Stern-Gerlach experiment

From Physics wiki

Jump to: navigation, search

In the Stern-Gerlach experiment a beam of silver atoms (or other particles) passes through an inhomogeneous magnetic field and strikes a photographic plate. Otto Stern and Walther Gerlach devised and performed the experiment in 1922 in order to test the Bohr-Sommerfield model of the atom.

Contents

Experimental setup

Predictions

The potential energy of a particle with a magnetic dipole moment \boldsymbol{\mu}\, in a magnetic field \mathbf{B}\, is

U = -\boldsymbol{\mu}\cdot\mathbf{B}\,.

Ignoring variations in the field axes other than the z\, axis, we see that the force on the particle is

F_z = -\frac{\partial U}{\partial z} = \mu_z \frac{\partial B_z}{\partial z}\,.

In Larmor's theory \mu_z\, could take on any value, and so we would expect the intensity of the beam to reach a maximum near the center (no deflection) as well as some spread due to the thermal nature of the problem.

According to the Bohr-Sommerfield model, \mu_z\, could take on one of two values, and so we would expect the intensity to have two distinct peaks.

Results

Gerlach's postcard, dated 8 February 1922, to Niels Bohr. It shows a photograph of the beam splitting, with the message, in translation: "Attached [is] the experimental proof of directional quantization. We congratulate [you] on the confirmation of your theory." (Physics Today December 2003)
Gerlach's postcard, dated 8 February 1922, to Niels Bohr. It shows a photograph of the beam splitting, with the message, in translation: "Attached [is] the experimental proof of directional quantization. We congratulate [you] on the confirmation of your theory." (Physics Today December 2003)

In the experiment the beam was clearl split into two components, with a minimum intensity in the center, as can be seen in the postcard from Gerlach to Bohr. Although not providing a conclusive explanation for the origin of the splitting, the experiment did prove that the magnetic dipole moment of the silver atoms could take one of two discrete values.

Goudsmit and Uhlenbeck's interpretation

In 1925, Samuel A. Goudsmit and George E. Uhlenbeck postulated that the electron had an intrinsic angular momentum \mathbf{S}\,, independent of its orbital angular momentum \mathbf{L}\,, in order to explain observations of atomic spectra, although they made no mention of the results of Stern and Gerlach. The instrinsic angular momentum, or spin, gives rise to an independent magnetic dipole moment equal to

\boldsymbol{\mu}_s = -\frac{e}{2m}g \mathbf{S}\,.

A measurement of S_z\,, as in the Stern-Gerlach experiment, would yield one of two values: S_z = +\tfrac{\hbar}{2}\,, or S_z = -\tfrac{\hbar}{2}\,.

Sequential experiments

By placing a filter in the path of the beam, the Stern-Gerlach apparatus may be used to produce a beam of atoms (or electrons) with specific values of S_z\,, or any other orientation of \mathbf{S}\,, for that matter. The situation becomes more interesting when we place several apparatuses (with different field orientations) in sequence.

Suppose we set up the experiment so that the first apparatus only allows atoms with S_z = \tfrac{\hbar}{2}\, to pass through. If this beam then passes through a second apparatus with the same orientation, then a single bright spot is seen on the photographic plate, as expected. If, on the other hand, the second apparatus is oriented along, for example, the x\,-axis, then the beam is again split into two polarizations as viewed on the photographic plate. If we again select one of the beams from the second apparatus, for example, the one corresponding to S_x = -\tfrac{\hbar}{2}\,, and allow it to pass through a third apparatus oriented along the z\,-axis, then we observe two polarizations on the photographic plate. Evidently, the selection S_z = \tfrac{\hbar}{2}\, of the first apparatus was not preserved throughout the experiment. Furthermore, the change in the value of S_z\, resists any classical explanation, as the effect is necessarily non-deterministic.

Quantum mechanical explanation

References

[1]

  1. ↑ Franklin, Allan, "Experiment in Physics", The Stanford Encyclopedia of Philosophy (Fall 2007 Edition), Edward N. Zalta (ed.), <http://plato.stanford.edu/entries/physics-experiment/app5.html>.
Retrieved from "http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/Stern-Gerlach_experiment"
Personal tools