cphil_qm2
QUANTUM MECHANICS
CONTINUED
by Ray Shelton
In the case of a single point particle, the wave function may be thought of as an oscillating field spread throughout physical space. At each point in this space it has an amplitude and a wavelength. The square of the amplitude is proportional to the probability of finding the particle at that position; the wavelength, for a constant amplitude wave function, is related to the momentum of the particle ( equation [9b]). The particle will therefore have a definite position if the wave function is tightly bunched about a particular point in space; and it will have definite momentum if the wavelength and amplitude of the wave function are uniform throughout all of space. Typical wave functions for a system will not be of either of these types and there will be a certain amount of indefiniteness, or uncertainty, in both position and momentum. In particular, because of the mutually exclusive types of wave functions required for definite position and definite momentum, position and momentum cannot be definite simultaneously. This is known as the Heisenberg’s Uncertainty Principle, and is an elementary consequence of the wavelike character of particles. In a “coherent” state, which is a compromise between definite position and definite momentum, there is uncertainty in both position and momentum. This means that the laws of physics are no longer deterministic and the phenomena that they describe are no longer subject to a rigorous determinism; they only obey the laws of probability. Heisenberg’s Principle of Uncertainty gave an exact formulation to this fact.
In general, the uncertainty principle states that there is a real, physical, and absolute limit to the delicacy with which observations can be made. Consequently, any observation necessarily produces an inherently uncontrollable and unknowable disturbance in the subsequent behavior of the system. Heinsenberg’s original statement of the uncertainty principle was more specific and more quantitative than the one we have just given. It stated that if the x-coordinate of a particle is measured to within an uncertainty Δx, then its x component of momentum will necessarily be disturbed in such a way that it becomes uncertain by an amount at least
Δpx, where
That is, the product of the uncertainties ΔxΔpx is a lower bound of h/2π, Planck’s constant divided by 2π. If the position of an electron is determined to within a small distance, so that Δx is small, then the corresponding uncertainty in momentum,
Δpx ≥ (h/2π)/Δx, is large.
Conversely, if Δpx is small, then
Δx ≥ (h/2π)/Δpx
and there is a large uncertainty in position. Thus it is not possible for both Δx and Δpx to be arbitrarily small. The careful derivation of this uncertainty principle was done in 1927 by the German physicist Werner Heisenberg (1901-1976). He derived it not just for the x-dimension but also for the y– and z-dimensions; that is,
ΔxΔpx ≥ h/2π
ΔyΔpy ≥ h/2π
ΔzΔpz ≥ h/2π
or, more generally,
ΔqΔp ≥ h/2π
where Δq is the uncertainity in its location in any direction, x, y or z, and Δp is the uncertainity of the momentum in the corresponding direction.
The double slit experiment describe earlier in which the slit through which an electron emerges is observed, is a good example of the uncertainty principle. Consider one electron of an monoenergetic beam that is incident on a single slit. This arrangement can be regarded as a way to measure the vertical coordinate y of an electron passing through the slit. An electron that emerges from the slit has a position that is uncertain by Δy equal to the width of the slit d. This observation corresponds to a measurement of the vertical coordinate of the position of the electron to within an uncertainty equal to the width of the slit. Since we don’t know beforehand where the electron will hit the screen, the vertical component of the momentum py is uncertain Δpy, which we now can estimate. Using the wave nature of an electron, we can expect that an electron is likely to hit the screen somewhere between the two minima of the single-slit diffraction pattern. The condition for the minimum for a wave of wavelength λ is d sin θ = λ. The uncertainty in momentum that corresponds to an electron hitting anywhere between these minima is
|Δpy| = 2p sin θ = 2(h/λ) sin θ.
The uncertainty product is
ΔyΔpy = d[2(h/λ) sin θ] = 2h,
since d sin θ = λ.
The measuring process introduces uncertainties consistent with the Heisenberg uncertainty relation. In this example, the estimate of the uncertainties Δy, Δpy the particlelike quantities are connected with the wave nature of an electron and that is generally true. In the double slit experiment the resulting uncertainty in the vertical component of the momentum of the electron is reflected in a redistribution that destroy the interference pattern.
