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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
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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: 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 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.
