| Part of a series of articles about |
| Quantum mechanics |
|---|
Scientists |
Inquantum physics, theStern–Gerlach experiment demonstrated that the spatial orientation ofangular momentum isquantized. Thus anatomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silveratoms were sent through a spatially-varyingmagnetic field, whichdeflected them before they struck a detector screen, such as aglass slide. Particles with non-zeromagnetic moment were deflected, owing to the magnetic fieldgradient, from a straight path. The screen revealed discrete points of accumulation, rather than a continuous distribution,[1] owing to their quantizedspin. Historically, this experiment was decisive in convincing physicists of the reality of angular-momentum quantization in all atomic-scale systems.[2][3][4]
After its conception byOtto Stern in 1921, the experiment was first successfully conducted withWalther Gerlach in early 1922.[1][5][6]
The Stern–Gerlach experiment involves sendingsilver atoms through aninhomogeneousmagnetic field and observing their deflection. Silver atoms were evaporated using an electric furnace in a vacuum. Using thin slits, the atoms were guided into a flat beam and the beam sent through an inhomogeneous magnetic field before colliding with a metallic plate. The laws of classical physics predict that the collection of condensed silver atoms on the plate should form a thin solid line in the same shape as the original beam. However, the inhomogeneous magnetic field caused the beam to split in two separate directions, creating two lines on the metallic plate.[3]
The results show that particles possess an intrinsicangular momentum that is closely analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values. Another important result is that only one component of a particle's spin can be measured at one time, meaning that the measurement of the spin along the z-axis destroys information about a particle's spin along the x and y axis.
The experiment is normally conducted using electricallyneutral particles such as silver atoms. This avoids the large deflection in the path of a charged particle moving through a magnetic field and allows spin-dependent effects to dominate.[7][8]
If the particle is treated as a classical spinningmagnetic dipole, it willprecess in a magnetic field because of the torque that the magnetic field exerts on the dipole (seetorque-induced precession). If it moves through 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. However, if the magnetic field is inhomogeneous then the force on one end of the dipole will be slightly greater than the opposing force on the other end, so that there is a net force which deflects the particle's trajectory. If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to berandom andcontinuous. Each particle would be deflected by an amount proportional to thedot product of its magnetic moment with the external field gradient, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount. This was a measurement of the quantumobservable now known asspin angular momentum, which demonstrated possible outcomes of a measurement where the observable has a discrete set of values orpoint spectrum.[9]
Although some discrete quantum phenomena, such asatomic spectra, were observed much earlier, the Stern–Gerlach experiment allowed scientists to directly observe separation between discrete quantum states for the first time.
Theoretically,quantum angular momentumof any kind has a discrete spectrum, which is sometimes briefly expressed as "angular momentum is quantized".
Spin-1/2 particles have only two possible spin angular momentum values measured along any axis, or, a purely quantum mechanical phenomenon. Because its value is always the same, it is regarded as an intrinsic property and is sometimes known as "intrinsic angular momentum" (to distinguish it from orbital angular momentum, which can vary and depends on the presence of other particles). If one measures the spin along a vertical axis, the states are described as "spin up" or "spin down", based on the magnetic moment pointing up or down, respectively.
To mathematically describe the experiment with spin-1/2 particles, it is easiest to useDirac'sbra–ket notation. As the particles pass through the Stern–Gerlach device, they are deflected either up or down, and observed by the detector which resolves to either spin up or spin down. These are described by the angular momentum quantum number, which can take on one of the two possible allowed values, either +1/2 or -1/2. The act of observing (measuring) the momentum along the axis corresponds to the-axisangular momentum operator, often denoted. In mathematical terms, the initial state of the particles is
where constants and are complex numbers. This initial state spin can point in any direction. The squares of theabsolute values and are respectively the probabilities for a system in the state to be found in and after the measurement along axis is made. The constants and must also be normalized in order that the probability of finding either one of the values be unity, that is we must ensure that. However, this information is not sufficient to determine the values of and, because they are complex numbers. Therefore, the measurement yields only the squared magnitudes of the constants, which are interpreted as probabilities.