In addition to the uncertainty relation equation [24] between a position coordinate and the corresponding momentum, there is an uncertainty relation between time and energy. Suppose that not only the energy of a particle is to be measured, but also the time at which the particle has such energy. If Δt and ΔE are the uncertainties in the values of these quantities, the following relation holds:
This relation is to be understood in the following way. If the time at which a particle passes through a given point is to be defined, the particle must be represented by a pulse or wave packet having a very short duration Δt. But to build such a pulse it is necessary to superpose waves which have different frequencies, with an amplitude appreciable only in a frequency range Δf centered around the frequency f. The theory of waves requires that several oscillations must be observed before a reasonable estimate of the frequency is determined. For wave motion it can be shown that a measurement carried out in a time interval Δt will yield a frequency measurement with uncertainty Δf such that
Multiplying this equation by h and using the differential equation obtained from equation [9a]: hf = E,
hΔf = ΔE,
we get the Heisenberg energy-time principle of uncertainty, given by equation [25]:
ΔtΔE ≥ h/2π.
Thus the product of the uncertainties in energy and in time is at least as Planck’s constant h/2π. The meaning of equation [25] is as follows. If an object — an electron or a photon — is known to have an energy state E over a limited period of time Δt, then the energy will be uncertain by at least an amount h/(2πΔt); thus the energy of the object can be given with infinite precision (ΔE = 0) only if the object exists for an infinite time (Δt = ∞).
Now the uncertainty relation between the position and the momentum of a particle, equation [24], can be derived. For an particle moving along the x-axis, the relation between the uncertainty Δpx in momentum and the uncertainty Δλ in wavelength can be obtained by taking the differential of equation [9b]:
p = h/λ;
we get
Suppose that the wave has been observed only over the finite time Δt; then during this time the wave will have traveled the distance
Δx = wΔt,
where w is the speed of the wave. Therefore,
w = Δx/Δt.
But since w = fλ, where f is the frequency of the wave and λ is its wavelength, then f = w/λ. Taking the differential of this relation, we get
Δf = w(-Δλ/λ2) = (Δx/Δt)(-Δλ/λ2).
Hence, ΔfΔt/Δx = -Δλ/λ2.
Substituting into equation [27], we get
Δpx = h(ΔfΔt/Δx), or ΔxΔpx = hΔfΔt,
But substituting equation [26] into this, we get equation [24].
ΔxΔpx ≥ h/2π. [24]
Both wave and particle characteristics are attributed to electromagnetic radiation and to material particles. The concepts of wave and particle are basic to physics; they represent the only two possible modes of energy transport. Classical physics always successfully describes any large-scale phenomenon of energy transport by applying one of these descriptions. For example, a disturbance that travels across the surface of a pond is certainly a wave, and a thrown baseball is an example of energy transport by a particle. There is no doubt about which description that should be applied in each case. But less direct example of wave motion is the propagation of sound through air. We do not “see” the waves, as we did the water waves; but the wave description is confidently applied to sound propagation, because it alone describes diffraction and interference. The wave model for the propagation of sound agrees with all experimental observations of sound. But even less direct is the explanation of the behavior of gases by the kinetic theory of gases. Even though the molecules can not be seen, the explanation by very small, hard spheres account for the various experiments performed upon gases. A particle model is the only appropriate way of explaining gas behavior. When explaining new phenomena, either one of these two models is applied, because one of them is always successful in accounting for the experimental data.
