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The Wave Field. Landscape Sculpture by Maya Lin on North Campus, the University of Michigan, Ann Arbor.
CHAPTER 2
WAVES AND PARTICLES
Quantum mechanics is the theoretical framework which describes the be- havior of matter on the atomic scale. It is the most successful quantitative theory in the history of science, having withstood thousands of experimen- tal tests without a single verifiable exception. It has correctly predicted or explained phenomena in fields as diverse as chemistry, elementary-particle physics, solid-state electronics, molecular biology and cosmology. A host of modern technological marvels, including transistors, lasers, computers and nuclear reactors are offspring of the quantum theory. Possibly 30% of the US gross national product involves technology which is based on quantum mechanics. For all its relevance, the quantum world differs quite dramat- ically from the world of everyday experience. To understand the modern
theory of matter, conceptual hurdles of both psychological and mathemat- ical variety must be overcome. A paradox which stimulated the early development of the quantum theory concerned the indeterminate nature of light. Light usually behaves as a wave phenomenon but occasionally it betrays a particle-like aspect, a schizoid tendency known as the wave-particle duality. We consider first the wave theory of light.
The Double-Slit Experiment
Fig. 1 shows a modernized version of the famous double-slit diffraction ex- periment first performed by Thomas Young in 1801. Light from a monochro- matic (single wavelength) source passes through two narrow slits and is pro- jected onto a screen. Each slit by itself would allow just a narrow band of light to illuminate the screen. But with both slits open, a beautiful inter- ference pattern of alternating light and dark bands appears, with maximum intensity in the center. To understand what is happening, we review some key results about electromagnetic waves.
Figure 1. Modern version of Young’s interference experiment using a laser gun. Single slit (left) produces an intense band of light. Double slit (right) gives a diffraction pattern. See animated applet at http://www.colorado.edu/physics/2000/applets/twoslitsa.html
Maxwell’s theory of electromagnetism was an elegant unification of the diverse phenomena of electricity, magnetism and radiation, including light.
such that the intensity is given by
ρ(r, t) = [Ψ(r, t)]^2 (5)
The function Ψ(r, t) will, in some later applications, have complex values. In such cases we generalize the definition of intensity to
ρ(r, t) = |Ψ(r, t)|^2 = Ψ(r, t)∗^ Ψ(r, t) (6)
where Ψ(r, t)∗^ represents the complex conjugate of Ψ(r, t). In quantum- mechanical applications, the function Ψ is known as the wavefunction.
+
+
+
1 +^1 +
Figure 3. Interference of two equal sinusoidal waves. Top: constructive interference. Bottom: destructive interference. Center: intermediate case. The resulting intensities ρ = Ψ^2 is shown on the right.
The electric and magnetic fields, hence the amplitude Ψ, can have either positive and negative values at different points in space. In fact constructive and destructive interference arises from the superposition of
waves, as illustrated in Fig. 3. By Eq (5), the intensity ρ ≥ 0 everywhere. The light and dark bands on the screen are explained by constructive and destructive interference, respectively. The wavelike nature of light is con- vincingly demonstrated by the fact that the intensity with both slits open is not the sum of the individual intensities, ie, ρ 6 = ρ 1 + ρ 2. Rather it is the wave amplitudes which add:
with the intensity given by the square of the amplitude:
ρ = Ψ^2 = Ψ^21 + Ψ^22 + 2Ψ 1 Ψ 2 (8)
The cross term 2Ψ 1 Ψ 2 is responsible for the constructive and destructive interference. Where Ψ 1 and Ψ 2 have the same sign, constructive interference makes the total intensity greater than the the sum of ρ 1 and ρ 2. Where Ψ 1 and Ψ 2 have opposite signs, there is destructive interference. If, in fact, Ψ 1 = −Ψ 2 then the two waves cancel exactly, giving a dark fringe on the screen.
Wave-Particle Duality
The interference phenomena demonstrated by the work of Young, Fresnel and others in the early 19th Century, apparently settled the matter that light was a wave phenomenon, contrary to the views of Newton a century earlier—case closed! But nearly a century later, phenomena were discov- ered which could not be satisfactorily accounted for by the wave theory, specifically blackbody radiation and the photoelectric effect. Deviating from the historical development, we will illustrate these ef- fects by a modification of the double slit experiment. Let us equip the laser source with a dimmer switch capable of reducing the light intensity by sev- eral orders of magnitude, as shown in Fig. 4. With each successive filter the diffraction pattern becomes dimmer and dimmer. Eventually we will be- gin to see localized scintillations at random positions on an otherwise dark screen. It is an almost inescapable conclusion that these scintillations are caused by photons, the bundles of light postulated by Planck and Einstein to explain blackbody radiation and the photoelectric effect.
of momentum in a photon-electron collision, the photon is found to carry a momentum p, given by p = h/λ (9)
Eqs (8) and (9) constitute quantitative realizations of the wave-particle duality, each relating a particle-like property—energy or momentum—to a wavelike property—frequency or wavelength.
Figure 5. Compton effect. The momentum and energy carried by the inci- dent x-ray photon are transferred to the ejected electron and the scattered photon.
According to the special theory of relativity, the last two formulas are actually different facets of the same fundamental relationship. By Einstein’s famous formula, the equivalence of mass and energy is given by
E = mc^2 (10)
The photon’s rest mass is zero, but in travelling at speed c, it acquires a finite mass. Equating Eqs (8) and (10) for the photon energy and taking the photon momentum to be p = mc, we obtain
p = E/c = hν/c = h/λ (11)
Thus, the wavelength-frequency relation (1), implies the Compton-effect formula (9).
