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## The Stern-Gerlach Experiment

This text constitutes a short introduction to a JAVA applet designed to illustrate the measurement process in quantum mechanics, through a visualization of the classic Stern-Gerlach experiment. Its contents are:
• ### Description of the apparatus

The Stern-Gerlach apparatus consists essentially of a magnet producing a non-uniform magnetic field. A beam of atoms enters the magnet in a direction perpendicular to the field gradient. As a consequence of the interaction of their spin with the magnetic field, the atoms undergo a deflection as they go through the field. Beyond the magnet, the atoms are detected by counters, possibly acting also as filters.

• ### Interaction of the atomic spin with the magnetic field

To the spin (or intrinsic angular momentum) of the atom is associated a magnetic moment, proportional to the spin. It can be shown that a non-uniform magnetic field applies upon a magnetic moment a force aligned with the direction of the field gradient. The value of the force is proportional to the field gradient and to the component of the magnetic moment in the direction of this gradient.

Thus if, as is the case in the applet, the field gradient is vertical and the initial direction of the beam is horizontal, the atoms will be deflected upwards or downwards, according to the value of the component of their spin in the vertical direction. To be specific, we assume that atoms whose vertical spin component is positive are deflected upwards and those whose vertical spin component is negative are deflected downwards.

• ### Spin quantization

In quantum mechanics, spin is quantized. The spin of a given atom can be characterized by a quantum number s which may assume integer or half-integer values. For a given value of s, the projection of the spin on any axis may assume 2s+1 values ranging from -s to +s by unit steps (in units of the rationalized Planck constant). We shall deal with atoms of spin s=1/2, in which case the spin projection can assume the values +1/2, which we shall refer to as spin up and -1/2, which we shall refer to as spin down.

According to the discussion of the interaction of the spin with the magnetic field, atoms with spin up will be deviated upwards when passing through the magnet, and atoms with spin down will be deviated downwards. Therefore, by observing which way an atom goes, we can measure its spin component along the vertical axis.

• ### Measurement in quantum mechanics

In quantum mechanics, the value of a physical quantity (usually called an observable) cannot in general be predicted with certainty. What this theory permits is to attribute a probability to every possible value of an observable. In order to compare these theoretical probabilities to data, one must perform measurements on a number N of identical systems and count the number of times Nv a particular value v of the observable is obtained. In the limit of a very large number N of measurements, the relative frequency Nv/N of occurrence of the value v approaches the probability Pv associated to this value.

For example, in the case of the Stern-Gerlach experiment, if N is the number of atoms in the beam and N+ the number of atoms that are deflected upwards, then N+/N approaches the probability P+ of an atom having spin up.

• ### Pure states

In the discussion of the measurement of a given observable, the states for which the probability of obtaining a particular result is unity play a central role. In the case of the component of a spin s=1/2, there are two such states, which we shall denote |+) and |-), according to the sign of the corresponding value. More general states, for which both positive and negative results are possible, can be constructed by linear superposition of the above states, i.e.:

|general state) = c+|+) + c-|-),

where c+ and c- are arbitrary complex numbers, subject to the normalization condition

|c+|2 + |c-|2 =1.

 P+ =|c+|2 and P- =|c-|2.

In the applet, the coefficients c+ and c- are assumed to be real and are parametrized in terms of an angle a through

 c+ = cos(a/2) and c- = sin(a/2).

Although the motion through the Stern-Gerlach apparatus of an atom whose spin component is known with certainty [i.e. an atom in a state |+) or |-)] can be described approximately in terms of the classical mechanics of a point particle, a more precise description associates to the atom a wave packet, i.e. a probability distribution whose center describes the classical trajectory. For an atom in the general spin state, the wave packet is split by the magnet in two components, one of which is deflected upwards and the other downwards. The probabilities associated to these components are P+ and P-, respectively.

• ### Statistical mixtures

Pure states correspond to the maximum knowledge which can be acquired about a quantum system. Such knowledge is not always available. In many cases, one is only able to attribute probabilities to the various possible states. For instance, in the case of the atomic beam entering the Stern-Gerlach apparatus, all we might know are the probabilities of the state being |+) or |-). We then have a statistical mixture. Although we shall also use the notation P+ and P- for these probabilites, their nature is quite different from that of the probabilities associated to components of a pure state.

In the case of a beam of atoms, a pure spin state corresponds to a completely polarized beam while a statistical mixture corresponds to a partially polarized beam if the probabilities for the various possible states are unequal and to an unpolarized beam if the probabilities are equal.