But the wave and particle models are mutually incompatible and contradictory. If a wave is to have its frequency or its wavelength given with infinite precision, then it must have an infinite extension in space. On the contrary, if it is confined in some limited region of space, it resembles a particle by its localizability, but it cannot be characterized by a single frequency and wavelength. An ideal wave, one whose frequency and wavelength are known with certainty, is altogether incompatible with an ideal particle, which is localized in space. The photoelectric effect, the Compton effect, and other experiments on X-rays have placed the particle theory of light on firm experimental basis. But the classic experiments of Young and others on diffraction and interference showed that the wave theory of light rests on firm basis. The wave theory describes experiments in interference and diffraction, in which there are alternate light and dark bands that are predicted and explained by the wave theory. The particle theory does not explain interference or diffraction. Maxwell’s classical electromagnetic theory uses the wave model to explain the propagation of light. But it cannot account for the quantum effects of light. These quantum effects of light are shown in the interaction of electromagnetic radiation with particles. The interaction between radiation and matter requires the particle model, and such interactions are best described as collisions between particles. Thus radiation must consist of photons, each with specific energy and momentum, with these electromagnetic particles localized at a particular point in space, namely at the site of the interaction. The photon-interaction experiments can only be explained if the electromagnetic radiation is assumed to consist of particles in collision. That is, the photon-electron interaction can only be explained by the particle model, not by the wave model.
Experiments show that electromagnetic radiation have both wave and particle characteristics, but not in the same experiments. Interference or diffraction experiments require a wave interpretation, and it is impossible to apply simultaneously a particle interpretation; a photon-interaction experiment requires a particle interpretation, and it is impossible to apply simultaneously the wave interpretation. Both the wave and particle aspects are essential to explain all the experimental data. This is referred to as the wave-particle duality. Apparently, light is a more complex phenomenon than just a simple wave or a beam of particles. In order to resolve this dilemma the Danish physicist, Niels Bohr, proposed in 1927 his famous principle of complementarity. According to this principle the wave and particle aspects of electromagnetic radiation are complementary. To interpret the behavior of electromagnetic radiation in any one experiment, one must choose either the particle or the wave description. But choosing of one description must preclude simultaneously choosing of the other. The experimental arrangement determines which description that is chosen.
The complementarity principle applies also to the wave-particle duality of particles, such as electrons. For example, electrons in a cathode ray tube follow well-defined path and indicate their collisions with a fluorescent screen by very small, bright flashes. Electrons appear as particles in a cathode ray experiments, because all of the energy, momentum, and electric charge is assigned at any one time to a small region of space. When electrons interact with other objects, they behave like particles. The particle nature of electrons is shown in the cathode ray experiments, and therefore, according to the principle of complementarity, the wave nature of electrons is suppressed.
On the hand, the wave nature of electrons appears in the experiments showing electron diffraction. Electrons are propagated as waves with an indefinite extension in space, and it is impossible to specify the location of any one electron. That is, the electron diffraction experiments show the wave nature of electrons, and according to the principle of complementarity, the particle nature is suppressed in these experiments.
The following table summarizes the experimental data for the wave-particle duality of both matter and radiation.
| Matter | Radiation | |
|---|---|---|
| Wave nature | Davisson and Germer’s electron diffraction experiments. | Young’s double-slit interference experiment. |
| Particle nature | J.J. Thomson’s measurement of e/m of the electron. | The Compton effect. |
During 1927, Bohr worked on a paper expressing his ideas on complementarity, with many discussions with Heisenberg and Pauli. He was planing to present it in Como, Italy, at a meeting in honor of the Italian physicist Alessandro Volta and later at the fifth Solvay Conference in Brussels, where many of the leaders of physics would be present, including Einstein. Bohr had first met Einstein in 1920, when they both were already physicists of international reknown. More than a professional relation developed; they came to have a deep respect and love for each other. Bohr hoped that the principle of complementarity would convince Einstein of the correctness of the quantum theory. When Bohr presented the Copenhagen interpretation of quantum mechanics at the Solvay Conference, most physicists present accepted Bohr’s new work, but not Einstein. Einstein rejected the Copenhagen interpretation that rejected the determinism of classical physics to which Einstein firmly held. Einstein rejected the statistical interpretation of nature and opposed the rejection of objectivity and the view that held that material reality depended in part on how it was observed. Einstein devised thought experiments in which he attempted to show the flaws of the Copenhagen interpretation. Every time he thought he had found a flaw, Bohr would find an error in his reasoning. Nevertheless, Einstein persisted. Finally, Paul Ehrenfest said to him, “Einstein, shame on you! You are beginning to sound like the critics of your own theories of relativity. Again and again your arguments have been refuted, but instead of applying your own rule that physics must be built on measurable relationships and not on preconceived notions, you continue to invent arguments based on those same preconceptions.” The meeting ended with Einstein unconvinced, and this was a profound disappointment to Bohr.
