model ► Represent states of definite results (eigenstates) as a set of orthonormal basis vectors. ► Represent physical states as normalised vectors. ► Probability amplitude for result a i from state ψ: c i =. ► Probability amplitude for result a i from state ψ: c i =  a i |ψ .  zero amplitude to get anything but a i in “definite a i ” 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  c x |x   c y |y   c z |z  |ψ |ψ 

8. 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:  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 e iθ, i.e. e iθ  Ambiguous up to factor of e iθ, i.e. |ψ  and e iθ |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.

9. 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.

10. Classical Mechanics

11. Quantum Mechanics