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.
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.
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
Δpx =
h(-Δλ/λ2). [27]
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]
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. |
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.
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 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
p → (h/2πi)∇, [30]
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:
E2 = p2c2 +
m2c4. [33]
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.
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.
(1) Max Born and Albert Einstein, The Born-Einstein Letters
(New York: Walker & Company, 1971), p. 91.
(2) Albert Einstein, Boris Podolsky, and Nathan Rosen,
"Can Quantum-Mechanical Description of Physical Reality Be
Considered Complete?"
Physical Review 47, 1935, 777ff.
| Small → | ||
|---|---|---|
| Fast ↓ | Classical mechanics | Quantum mechanics |
| Relativistic mechanics | 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 familar 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.