If the beam consists of a statistical mixture of the states |+) and |-), those atoms that are in the state |+) will be deflected upwards by the magnet, while those atoms that are in the state |-) will be deflected downwards.

• ### Quantum-state reduction

When a measurement is performed on a quantum system, its result provides new information about the system. This means that the probabilities associated with possible results of subsequent measurements are modified. Since these probabilities are derived from the quantum state, one may conclude that a measurement affects the quantum state, in a way which depends on the result obtained. In fact, in the simplest case, the result obtained in the measurement determines completely the subsequent quantum state. This effect of the measurement on the quantum state is called reduction or collapse of the quantum state.

In the case of the Stern-Gerlach experiment, if the detector located on the upwards-deflected path clicks, then after the click we know that the atom has spin up, and therefore is described by the state |+). Similarly, if the detector located on the downwards-deflected path clicks, the subsequent spin state is |-). So the initial general pure state is reduced to |+) or |-), depending on which detector clicks.

In fact, there is not even need for two detectors. If for instance we have a detector on the downward-deflected path only, then if the detector does not click, we may conclude that the atom was not deflected downwards. Since it had to go somewhere, we may conclude that it went upwards and therefore has spin up and can be attributed the state |+). In that case, the state of the atom gets reduced without any apparent interaction between atom and detector!

From the above discussion, it is clear that state reduction should be conceived as a logical process associated with the acquisition of new information, rather than as a truly physical process. For this reason, the interpretation of quantum states as representing an underlying objective reality - a philosophical stand commonly characterized as realism - is not easily reconciled with quantum mechanics as it is usually formulated.

These considerations should be kept in mind when interpreting the visualization presented by the applet, in which state reduction is shown in a rather concrete fashion.

In the case of a statistical mixture of the states |+) and |-), the detection of the atoms reveals in which of these two states a particular atom really was. This is the familiar process of acquiring the information we were missing to be able to describe with certainty a situation which already existed prior to the measurement. No state reduction is implied.

• ### Running the applet

• #### Controls and inputs

In the applet, you can choose between two kinds of atomic beams, distinguished by the description of their spin: pure state or statistical mixture.

You can also choose between two experimental setups. In the first, referred to as measurement, there are two detectors which "observe" the passage of the atoms without stopping them. In the second, called filtering, there is only one detector, on the downwards path, and it acts as a filter, i.e. it stops an atom upon detection.

If you choose the pure state option, the input parameter is the angle a for which you can enter any integer value between 0 and 180 degrees. The corresponding probabilities P+ and P- will then automatically appear on the readers of the detectors (if you choose the filtering option, only P- appears; of course P+=1-P-).

If you choose the statistical mixture option, the input parameter is the probability P+, for which you should enter a number between 0 and 1 with no more than 2 significant digits.

Note that you can enter the parameters either by typing them as text or by adjusting the slider.

Beside the options and parameter inputs described above, the applet possesses the following controls:

• A Start-Stop button which starts and stops a given experiment. You should note that if you stop an experiment, the counters are reset to zero. You should use the button described below to interrupt temporarily an experiment.
• A Pause-Go on button which allows you to suspend temporarily and then resume a given experiment.
• An Animation Speed button, which allows you to speed up or slow down the animation.

• #### Hints

Notice the flash (and click if your computer produces sound) when an atom is detected. To test your reflexes, try to freeze the animation with the Pause button at the instant of the flash.

If you are watching an experiment on a pure state, notice that the wave packet divides into two components as the atom enters the magnet. The shade of a component gives a visual measure of the probability associated with it.

If you are watching an experiment on a pure state, observe the reduction of the state vector (or collapse of the wave packet): when a detector flashes, one of the components of the wave packet disappears and the other acquires the darkest possible shade, associated with unit probability.

In a filtering experiment, catch the collapse of the wave packet of an atom which is not detected.

Notice how the relative frequencies slowly approach the theoretical probabilities as the number of detected atoms increases. Since this process of accumulating events can become rather tedious, a good idea is to speed up the animation after observing carefully the aspects mentionned above, with the animation running slowly.

If your computer is equipped with loudspeakers, you can turn on the Sound checkbox to hear noises produced by the firing of the detectors. However, with this option on, the animation may become rather slow and unsmooth.

Note that the applet runs in a separate window, so that you can come back to consult this text without interrupting the animation. Homepage do autor Homepage do IFURGS

Michel Betz betz@if.ufrgs.br
Instituto de Física
Universidade Federal do Rio Grande do Sul