6-1
Commuting and non-commuting operators
• In classical physics, there is no limit to what can
be known about system (x, p, Etotal, Ekinetic )
• In QM, two observables can be known
simultaneously only if a measurement doesn’t
change the state of the system.
• This is only the case if (x) is an eigenfunction
of both operators
• Two observables can be known simultaneously
only if they have a common set of EF
Consider operators
• Measuring the observable corresponding to
first and then that corresponding to
equivalent to
• Assume that n(x) is an EF of
• Also true that
System unchanged by measurement if n(x) EF of
both operators!
6-2
• In this case
• Two operators that have a common set of EF are
said to commute.
• To test this for an arbitrary function f(x). If
then the operators commute.
is called the commutator.
• Example: Do
NO!
commute?
6-3
Example: Do
commute?
Evaluate commutator =
=0 only for free particle
6-4
The Stern-Gerlach Experiment
Silver atom beam in inhomogeneous magnetic field
Atoms are deflection determined by z component
of magnetic moment, mz.
z component is either “up” or “down”
•Note that spin comes out of relativistic QM
No
field
With
field
6-5
• Classically, all mz values allowed.
• QM operator
“measure z component of magnetic moment, mz”
• Experiment shows mz is quantized
Only two values possible (up & down only)
• Means there are two eigenfunctions,  and 
• Eigenvalues: equal magnitude; opposite sign
• Most general wavefunction is a linear combination
of these EF’s
Review: page 40, equation 3.6 and discussion
Note: misprint on equations 6.2 and 6.3, take out
the
Experiment shows equal number of “spin up” and
“spin down” so,
Before entering magnet
(superposition of states)
After measurement wavefunction “collapses” to
either  or .
6-6
Now add second magnet at 90º
Operator
“measure x component of magnetic moment, mx”
Experimental results
Other than quantization, this result seems reasonable
Since there was no selection for mx in the first
magnetic.
Now for the weird part.
Add a 3rd magnet, back in the z direction.
What does your intuition tell you the result will be?
6-7
Might think that since the mz’s were separated by the
first magnet that these components would stay the
same throughout.
You’d be wrong.
Experiment shows:
The second measurement in the x direction changes
the system.
Wavefunction again becomes superposition of both
possible eigenfunctions  and .
So both eigenvalues, “spin up” and “spin down” will
be measured.
6-8
Conclusions from S-G experiment
• Operators “measure the z component of the
magnetic moment” and “measure the x component of
the magnetic moment” do not commute.
• Can’t know both components simultaneously.
• Unlike classical physics, measurement process
changed the state of system.
Heisenberg Uncertainty Principle
• Recall we showed that position and momentum
operators do not commute.
• We cannot know momentum and position
simultaneously and exactly.
• Consider free particle for which momentum
exactly known ( is an EF of
).
6-9
Recall we normalized this is chapter 4.
Probability of finding particle at x is given by
• Probability is independent of x
• Equally likely to find particle anywhere.
• Complete uncertainty about position; total certainty
about momentum.
• Can’t calculate trajectory like in classical physics!
6-10
Create particle with limited certainty in momentum.
(Some uncertainty in the momentum.)
Do so by superposing plane waves
This function is not an EF of the momentum operator.
The momentum is somewhere in range
What is P(x) for this wave function?
Result is shown by red curve.
21 waves
shown
Red curve is
their sum
squared
Now the position is more certain!
6-11
If you added an infinite number of waves together,
the probability for the position would go to the
outline of the peaks.
Heisenberg quantified the relationship.
Heisenberg uncertainty principle
6-12