Three years later at the next Solvay Conference, Einstein arrived with a new thought experiment, the “clock in box” experiment. Einstein said, imagine one had a clock in a box preset so that it would open and close very quickly a shutter on the light-tight box. Inside the box was a gas of photons. When the shutter opened, a single photon would escape. By weighing the box before and after the shutter is opened and closed, one could determine the mass and hence the energy of photon that escaped. Consequently, it would be possible to determine both the energy and the time of escape of the photon with arbitrary precision. This violated the Heisenberg energy-time principle of uncertainty, given by equation [25], and hence, Einstein concluded, quantum theory must be wrong.
Bohr spent a sleepless night thinking about the problem. If Einstein’s reasoning is correct, quantum mechenics must be wrong. But by the morning Bohr discovered a flaw in Einstein’s reasoning. The photon that escaped from the box, imparts an unknown momentum to the box, causing it to move in the gravitational field which is being used to weigh it. However, according to Einstein’s own General Theory of Relativity, the rate of the clock being used to measure time depends upon its position in the gravity field. Since the position of the clock is uncertain due to the “kick” it gets when the photon escapes, so is the time which it measures. Thus Bohr showed that the thought experiment devised by Einstein did not in fact violate the uncertainty relation but rather confirmed it.
After this, Einstein never disputed the consistency of the new quantum theory. He continued to object that it gave a incomplete and nonobjective view of nature. But this objection was a philosophical issue and not one of theoretical physics. The debate continue throughout their lives and was never resolved. And it could not have been. Once that the debate had departed from the basis that a theory of nature is determined by experiment and became a difference of the appreciation of nature of reality, there could be no resolution, and there was none. Einstein objected to quantum mechanics for several reasons.
First, Einstein did not see probabilities as a valid basis for any physical theory. He could not accept a pure-chance that was built into a theory of probabilities. He wrote to Max Born, “Quantum mechanics is very impressive, … but I am convinced that God does not play dice.” (1)
Second, Einstein did not believe that the quantum theory was complete. He argued, “The following requirement for a complete theory seems to be necessary one: every element of the physical reality must have a counterpart in physical theory.” (italics in original) (2) Einstein believed that quantum mechanics fails in this regard; in dealing with group behavior as a theoretical system, if it cannot account in detail for individual happenings, it is an incomplete theory. For this reason, Einstein objected that quantum mechanics was tentative and incomplete.
Einstein was a firm believer in causality and he could not accept a nonobjective view of the natural world. In response to the experimental success of quantum mechanics, Einstein wrote to Born, “I am convinced of [the objective reality] although, up to now success is against it.” (3) It should be pointed out that Einstein did accept the mathematical equations of quantum mechanics. But he believed that quantum mechanic was an incomplete manifestation of an underlying theory (the unified field theory) where an objectively real description is possible. He never abandoned his search for a theory that would merge quantum mechanics with relativity. Of course, he never lived long enough to see that.
One of the pivotal figures of the quantum revolution of the 1920’s was the British theoretical physicist Paul Dirac, who was born in 1902 and died in 1984. He combined the first version of quantum theory, developed by Werner Heisenberg (1901-1976) in 1925, with Einstein’s special theory of relativity, introducing the concept of quantum spin for the electron (an idea immediately applied to other particles); he developed a complete mathematical formulation of the quantum theory; he wrote a widely influential text book on the subject, The Principles of Quantum Mechanics, still used by students and researchers; and he played a major part in the development of QED ( Quantum ElectroDynamics), although to the end of his life in 1984 he remained deeply dissatisfied with the practice of renormalization, which he thought did more to paper over the cracks of a flawed theory. Outside the inner circle of physics, he is best known for his contribution in 1928 to our knowledge of Universe that the particles of the material world have counterparts in the form of antimatter.
