3' Quantum Mechanics and Vector Spaces

1
3. Quantum Mechanics and Vector Spaces

• 3.1 Physics of Quantum mechanics
• Principle of superposition
• Measurements
• 3.2 Redundant mathematical structure
• 3.3 Time evolution
• The Schrödinger equation
• Time evolution operator
• Example Nuclear magnetic resonance

2
3.1.1 Principle of Superposition

• 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)

Credit Tonomura et al, Hitachi Corp.
3
Interference experiments
??
??
SG-x
??
SG z
f
??
??
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.

4
Why not stick with wave functions?

• Dont take vector too seriously
• its a metaphor
• Really a general theory of superposables
• So you can always think of waves instead if that
  helps.
• Often were interested in quantum numbers, not
  the wave pattern vector approach avoids
  calculating wave functions when not needed.
• Wave function picture incomplete
• If you know ?(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
3.1.2 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.

6
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 valuesnever
  an intermediate result.
• Probability of result a, Prob(a) ? amplitude2
  in superposition.
• We always get some result ? Probs 1.

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 ai from state ?
  ci ?ai ? ?.
• 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

8
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 ?
  ci ?ai ? ?.
• 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!

9
3.2 Redundant Mathematical Structure

• A mathematical model for a physical process may
  contain things that dont 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 ei?, i.e. ? ? and
  ei?? ? represent the same state.
• Normalised vectors do not make a vector
  spacemaths 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.

10
Redundancy (continued)

• Hilbert space includes unphysical vectors
• all those with infinite energy, i.e. outside the
  domain of the energy operator, H, (e.g.
  discontinuous wave functions).
• Should other operators (x ? p ?) have finite
  expected values?
• Perhaps a good reason for ignoring Hilbert space
  as such!
• 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.