Transcript Slides from Lecture 9-11

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
2.1.1 Principle of Superposition
We can produce interference
between different components
of a quantum state, e.g.
► Two-slit experiment
 photons, electrons, buckyballs
(C60)…
►
“The most beautiful
experiment in physics”
 …according to Physics World
readers (2002)
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.
Credit: Tonomura et al, Hitachi Corp.
Interference experiments
|
|
|
SG−x
SG−x
SG x
|
φ
SG x
SG z
|
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.
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).
►
Ag
SG z
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
SG z
Measurement (continued)
|
Ag
SG z
SG−z
SG−z
SG z
SG z
|
►
►
►
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.
Mathematical model
►
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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:
   ci ai   ai ai 
i
i
►
Sum of probabilities = 1 is
Pythagoras rule in N-D vector
space!
1
|ψ 
cz|z
cx|x
cy|y
2
  1     cx  c y  cz
2
2
2
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:
Proba  
a
2
a a  
 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.
Classical Mechanics
Quantum Mechanics