1. Physical Chemistry 2nd Edition
Chapter 17
Commuting and Noncommuting Operators and
the Surprising Consequences of Entanglement
Physical Chemistry 2nd Edition
Thomas Engel, Philip Reid

2. Objectives Introduction of Stern-Gerlach Experiment
Understanding of Heisenberg Uncertainty Principle

3. Outline Commutation Relations The Stern-Gerlach Experiment
The Heisenberg Uncertainty Principle

4. 17.1 Commutation Relations
How can one know if two operators have a common set of eigenfunctions?
We use the following
If two operators have a common set of eigenfunctions, we say that they commute.
Square brackets is called the commutator of the operators.

5. Example 17.1
Determine whether the momentum and (a) the kinetic energy and (b) the total energy can be known simultaneously.

6. Solution
To solve these problems, we determine whether two operators commute by evaluating the commutator If the commutator is zero, the two observables can be determined simultaneously and exactly.

7. Solution a. For momentum and kinetic energy, we evaluate
In calculating the third derivative, it does not matter if
the function is first differentiated twice and then once
or the other way around. Therefore, the momentum
and the kinetic energy can be determined
simultaneously and exactly.

8. Solution b. For momentum and total energy, we evaluate
Because the kinetic energy and momentum operators commute, per part (a), this expression is equal to

9. Solution We conclude the following:
Therefore, the momentum and the total energy cannot be known simultaneously and exactly.

10. 17.2 Wave Packets and the Uncertainty Principle

11. 17.2 The Stern-Gerlach Experiment
In Stern-Gerlach experiment, the inhomogeneous magnetic field separates the beam into two, and only two, components.
The initial normalized wave function that describes a single silver atom is

12. 17.2 The Stern-Gerlach Experiment
The conclusion is that the operators A, “measure the z component of the magnetic moment,” and B, “measure the x component of the magnetic moment,” do not commute.

13. Example 17.2
Assume that the double-slit experiment could be carried out with electrons using a slit spacing of b=10.0 nm. To be able to observe diffraction, we choose , and because diffraction requires reasonably monochromatic radiation, we choose Show that with these parameters, the uncertainty in the position of the electron is greater than the slit spacing b.

14. 17.3 The Heisenberg Uncertainty Principle
As a result of the superposition of many plane waves, the position of the particle is no longer completely unknown, and the momentum of the particle is no longer exactly known.

15. 17.3 The Heisenberg Uncertainty Principle
Heisenberg uncertainty principle quantifies the uncertainty in the position and momentum of a quantum mechanical particle.
It is concluded that if a particle is prepared in a state in which the momentum is exactly known, then its position is completely unknown.
Superposition of plane waves of very similar wave vectors given by

16. 17.3 The Heisenberg Uncertainty Principle
Both position and momentum cannot be known exactly and simultaneously in quantum mechanics.
Heisenberg famous uncertainty principle is

17. Solution Using the de Broglie relation, the mean momentum is given by
And

18. Solution The minimum uncertainty in position is given by
which is greater than the slit spacing. Note that the
concept of an electron trajectory is not well defined
under these conditions. This offers an explanation
for the observation that the electron appears to go
through both slits simultaneously!