If the experiment is conducted using charged particles like electrons, there will be aLorentz force that tends to bend the trajectory in a circle. This force can be cancelled by an electric field of appropriate magnitude oriented transverse to the charged particle's path, but by developing arguments proposed by N. Bohr, N. F. Mott claimed that theuncertainty principle would make a Stern–Gerlach type experiment with electrons impossible.[10] However, whenHans Georg Dehmelt measured the anomalous g-factor of the electron in apenning trap, scientists likeRudolf Peierls asserted that Bohr's claim was incorrect .[11][12]
If we link multiple Stern–Gerlach apparatuses (the rectangles containingS-G), we can clearly see that they do not act as simple selectors, i.e. filtering out particles with one of the states (pre-existing to the measurement) and blocking the others. Instead they alter the state by observing it (as inlight polarization). In the figure below, x and z name the directions of the (inhomogenous) magnetic field, with the x-z-plane being orthogonal to the particle beam. In the three S-G systems shown below, the cross-hatched squares denote the blocking of a given output, i.e. each of the S-G systems with a blocker allows only particles with one of two states to enter the next S-G apparatus in the sequence.[13]
The top illustration shows that when a second, identical, S-G apparatus is placed at the exit of the first apparatus,only z+ is seen in the output of the second apparatus. This result is expected since all particles at this point are expected to have z+ spin, as only the z+ beam from the first apparatus entered the second apparatus.[14]
The middle system shows what happens when a different S-G apparatus is placed at the exit of the z+ beam resulting of the first apparatus, the second apparatus measuring the deflection of the beams on the x axis instead of the z axis. The second apparatus produces x+ and x- outputs. Now classically we would expect to have one beam with the x characteristic oriented + and the z characteristic oriented +, and another with the x characteristic oriented - and the z characteristic oriented +.[14]
The bottom system contradicts that expectation. The output of the third apparatus which measures the deflection on the z axis again shows anoutput ofz- as well as z+. Given that the input to the second S-G apparatus consistedonly of z+, it can be inferred that a S-G apparatus must be altering the states of the particles that pass through it. This experiment can be interpreted to exhibit theuncertainty principle: since the angular momentum cannot be measured on two perpendicular directions at the same time, the measurement of the angular momentum on the x direction destroys the previous determination of the angular momentum in the z direction. This is why the third apparatus measures both z+ and z- beams; the x measurement "makes a clean slate" of the z+ output.[14]
The Stern–Gerlach experiment was conceived byOtto Stern in 1921 and performed by him andWalther Gerlach inFrankfurt in 1922.[13] At the time of the experiment, the most prevalent model for describing theatom was theBohr-Sommerfeld model,[15][16] which describedelectrons as going around the positively chargednucleus only in certain discreteatomic orbitals orenergy levels. Since the electron wasquantized to be only in certain positions in space, the separation into distinct orbits was referred to asspace quantization. The Stern–Gerlach experiment was meant to test theBohr–Sommerfeld hypothesis that the direction of the angular momentum of a silver atom is quantized.[17]
The experiment was first performed with an electromagnet that allowed the non-uniform magnetic field to be turned on gradually from a null value.[1] When the field was null, the silver atoms were deposited as a single band on the detecting glass slide. When the field was made stronger, the middle of the band began to widen and eventually to split into two, so that the glass-slide image looked like a lip-print, with an opening in the middle, and closure at either end.[18] In the middle, where the magnetic field was strong enough to split the beam into two, statistically half of the silver atoms had been deflected by the non-uniformity of the field.
The experiment was performed several years beforeGeorge Uhlenbeck andSamuel Goudsmit formulated their hypothesis about the existence ofelectron spin in 1925.[19] 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 experimental result was also consistent with theBohr–Sommerfeld theory.[20]
In 1927, Thomas Erwin Phipps andJohn Bellamy Taylor [de] reproduced the effect usinghydrogen atoms in theirground state, thereby eliminating any doubts that may have been caused by the use ofsilver atoms.[21] However, in 1926 the non-relativistic scalarSchrödinger equation had incorrectly predicted themagnetic moment of hydrogen to be zero in its ground state. To correct this problemWolfgang Pauli considered a spin-1/2 version of the Schrödinger equation using the 3Pauli matrices which now bear his name, which was later shown byPaul Dirac in 1928 to be a consequence of his relativisticDirac equation.
In the early 1930s Stern, together withOtto Robert Frisch andImmanuel Estermann improved themolecular beam apparatus sufficiently to measure the magnetic moment of theproton, a value nearly 2000 times smaller than the electron moment. In 1931, theoretical analysis byGregory Breit andIsidor Isaac Rabi showed that this apparatus could be used to measure nuclear spin whenever the electronic configuration of the atom was known. The concept was applied by Rabi and Victor W. Cohen in 1934 to determine the spin ofsodium atoms.[22]
In 1938 Rabi and coworkers inserted an oscillating magnetic field element into their apparatus, inventingnuclear magnetic resonance spectroscopy.[23][24] By tuning the frequency of the oscillator to the frequency of the nuclear precessions they could selectively tune into each quantum level of the material under study. Rabi was awarded the Nobel Prize in 1944 for this work.[25]
The Stern–Gerlach experiment was the first direct evidence of angular-momentum quantization in quantum mechanics,[26] and it strongly influenced later developments inmodern physics:
{{cite book}}:ISBN / Date incompatibility (help)
