130_notes.dvi

The probability to find a particle at a position~xat some time tis the absolute square of the
probability amplitudeψ(~x,t).
P(~x,t) =|ψ(~x,t)|^2

To compute the probability to find an electron at our thought experiment detector, we add the
probability amplitude to get to the detector through slit 1 to the amplitude to get to the detector
through slit 2 and take the absolute square.

Pdetector=|ψ 1 +ψ 2 |^2

Quantum Mechanics completely changes our view of the world.Instead of a deterministic world,
we now have only probabilities. We cannot even measure both the position and momentum of a
particle (accurately) at the same time. Quantum Mechanics will require us to use the mathematics
of operators, Fourier Transforms, vector spaces, and much more.

1.4 Wave Packets and Uncertainty

The probability amplitude for a free particle with momentum~pand energyE=

√

(pc)^2 + (mc^2 )^2
is the complex wave function
ψfree particle(~x,t) =ei(~p·~x−Et)/ ̄h.

Note that|ψ|^2 = 1 everywhere so this does not represent a localized particle. In fact we recognize
the wave property that, to have exactly one frequency, a wave must be spread out over space.

We can build up localized wave packets that represent single particles(See section 5.1) by adding
up these free particle wave functions (with some coefficients).

ψ(x,t) =

1

√

2 π ̄h

+∫∞

−∞

φ(p)ei(px−Et)/ ̄hdp

(We have moved to one dimension for simplicity.) Similarly we can computethe coefficient for each
momentum

φ(p) =

1

√

2 π ̄h

∫∞

−∞

ψ(x)e−ipx/ ̄hdx.

These coefficients,φ(p), are actually the state function of the particle in momentum space. We can
describe the state of a particle either in position space withψ(x) or in momentum space withφ(p).
We can useφ(p) to compute the probability distribution function for momentum.

P(p) =|φ(p)|^2

We will show that wave packets like these behave correctly in the classical limit, vindicating the
choice we made forψfree particle(~x,t).

The Heisenberg Uncertainty Principle (See section 5.3) is a propertyof waves that we can deduce
from our study of localized wave packets.

∆p∆x≥

̄h

2