For a theorist who accomplished so much, Dirac’s prediction of antimatter came about almost by accident and which he presented to the world in an imprecise way. As he was working with Einstein’s relativistic mass-energy equation, that is,
E2 = p2c2 + m02c4, [28]
he noticed that this equation for E had two solutions, not one. That is,
E = ±√(p2c2 + m02c4). [29]
Just as a simple quadratic equation, one that involves the square of an unknown quantity, has two solutions, one positive and one negative. For example, the simple quadratic equation x2 = 4, has two solutions: +2 and -2. Since the product of these two solutions with themselves (-2)(-2) and (+2)(+2) equals 4, they are the solutions to the simple quadratic equation x2 = 4. So Einstein’s mass-energy equation has two solutions and both solutions are correct. Dirac was using this equation to describe an electron. One of these answers clearly applied to the electron (which has a negative charge). But to what particle did the other solution apply? In 1928, physicists only knew of two elementary particles, the electron and the proton (though some suspected a third, the neutron). So Dirac at first thought that the positive solution referred to the proton. The two solutions of equation [29] did not seem to require the same mass. The solutions of the equation [29] referred not to mass but to energy. And there was at that time no knowledge of a positive charged particle that had the same mass of the electron. So Dirac proposed a resolution to this problem; he postulated that there was negative energy states and that these negative energy states are filled by an infinite “sea” of electrons. When enough energy is imparted to an electron to knock it into a positive state, the absence of the electron would appear as an ordinary particle with a positive charge. Thus the “hole in the sea” would function as an ordinary particle with positive energy and positive charge. This was a brilliant solution but it had one problem: the positive particle should have the same mass as an electron.
But in 1932, the American physicist, Carl Anderson, was studying cosmic rays with a cloud chamber, a device in which a cosmic ray particles leave behind them cloud trails, like the condensation trails produced by high-flying aircraft. Using a magnetic field in which the path of the particle was curved, he found a new particle. By measuring the curvature of the track and its texture, he found that the new particle had a mass nearly equal to that of the electron. This new particle was named positron, the positive charged twin of the electron. There was no need for the “sea of electrons” or its mysterious “holes.” The dualism of the solution of Dirac’s equation was interpreted as to point to a profound and universal character of quantum field theory: for every kind of particle there must exist a corresponding antiparticle with the same mass but opposite electric charge. The positron, then, was an antielectron. (Really it is rather arbitrary which one we call “particle” and which we call “antiparticle”; we could just as well referred to the electron as the antipositron. But since there is a lot more electrons around, and not so many positron, it seems appropriate to think of the electron as “matter” and the positron as “antimatter”).
The evidence for these positively charged particles had been around in cosmic ray tracks for some time, but had been mistaken for tracks of electrons moving in the opposite way. The neutron was discovered in the same year as the positron. Dirac’s calculations were extended to all atomic particles and this gave physicists six particles (plus the photon) to deal with: the electron and positron; the proton and a negatively charged antiproton (not yet detected); and the neutron and a (presumed) antineutron. The laws of physics require that when a particle meets its antiparticle counterpart, the two will annihilate each other in a burst of energy (gamma rays). The positron and electron cancel out each other, being converted into energy. And the process can be reversed. If enough energy is available, the electron-positron pairs, or other particle-antiparticle pairs can be produced. Only particle-antiparticle pairs of the same particle are involved in these processes, never, for example, a proton and an antineutron. All the predictions of the antimatter theory were confirmed by experiment, although the antiproton and antineutron were not dectected until the middle of the 1950’s.
The discovery of antimatter by Dirac was incidental to his main task, that is, to make quantum mechanics invariant under Lorentz transformations, so that it conforms to the Special Theory of Relativity. This was necessary to order that the quantum theory may apply to high-speed particles. Dirac did not attempt to make the quantum theory conform to the general relativity, since general relativity is required only when one is dealing with gravitation, and gravitational forces are quite unimportant in atomic phenomena. Quantum mechanics, just like ordinary mechanics and electrodynamics, must be made to conform to the principles of special relativity. Because the entities (electrons, etc.) described by quantum theory quite often travel at speeds at or near the speed of light, c, this becomes an essential requirement for quantum mechanics. As special relativity was necessary to correct classical Newtonian mechanics, so special relativity must correct the quantum mechanics of Schrodinger and Heisenberg. In 1926 two German physicists, Oskar Klein and Walter Gordon had proposed a relativistic version of Schrodinger’s equation for the electron that was called the Klein-Gordon equation. The Schrodinger’s Equation is derived by starting with the classical Hamiltonian energy-momentum relation:
p2/2m + V = E.
