Download Review PH301 -- duality, wavefunction, probability

PH 401
Dr. Cecilia Vogel
Lecture 1
Review
Review 301 quantum section
Outline
light waves
matter waves
duality, complementarity
wave function
probability
Light and Quantum
Quantum Mechanics began historically with
light
Light was known to be an EM wave
obey wave equation 2E=me∂2E/∂t2.
solutions sin(kx-wt) and cos(kx-wt) and
any combo thereof
Light also found to have particle nature
individual, indivisible photons
Each photon has E=hf=w
and p = h/l = k
Where EM wave has large amplitude
bright, many photons
any one photon likely to be ther
Wave-particle duality
Light (and matter( have both wave and
particle properties
gives people issues due to perceptions:
IF you perceive a wave as having a wavelength
IF you perceive a particle as having a position
that’s not the whole story
Reality is a gray area between these extremes
the wave/particle has a spread of positions
and a spread of wavelengths
All well-behaved functions are like this:
DxDk>1/2
Matter
Matter particles, like electrons,
have particle properties (of course)
individual, indivisible particles
energy & momentum
Matter particles, also have wave properties!
They diffract!
They interfere!
Obey a wave equation = Time-dependent
Schroedinger eqn
Duality equations
wave and particle properties are
related
E  hf  h
p  h/l
E w
p k
wavefunction is a function of
(kx-wt)
Complex Wavefunction
If you have issues with a physical
quantity like wavefunction being complex
all measureable quantities will be real
like probability density = |Y|2.
OR you can think of wavefucntion as
having two components
like light has E-field and B-field
each component will be real
but you will have two components to
calculate with two coupled differential eqns
complex functions make the math easier!
Wave Function
Wave nature described by wavefunction
NOT like water or sound wave
where matter actually moves
More like light, wavefunction is a field
electric field (and B(x,t) = magnetic field).
has a value for every point in space
For matter the wave function is Y(x,t)
like nothing we’ve encountered before.
Not an EM wave.
does not have a direction in space.
Wavefunction Interpreted
For light beam, where the wave function
(E-field) is large,
the light is bright
there are lots of photons
For beam of matter particles, where the
wave function is large
there are lots of particles.
The “bright” spots in interference pattern
are where lots of photons or matter particles
strike.
Probability Interpretation
If you have one particle, rather than a
beam,
the wavefunction only gives probability
density
P(x,t) = |Y(x,t)|2.
there is no way to predict precisely where it
will be.
Where the wave function is large
the particle is likely to be.
The “bright” spots in interference pattern
are where a photon or matter particle is
likely to strike.
Probability and Probability
Density
P(x,t) = |Y(x,t)|2 is the probability
density at position x at time t
like mass density.
To get probability, must have finite
region of space
the probability of the particle being in a
volume of space
P
volume
|Y(r,t) | dV
2