Chapter 17 Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement. Physical Chemistry 2 nd Edition. Thomas Engel, Philip Reid. Objectives. Introduction of Stern-Gerlach Experiment Understanding of Heisenberg Uncertainty Principle. Outline.
Commuting and Noncommuting Operators and
the Surprising Consequences of Entanglement
Thomas Engel, Philip Reid
Determine whether the momentum and (a) the kinetic energy and (b) the total energy can be known simultaneously.
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.
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.
b. For momentum and total energy, we evaluate
Because the kinetic energy and momentum operators commute, per part (a), this expression is equal to
We conclude the following:
Therefore, the momentum and the total energy cannot be known simultaneously and exactly.
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.
Using the de Broglie relation, the mean momentum is given by
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!