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### Physical Chemistry 2nd Edition

Commuting and Noncommuting Operators and

the Surprising Consequences of Entanglement

Thomas Engel, Philip Reid

Objectives

- Introduction of Stern-Gerlach Experiment
- Understanding of Heisenberg Uncertainty Principle

Outline

- Commutation Relations
- The Stern-Gerlach Experiment
- The Heisenberg Uncertainty Principle

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.

Example 17.1

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

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.

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.

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

Solution

We conclude the following:

Therefore, the momentum and the total energy cannot be known simultaneously and exactly.

17.2 Wave Packets and the Uncertainty Principle

- 17.2 Wave Packets and the Uncertainty Principle

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

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.

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.

17.3 The Heisenberg Uncertainty Principle

- 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.

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

17.3 The Heisenberg Uncertainty Principle

- Both position and momentum cannot be known exactly and simultaneously in quantum mechanics.
- Heisenberg famous uncertainty principle is

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!

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