Stern-Gerlach experiment

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In quantum mechanics, the Stern–Gerlach experiment,[1] named after Otto Stern and Walther Gerlach, is an important 1922 experiment on the deflection of particles, often used to illustrate basic principles of quantum mechanics. It can be used to demonstrate that electrons and atoms have intrinsically quantum properties, and how measurement in quantum mechanics affects the system being measured.

Contents

Basic theory and description

See also: Spin quantum number#Electron spin
Basic elements of the Stern–Gerlach experiment.

Otto Stern and Walther Gerlach devised an experiment to determine whether particles possess intrinsic angular momentum. In a classical system, such as the earth orbiting the sun, the earth has angular momentum from both its revolution around the sun, and the rotation about its axis. If the electron is treated like a classical dipole with two halves of charge spinning quickly, it will begin to precess in a magnetic field, because of the torque that the magnetic field exerts on the dipole (see torque-induced precession).

If the particle travels in a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. If the experiment is conducted using ions, an electric field of appropriate magnitude and oriented transverse to the charged particle's path is used to compensate for the tendency of any charged particle to curl in its path through a magnetic field (see cyclotron motion), and the fact that ions are charged can safely be ignored. The Stern–Gerlach experiment is normally conducted using electrically neutral particles to prevent the gross deflection of the orbit of a charged particle in a magnetic field and bring out the spin-dependent effect. Therefore a beam of neutral atoms each having an unpaired electron is used. Singly charged ions have also been used.

If the particle travels through an inhomogeneous magnetic field, then the force on one end of the dipole will be slightly greater than the opposing force on the other end of the dipole. This leads to the particle being deflected in the inhomogeneous magnetic field. The direction in which the particles are deflected is typically called the "z" direction.

If the particles are classical, "spinning" particles, then the distribution of their spin angular momentum vectors is taken to be truly random and each particle would be deflected up or down by a different amount, producing an even distribution on the screen of a detector. Instead, the particles passing through the device are deflected either up or down by a specific amount. (Assuming that the precessing atoms have not had sufficient time to radiate away a significant amount of energy) this can only mean that spin angular momentum is quantized (i.e., it can only take on discrete values). There is not a continuous distribution of possible angular momenta.

Spin values for fermions.

Electrons are spin-1⁄2 particles. These have only two possible spin angular momentum values, called spin-up and spin-down. The exact value in the z direction is +ħ/2 or −ħ/2. If this value arises as a result of the particles rotating the way a planet rotates, then the individual particles would have to be spinning impossibly fast. Even if the electron radius were as large as 14 nm (classical electron radius) then it would have to be rotating at 2.3×1011 m/s. The speed of rotation would be in excess of the speed of light, 2.998×108 m/s, and is thus impossible.[2] Thus, the spin angular momentum has nothing to do with rotation and is a purely quantum mechanical phenomenon. That is why it is sometimes known as the "intrinsic angular momentum."

For electrons, two possible values for spin exist, as well as for the proton and the neutron, which are composite particles made up of three quarks each, which are themselves spin-1⁄2 particles. Other particles may have a different number of possible values. Delta baryons (Δ++, Δ+, Δ0, Δ−), for example, are spin +3⁄2 particles and have four possible values for spin angular momentum. Vector mesons, as well as photons, W and Z bosons and gluons are spin +1 particles and have three possible values for spin angular momentum.

To describe the experiment with spin +1⁄2 particles mathematically, it is easiest to use Dirac's bra-ket notation. As the particles pass through the Stern–Gerlach device, they are "being observed." The act of observation in quantum mechanics is equivalent to measuring them. Our observation device is the detector and in this case we can observe one of two possible values, either spin up or spin down. These are described by the angular momentum quantum number j, which can take on one of the two possible allowed values, either +ħ/2 or −ħ/2. The act of observing (measuring) corresponds to the operator Jz. In mathematical terms,

|\psi\rangle = c_1\left|\psi_{j = +\frac{\hbar}{2}}\right\rangle + c_2\left|\psi_{j = -\frac{\hbar}{2}}\right\rangle.

The constants c1 and c2 are complex numbers. The squares of their absolute values (|c1|2 and |c2|2)determine the probabilities that in the state |\scriptstyle \psi\rangle one of the two possible values of j is found. The constants must also be normalized in order that the probability of finding either one of the values be unity. However, this information is not sufficient to determine the values of c1 and c2, because they may in fact be complex numbers. Therefore the measurement yields only the absolute values of the constants.

Sequential experiments

If we link multiple Stern–Gerlach apparatuses, we can clearly see that they do not act as simple selectors, but alter the states observed (as in light polarization), according to quantum mechanical laws [3]:

Sg-seq.svg


History

A plaque at the Frankfurt institute commemorating the experiment

The Stern–Gerlach experiment was performed in Frankfurt, Germany in 1922 by Otto Stern and Walther Gerlach. At the time, Stern was an assistant to Max Born at the University of Frankfurt's Institute for Theoretical Physics, and Gerlach was an assistant at the same university's Institute for Experimental Physics.