Applying to this relation the quantum operators
E → ih/2π(∂/∂t), [31]
and letting the resulting operator act on the “wave function,” ψ, we get the Schrodinger equation:
-(h2/8π2)∇2ψ + Vψ = i(h/2π)(∂ψ/∂t). [32]
The Schrodinger equation describes the nonrelativistic quantum mechanics of a particle. Now the Klein-Gordon equation can be obtained in exactly the same way as the Schrodinger equation except by starting with the relativistic energy-momentum relation:
which may be written as
E2/c2 – p2 = m2c2.
This may be expressed in terms of a four-vector, that is, the energy-momentum four-vector
pμ = [(E/c), px, py, pz].
Taking the product of this four-vector with itself gives
pμpμ = E2/c2 – p2,
so that the energy-momentum relation may be expressed as
pμpμ = m2c2, or pμpμ – m2c2 = 0. [34]
(Potential energy is left out for now, since a free particle, not moving in an energy field, is being considered.) Since the quantum operator [30] require no relativistic modifications, it can be written in four-vector notation as
pμ → (ih/2π) ∂μ, and pμ → (ih/2π) ∂μ, [35]
where
∂μ = ∂/∂xμ, and ∂μ = ∂/∂xμ, [36]
with μ = 0, 1, 2, 3; and
x0 = ct, x1 = x, x2 = y, x3 = z;
and
x0 = ct, x1 = x, x2 = y, x3 = z;
that is,
∂0 = (1/c)∂/∂t, ∂1 = ∂/∂x, ∂2 = ∂/∂y, ∂3 = ∂/∂z,
and
∂0 = (1/c)∂/∂t, ∂1 = ∂/∂x, ∂2 = ∂/∂y, ∂3 = ∂/∂z. [37]
Placing the right side of equation [35] into equation [34], and letting the derivatives act upon the wave function ψ, we get
-(h2/4π2) ∂μ∂μψ = m2c2ψ, or [38]
-∂μ∂μψ = (4π2m2c2/ h2)ψ, or
-(1/c2)(∂2ψ/∂t2) + ∂2ψ/∂x2 + ∂2ψ/∂y2 + ∂2ψ/∂z2 = (2πmc/h)2ψ, or
-(1/c2)(∂2ψ/∂t2) + ∇2ψ = (2πmc/h)2ψ. [39]
This equation is the Klien-Gordon equation; apparently Schrodinger discovered this equation even before he discovered the nonrelativistic one bearing his name; it was rejected on the ground that it was incompatible with statistical interpretation of the wave function ψ [which says the |ψ|2 is the probability of finding a particle at the point (x, y, z)]. The source of the difficulty was blamed on the fact that the Klein-Gordon equation is second order in t; note that the Schrodinger Equation [32] is first order in t. Later in 1934, Pauli and Weisskopf showed that the statisical interpretation is itself flawed in the relativistic quantum theory and thus restored the Klein-Gordon equation for particles of spin 0, while keeping Dirac’s equation [40] for particles of spin ½.
Dirac had set out to find an equation that was consistent with the relativistic energy-momentum relation and that was first order in time. His basic strategy to find such a equation was to “factor” the energy-momentum equation [34]. But he ran into a difficulty, which he solved by treating the wave functions as matrices instead of as just numbers.
As a 4 × 4 matrix equation, then, the relativistic energy-momentum relation [34] does factor:
pμpμ – m2c2 = (γκpκ + mc) (γλpλ – mc) = 0,
where γμ is a set of 4 × 4 “gamma matrices”. To obtain the Dirac equation one of the factors is chosen, it does not matter which one, but usually the one on the right is chosen,
γμpμ – mc = 0,
then this equation is converted into an operator equation, using equation [35], substituting
pμ → (ih/2π)∂μ.
