Transcript Wave Functions

Wavefunctions and
Uncertainty
Chapter 41
Radical view of light
• Planck and Einstein
re-introduced the
particle notion for
light
• a wave “becomes” a
particle and still a
wave … hmmm
de Broglie...
• If light (ie “a wave”)
can be a particle
then maybe a
particle (electron?)
can “be a wave” what are the
implications of this
“leap of reason”?
Compare and contrast: Waves & Particles
• Waves are extended
• Waves are
continuous
• Waves conform to
wave equations
• Waves diffract and
interfere
• Waves have
amplitude, frequency
and velocity
• Particles are points
• Particles are
discontinuous
• Particles obey
equations of
mechanics
• Particles “bounce”
• Particles have mass,
size(?) and velocity
ParticleWaves, Wavicles or just
Weirdness …
• If de Broglie is correct then we can ascribe a
wavelength to a particle:
E hf
p 
c
c
p  mv
h h
mv 

c 
Schroedinger...
Once at the end of a colloquium
I heard Debye saying something
like: “Schroedinger, you are not
working right now on very
important problems… why don’t
you tell us some time about that
thesis of de Broglie’s… in one of
the next colloquia, Schroedinger
gave a beautifully clear account
of how de Broglie associated a
wave with a particle, and how
he could obtain the quantization
rules… When he had finished,
Debye casually remarked that
he thought this way of talking
was rather childish… To deal
properly with waves, one had to
have a wave equation.
Felix Bloch, Address to the
American Physical Society,
1976
What’s a Wavefunction?
• Schroedinger’s
Equation:
Hy= E y
Wavefunction “psi” solves Schroedinger’s equation
and contains, in its components, all of the
information we need to determine values of
observables…
Wavefunctions are only part of the
story…
• The information in the wavefunction is “coded” in
its components. Actual values for observables
depend on “how you ask” the wavefunction
• Operators tell us what we want to know:
• Example: momentum
– Classical:
p  mv

– Quantum:
d
p  i
dx
This is not as strange as it seems –
you got used to thinking of
momentum as “em” times “vee” –
this is just another way to think
about it!
Schroedinger’s key assumptions concerning
a quantum-mechanical wave equation...
1 It must incorporate the relations:
  h p and f  E h
2
p
E
2m
V
2 Since normal waves “add” linearly (principle of
superposition), so too must the solutions to the
qm-wave equation. This means the solutions
must be linear.
What wavefunctions are (and are not!)
• Wavefunctions are
mathematical ideas that depict
probability distributions (Born
interpretation)
• Wavefunctions can be
described using the
mathematics of waves but are
not “real”
• Wavefunctions obey strict
mathematical rules:
– continuous, differentiable, finite
Key Ideas…
• Chp 41.1 – 41.3
– Wave Interference and how it applies to “quantons”
– Probability and Probability Density
– Wave Functions
• Chp 41.4 – 41.6
– Normalization
– Wave packets
– Uncertainty