At the time of the experiment, the most prevalent model for describing the atom was the Bohr model, which described electrons as going around the positively-charged nucleus only in certain discrete atomic orbitals or energy levels. Since the electron was quantized to be only in certain positions in space, the separation into distinct orbits was referred to as space quantization. The Stern–Gerlach experiment was meant to test the Bohr–Sommerfeld hypothesis that the direction of the angular momentum of a silver atom is quantized.[4]

Note that the experiment was performed several years before Uhlenbeck and Goudsmit formulated their hypothesis of the existence of the electron spin. Even though the result of the Stern−Gerlach experiment has later turned out to be in agreement with the predictions of quantum mechanics for a spin-1⁄2 particle, the experiment should be seen as a corroboration of the Bohr-Sommerfeld theory.[5].

In 1927, T.E. Phipps and J.B. Taylor reproduced the effect using hydrogen atoms in their ground state, thereby eliminating any doubts that may have been caused by the use of silver atoms.[6] (In 1926 the non-relativistic Schrödinger equation had incorrectly predicted the magnetic moment of hydrogen to be zero in its ground state. To correct this problem Pauli introduced "by hand" so to speak, the 3 spin matrices which now bear his name, but which were then later shown by Dirac in 1928 to be a natural consequence of his relativistic equation.)[7]

Impact

The Stern–Gerlach experiment had one of the biggest impacts on modern physics:

  • In the decade that followed, scientists showed using similar techniques, that the nuclei of some atoms also have quantized angular momentum. It is the interaction of this nuclear angular momentum with the spin of the electron that is responsible for the hyperfine structure of the spectroscopic lines.
  • In the thirties, using an extended version of the Stern–Gerlach apparatus, Isidor Rabi and colleagues showed that by using a varying magnetic field, one can force the magnetic momentum to go from one state to the other. The series of experiments culminated in 1937 when they discovered that state transitions could be induced using time varying fields or RF fields. The so called Rabi oscillation is the working mechanism for the Magnetic Resonance Imaging equipment found in hospitals.
  • Norman F. Ramsey later modified the Rabi apparatus to increase the interaction time with the field. The extreme sensitivity due to the frequency of the radiation makes this very useful for keeping accurate time, and it is still used today in atomic clocks.
  • In the early sixties, Ramsey and Daniel Kleppner used a Stern–Gerlach system to produce a beam of polarized hydrogen as the source of energy for the hydrogen Maser, which is still one of the most popular atomic clocks.
  • The direct observation of the spin is the most direct evidence of quantization in quantum mechanics.
  • The Stern–Gerlach experiment has become a paradigm of quantum measurement. In particular, it has been assumed to satisfy von Neumann projection. According to more recent insights, based on a quantum mechanical description of the influence of the inhomogeneous magnetic field[8], this can be true only in an approximate sense. Von Neumann projection can be rigorously satisfied only if the magnetic field is homogeneous. Hence, von Neumann projection is even incompatible with a proper functioning of the Stern–Gerlach device as an instrument for measuring spin.

See also

References

  1. ^ Gerlach, W.; Stern, O. (1922). "Das magnetische Moment des Silberatoms". Zeitschrift für Physik 9: 353–355. doi:10.1007/BF01326984. 
  2. ^ Tomonaga, S.-I. (1997). The Story of Spin. University of Chicago Press. p. 35. ISBN 0-226-80794-0. 
  3. ^ Sakurai, J.-J. (1985). Modern quantum mechanics. Addison-Wesley. 
  4. ^ Stern, O. (1921). "Ein Weg zur experimentellen Pruefung der Richtungsquantelung im Magnetfeld". Zeitschrift für Physik 7: 249–253. doi:10.1007/BF01332793. 
  5. ^ Weinert, F. (1995). "Wrong theory—right experiment: The significance of the Stern–Gerlach experiments". Studies in History and Philosophy of Modern Physics 26B: 75−86. doi:10.1016/1355-2198(95)00002-B. 
  6. ^ Phipps, T.E.; Taylor, J.B. (1927). "The Magnetic Moment of the Hydrogen Atom". Physical Review 29 (2): 309–320. doi:10.1103/PhysRev.29.309. Bibcode: 1927PhRv...29..309P. 
  7. ^ A., Henok (2002). Introduction to Applied Modern Physics. Lulu.com. p. 76. ISBN 1435705211 [Amazon-US | Amazon-UK]. 
  8. ^ Scully, M.O.; Lamb, W.E.; Barut, A. (1987). "On the theory of the Stern–Gerlach apparatus". Foundations of Physics 17: 575–583. doi:10.1007/BF01882788. 

Further reading

External links

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