Physical Chemistry 2nd Edition Chapter 17 Commuting and Noncommuting Operators and the Surprising Consequences of Entanglement Physical Chemistry 2nd Edition 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 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 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 Using the de Broglie relation, the mean momentum is given by And .

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!