Then let the result act on ψ giving us the Dirac Equation:
(ih/2π)γμ∂μψ – mcψ = 0. [40]
where ψ is now a four-element column matrix, which is called a “bi-spinor” or “Dirac spinor”; although it does have four components, it is not a four-vector. When Dirac solved his equation, assuming that ψ was independent of position, that is, for a particle at rest, he found that he had two solutions:
ψA = e-2πiEt/h and ψB = e+2πiEt/h,
each as having the characteristic time dependence of a quantum state with energy E. For a particle at rest, E = mc2, so ψA was exactly what it should be, when p = 0. But ψB represents a state where the energy is negative (E = –mc2). To account for this, Dirac first postulated his “sea” of negative energy particles. Then realizing that it could not be that, he interpreted the “negative energy” as representing antiparticles with a positive energy. Thus ψA describes electrons (for example) and ψB describes positrons. Each solution is a two component spinor, just right for a system of spin ½.
By 1925, physicists, who were attempting to explain the nature of atomic spectra, found that not all was correct. Where, according the Bohr model of the atom, just one spectral line should exist, there were found sometimes two very close together. To explain this and other similar puzzles, two Dutch physicists, Sam Goudsmit and George Uhlenbeck, of the Institute of Theoretical Physics in Leyden, in late 1925 proposed that the electron spins on its axis as it orbits about the nucleus (just as the earth spins on it axis as it revolves in its orbit about the sun).
The splitting of the spectral lines was explained by the existence of magnetic effects within the atom. As the electron orbits about the nucleus, its motion in its orbit sets up a small loop of electric current and so sets up a magnetic field; the atom behaves as a small magnet. The spin of the electron on its axis sets up even a smaller loop of electric current which is called the “magnetic moment of the electron”. This can either add to or subtract from the magnetic field of the atom. This will cause a small difference in the energy of the electronic orbit for different spins of the electrons. And the result is a splitting of the spectral lines associated by Bohr orbit.
This classical explanation has its limitations. The fact that the spectral lines splits into two components indicates that the electrons cannot be spinning about at just any arbitrary angle; its angular momentum must be such that there is only two values along the line of the atom’s magnetic field (or, in the case of free electron, along the line of any applied magnetic field). The component of the spin in this direction is referred to as the “z-components”, or the “third component” of spin and are measured to be quantized in half-integral units of Planck’s constant h (divided by 2π). That is,
sz = ±½(h/2π).
The 2π appears here because there are 2π radians in a complete rotation of 360°. Note that this picture is not just a model; this concept of electron spin is a quantum concept (it is proportional to h) that is instrinsic to the electron itself, and the electron has a quantum of spin angular momentum just as it has a quantum of electric charge.
In 1924, Wolfgang Pauli had proposed an explanation for the splitting of the spectra by assigning four separate quantum numbers to the electron. Three of these numbers were already included in Bohr’s model, and were thought of as describing the orbital angular momentum of the electron as it orbits in an atom (setting it distance from the nucleus), the shape of its orbit (circle or ellipse), and its orientation. The fourth number was used to describe the “spin”, which came in two values according to whether its spin angular momentum is pointing up or down with reference to magnetic field. Pauli at first opposed this interpetation of the fourth quantum number when it was proposed by a young American physicist, Ralph Kronig, who was visiting in Europe after receiving his PhD at Columbia University. Pauli could not reconcile the idea of the electron as a particle within the framework of relativistic theory. Kronig gave up his idea and never published it. Less than a year later, the same idea was published by Goudsmit and Uhlenbeck. As the theory of the spinning electron was refined to explain fully the splitting of the spectral lines, by March of 1926 Pauli had convinced himself and accepted the interpretation.
