130_notes.dvi

4 The Solution: Probability Amplitudes

ForEM waves, the intensity, and hence the probability to find a photon, is proportional to the
square of the fields. The fields obey the wave equation. The fields from two slits can add construc-
tively or destructively giving interference patterns. TheEandBfields are 90 degrees out of phase
and both contribute to the intensity.

We willuse the same ideas for electrons, although the details of the field will vary a bit because
electrons and photons are somewhat different kinds of particles. For both particles the wavelength
is given by

λ=

h

p

and the frequency by
E=hν= ̄hω.

We will use acomplex probability amplitudeψ(x,t) for the electron. The real and imaginary
parts are out of phase like the EM fields. The traveling wave with momentumpand energyEthen
is
ψ(x,t)∝ei(kx−ωt)=ei(px−Et)/ ̄h

Theprobability to find an electron is equal to the absolute squareof the complex probability
amplitude.
P(x,t) =|ψ(x,t)|^2

(We will overcome the problem that this probability is 1 everywhere for our simple wavefunction.)

We have just put in most of the physics of Quantum Mechanics. Much of what we do
for the rest of the course will be deduced from the paragraph above. Our input came
from deBroglie and Plank, with support from experiments.

Lets summarize the physics input again.

• Free particles are represented by complex wave functions with a relationship between their par-
  ticle properties – energy and momentum, and their wave properties– frequency and wavelength
  given by Plank and deBroglie.


• The absolute square of the wavefunction gives the probability distribution function. Quantum
  Mechanics only tells us the probability.


• We can make superpositions of our free particle wave functions to make states that do not
  have definite momentum. We will find that any state can be made fromthe superposition of
  free particle states with different momentum.

We now have a wave-particle duality for all the particles, however,physics now only tells us
the probabilityfor some quantum events to occur. We have lost the complete predictive power of
classical physics.

Gasiorowicz Chapter 1

Rohlf Chapter 5