Chapter 28 Quantum Mechanics of Atoms

28.2 The Wave Function and Its
Interpretation; the Double-Slit Experiment
This role is played by the wave function, .
The square of the wave function at any point is
proportional to the number of electrons
expected to be found there.
For a single electron, the square of the wave
function is the probability of finding the
electron at that point.
28.2 The Wave Function and Its
Interpretation; the Double-Slit Experiment
For example: the interference pattern is observed
after many electrons have gone through the slits.
If we send the electrons through one at
a time, we cannot predict the path any
single electron will
take, but we can
predict the overall
distribution.
28.3 The Heisenberg Uncertainty Principle
Fundamental: the position can only be
measured to about one wavelength.
But according to De Broglie:
28.3 The Heisenberg Uncertainty Principle
Combining, we find the combination of
uncertainties:
λ =
h
p
λ ⋅ p = h
This is called the Heisenberg uncertainty
principle.
It tells us that the position and momentum
cannot simultaneously be measured with
precision.
28.3 The Heisenberg Uncertainty Principle
This relation can also be written as a relation
between the uncertainty in time intervals and
the uncertainty in energy:
28.3 The Heisenberg Uncertainty Principle
Example: measure position of an electron to
10 -10 m. What is the velocity uncertainty?
∆p
h
∆v
= ≈ = 7⋅10
m m ∆x
6
m/s
This says that if an energy state only lasts for a
limited time, its energy will be uncertain. It also
says that conservation of energy can be
violated if the time is short enough.
What is the position uncertainty for ∆p=mc?
h
∆x
≈ =
∆p
h
mc
= 2.5⋅10
−12
m
2