The difficulty with the spin theory was that it was a property of a particle and it did not fit into the Schrodinger’s wave mechanics. Paul Dirac removed this difficulty by incorporating special relativity into quantum mechanics. And in doing this, Dirac introduced electron spin. He treated the electron as a true particle and associated with it is an orbital angular momentum and a spin angular momentum. In the nonrelativistic quantum mechanics, the Schrodinger equation [32] describes the mechanics of particles but says nothing about their spin angular momentum. It was treated in an ad hoc way. In the relativistic quantum mechanics of particles, the Klein-Gordon equation [39] describes particles with a spin of 0, the Dirac equation [40] describes particles of spin ½, and the Proca equation describes particles of spin 1. Particles with a half-intergal spin and obey the Pauli exclusion principle are called fermions, and particles such as the photons with an integral spin and do not obey the exclusion principle are called bosons.
ENDNOTES
The formulation of a model or guess in physics is guided by certain general principles, such as special theory of relativity and quantum mechanics. There are four areas of mechanics:
| Small → | ||
|---|---|---|
| Fast ↓ | Classical mechanics | Quantum mechanics |
| Relativistic mechanics | Quantum field theory | |
The everyday world is governed by classical mechanics. But for objects that travel very fast (at speeds comparable with the speed of light, c), the classical rules are modified by special relativity, and objects that are very small (comparable to the size of atoms), classical mechanics is superseded by quantum mechanics. Finally, for objects both fast and small, there is required a theory that incorporates both relativity and quantum principles: quantum field theory. Since elementary particles that are extremely small and are also typically very fast, they have to be treated by quantum field theory.
In this most sophisticated form of quantum theory, all entities are described by fields. Just as the photon is a manifestation of an electromagnetic field, so also is an electron taken to be the manifestation of an electron field and a proton of a proton field. In classical electrodynamics, the electrical repulsion of two electrons is attributed to the electric fields, each one responding to the others field. But in quantum field theory, the electric field is quantized (in the form of photons), and the interaction may be pictured as consisting of a stream of photons passing back and forth between the two charges, each electron continually emitting them and continually absorbing them. And the same goes for any noncontact force; where classically it is interpreted as “action at a distance” as “mediated” by a field. Now in quantum field theory it is mediated by an exchange of particles (the quanta of the field). In the case of electrodynamics, the mediator is the photon; for gravity it is called the graviton.
Once it is accepted that the electron wave function is extended throughout space (by virtue of Heisenberg’s uncertainty principle for a particle of a definite momentum), it is not too great of a leap to the idea of an electron field extending throughout space. Any one individual electron wavefunction may be thought of as a particular frequency excitation of the field and may be localized to a greater or lesser extent dependent upon the interaction. The electron field variable is then a Fourier sum over the individual wavefunction, where the coefficients multiplying each of the individual wavefunctions represent the probability of the creation or destruction of a quantum of that particular wavelength (momentum) at any given point. The representation of a field as a summation over its quanta, with coefficients specifying the probabilities of the creation and destruction of those quanta. This is referred to as second quantisation. First quantisation is the recognition of the particle nature of a wave or the wave nature of a particle (the Planck-Einstein and de Broglie hypotheses respectively). Second quantisation is the incorporation of the ablility to create and destroy the quanta in various reaction.
There is a relatively simple picture which can help us to appreciate the nature of quantum field and its connection with concept of a particle. A quantum field is equivalent, at least mathematically, to an infinite collection of harmonic oscillators. These oscillators can be thought of as a series of springs with masses attached. When some of the oscillators becomes excited, they oscillate (or vibrate) at particular frequencies. These oscillations correspond to a particular excitation of the quantum field and hence to the presence of particles, that is, field quanta.
We are familiar with electromagnetic and gravitational fields because, their quanta being bosons, there are no restriction on the number of quanta in any one energy level and so large assemblies of quanta may act together coherently to produce macroscoptic effects. Electron and proton fields are not at all evident because, being fermions, the quanta must obey the Pauli’s exclusion principle and this prevents them from acting together in a macroscopically observable fashion. So although we can concentrated beams of coherent photons (laser beams), we cannot produce similar beams of electrons. These instead must resemble ordinary incoherent lights (for example, torchlights) with a wide spread of energies in the beam.