The Stern-Gerlach Experiment

THE FOURTH ELECTRONIC DEGREE OF FREEDOM The Stern-Gerlach Experiment 

In 1921, Stern and Gerlach performed an experiment that later turned out to be a milestone in quantum mechanics.1,2 First, it provided an experimental basis for the concept of electron spin, introduced in 1925 by Goudsmit and Uhlenbeck.3,4 Second, it evolved into the quantum mechanical experiment par excellence. From this experiment, we easily learn basic concepts of quantum mechanics such as the additivity of probability amplitudes, basis states, projection operators, and the resolution of the identity.5 The latter concept relates to the fact that a complete set of basis states (i.e., the identity) can be inserted in any quantum mechanical equation without changing the result. 

0 is the acute angle between the orientation of the particle magnetic moment and the magnetic field vector. Before entering the magnet, the silver atoms are oriented randomly with respect to the magnetic field (i.e., cos(0) can adopt any value between -1 and 1). Classically, the interaction of 

The results of the Stern-Gerlach experiment were in complete contradiction to the classical interpretation and its predictions. Silver atoms turned out to possess a magnetic moment, but instead of a single, smeared-out distribution, two spots centered around Mup and Mdown were observed. Thus the magnetic moment of a silver atom is space-quantized by an inhomogeneous magnetic field, and this magnetic moment can adopt only two values, hz Hhl  

The origin of this magnetic moment was not clear in 1922. In its electronic ground state, a silver atom does not possess a spatial angular momentum, and the concept of an intrinsic electronic angular momentum (the electron spin) was yet to be created. In 1925, Goudsmit and Uhlenbeck introduced a fourth (spin) electron degree of freedom—in addition to the three spatial coordinates (x, y, z)—as a model to ease the explanation of the anomalous Zeeman effect.3,4 

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The Stern-Gerlach experiment demonstrated that electrons have an intrinsic angular momentum in addition to their orbital angular momentum, and the unfortunate term electron spin was coined to describe this pure quantum-mechanical phenomenon. Many nuclei also possess an internal angular momentum, referred to as nuclear spin. As in classical mechanics, there is a relationship between the angular momentum and the magnetic moment. For electrons, we write... 

Let us consider this case in some detail. If collisions are eliminated in a molecular beam, it is possible to orient molecules (their figure axis) by removing the particles possessing unwanted orientation (analogous to the Stern-Gerlach experiment with a magnetic field). Then, classically, the interaction energy with external electric field is simply... 

Figure 1 Schematic drawing of the Stern-Gerlach experiment.

From our present standpoint, we know that the deflection of a silver atom in the Stern-Gerlach experiment is caused by the interaction of its electronic spin angular momentum S with an inhomogeneous magnetic field. The projection of S on the direction of this field, Ms, is quantized. For a silver atom, Ms can take two values +vjh and — h, where h is Planck s constant h over 2ji and adopts a value of 1.054571596 x 10-34 Js in cgi units. 

Let us first consider the normal Zeeman effect, which applies to transitions between electronic states with zero total spin magnetic moment, so-called singlet states. Like the projection Ms of S in the Stern-Gerlach experiment, the projection Ml of the spatial angular momentum L is space quantized in the external magnetic field. We shall describe the quantization of the spatial angular momentum by means of quantum mechanical methods in detail later. Suffice it to say that each state with spatial angular momentum quantum number L splits into 2L + 1 components, i.e., a P state (L = 1) splits into three components with... 

The conceptual simplicity of the Stern-Gerlach experiment, coupled with the directness of its results, provided commanding evidence for the quantum theory. I. I. Rabi was a graduate student at Columbia University when the Stern results were announced. The Stern experiment changed forever Rabi s thinking about quantum mechanics. This convinced me once and for all, Rabi said later, that an ingenious classical mechanics was out and that we had to face the fact that the quantum phenomena required a completely new orientation. ... 

Shortly after the results of the Stern-Gerlach experiment appeared in the scientific literature. Stern received an invitation to join the faculty at the University of Hamburg, where, over the period 1922 to 1933, he continued his experimental work. In 1932, Stern decided to adapt the molecular beam method to a daunting experiment to measure the magnetic moment of the proton, the nucleus of the hydrogen atom. Joining him is this endeavor was Otto Robert Frisch. 

The Stern-Gerlach experiment demonstrates that the electron has a property called spin, which leads to a magnetic dipole moment. Spin is quantized with only two allowed values described by the quantum number m. Complete determination of the quantum state of the electron required values for all four quantum numbers (n, , m, m. ... 

Wolfgang Pauli helped to develop quantum mechanics in the 1920s by forming the concept of spin and the exclusion principle. According to Schrodinger s Equation, each electron is unique. The Pauli Exclusion Principle states that no two electrons may have the same set of quantum numbers. Thus, for two electrons to occupy the same orbital, they must have different spins so each has a unique set of quantum numbers. The spin quantum number was confirmed by the Stern-Gerlach experiment. 

The polarizability of some neutral atoms (H, Li, K, Cs) has been recently determined by a method similar to the Stern-Gerlach experiment ( 7, p. 166), namely, by measuring the deflexion of a beam of atoms in an inhomogeneous electric field (Stark, 1936). The results do not agree very well with theoretical computations from atomic models. 

Figure 1. The Stern-Gerlach experiment (highly schematic).

We focused on the size-selected cluster ions with small sizes in the above. On the other hand, the magnetic moments of neutral transition metal clusters in the size range between several and several hundreds are successfully measured by deflection in an inhomogeneous magnetic field. This method is basically the same as the Stern-Gerlach experiment known by the discovery of the electron spin by using the deflection of an atomic beam of silver. 

Overcoming these challenges, the two scientists carried out the stunning experiment now known by their names. The Stern-Gerlach experiment of 1922 proved that the magnetic moment of silver atoms is quantized, allowing a beam of the atoms to be separated by a magnetic field into... 

FIGURE 12.1 A diagram of the Stern-Gerlach experiment. A beam of silver atoms passed through a magnetic field splits into two separate beams. This finding was used to propose the existence of spin on the electron. 

Use the table of electron configurations to identify other elements that would yield the same result as Ag did in the Stern-Gerlach experiment. 

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