The best we can do is to describe the phenomena constituting the wave-particle duality. There is no widely accepted explanation in terms of everyday experience and common sense. Feynman referred to the “ex- periment with two holes” as the “central mystery of quantum mechanics.” It should be mentioned that a number of models have been proposed over the years to rationalize these quantum mysteries. Bohm proposed that there might exist hidden variables which would make the behavior of each photon deterministic, ie, particle-like. Everett and Wheeler proposed the “many worlds interpretation of quantum mechanics” in which each random event causes the splitting of the entire universe into disconnected parallel universes in which each possibility becomes the reality. Needless to say, not many people are willing to accept such a metaphysically unwieldy view of reality. Most scientists are content to apply the highly successful com- putational mechanisms of quantum theory to their work, without worrying unduly about its philosophical underpinnings. Sort of like people who enjoy eating roast beef but would rather not think about where it comes from. There was never any drawn-out controversy about whether electrons or any other constituents of matter were other than particle-like. Yet a variant of the double-slit experiment using electrons instead of light proves other- wise. The experiment is technically difficult but has been done. An electron gun, instead of a light source, produces a beam of electrons at a selected velocity, which is focussed and guided by electric and magnetic fields. Then, everything that happens for photons has its analog for electrons. Individual electrons produce scintillations on a phosphor screen–this is how TV works. But electrons also exhibit diffraction effects, which indicates that they too have wavelike attributes. Diffraction experiments have been more recently carried out for particles as large as atoms and molecules, even for the C 60 fullerene molecule. De Broglie in 1924 first conjectured that matter might also exhibit a wave-particle duality. A wavelike aspect of the electron might, for example, be responsible for the discrete nature of Bohr orbits in the hydrogen atom (cf. Chap. 7). According to de Broglie’s hypothesis, the “matter waves” associated with a particle have a wavelength given by
λ = h/p (12)
identical in form to Compton’s result (9) (which, in fact, was discovered later). The correctness of de Broglie’s conjecture was most dramatically
Accordingly, let us consider a very general instance of wave motion propagating in the x-direction. At a given instant of time, the form of a wave might be represented by a function such as
ψ(x) = f(2πx/λ) (15)
where f(θ) represents a sinusoidal function such as sin θ, cos θ, eiθ^ , e−iθ^ or some linear combination of these. The most suggestive form will turn out to be the complex exponential, which is related to the sine and cosine by Euler’s formula eiθ^ = cos θ + i sin θ (16)
Each of the above is a periodic function, its value repeating every time its argument increases by 2π. This happens whenever x increases by one wavelength λ. At a fixed point in space, the time-dependence of the wave has an analogous structure:
T (t) = f(2πνt) (17)
where ν gives the number of cycles of the wave per unit time. Taking into account both x- and t-dependence, we consider a wavefunction of the form
Ψ(x, t) = exp
[
2 πi
( (^) x
λ
− ν t
)]
representing waves travelling from left to right. Now we make use of the Planck and de Broglie formulas (8) and (12) to replace ν and λ by their particle analogs. This gives
Ψ(x, t) = exp[i(px − Et)/¯h] (19)
where
¯h ≡
h 2 π
Since Planck’s constant occurs in most formulas with the denominator 2π, this symbol, pronounced“aitch-bar,” was introduced by Dirac. Now Eq (17) represents in some way the wavelike nature of a particle with energy E and momentum p. The time derivative of (19) gives
∂Ψ ∂t
= −(iE/¯h) × exp[i(px − Et)/¯h] (21)
Thus
i¯h
∂t
= EΨ (22)
Analogously
−i¯h
∂x
= pΨ (23)
and
−¯h^2
∂^2 Ψ
∂x^2
= p^2 Ψ (24)
The energy and momentum for a nonrelativistic free particle are related by
E =
mv^2 =
p^2 2 m
Thus Ψ(x, t) satisfies the partial differential equation
i¯h
∂t
¯h^2 2 m
∂^2 Ψ
∂x^2
For a particle with a potential energy V (x),
E =
p^2 2 m
we postulate that the equation for matter waves generalizes to
i¯h
∂t
¯h^2 2 m
∂^2
∂x^2
For waves in three dimensions should then have
i¯h
∂t
Ψ(r, t) =
¯h^2 2 m
∇^2 + V (r)
Ψ(r, t) (29)
Here the potential energy and the wavefunction depend on the three space coordinates x, y, z, which we write for brevity as r. This is the time- dependent Schr¨odinger equation for the amplitude Ψ(r, t) of the matter waves associated with the particle. Its formulation in 1926 represents the
We construct thus the corresponding quantum-mechanical operator
Hˆ = − ¯h
2
2 m
∂^2
∂x^2
∂^2
∂y^2
∂^2
∂z^2
¯h^2 2 m
∇^2 + V (r) (36)
The time-independent Schr¨odinger equation (31) can then be written sym- bolically as Hˆ Ψ = E Ψ (37)
This form is actually more generally to any quantum-mechanical problem, given the appropriate Hamiltonian and wavefunction. Most applications to chemistry involve systems containing many particles—electrons and nuclei. An operator equation of the form
A ψˆ = const ψ (38)
is called an eigenvalue equation. Recall that, in general, an operator acting on a function gives another function [Eq (32)]. The special case (38) occurs when the second function is a multiple of the first. In this case, ψ is known as an eigenfunction and the constant is called an eigenvalue. (These terms are hybrids with German, the purely English equivalents being ‘character- istic function’ and ‘characteristic value.’) To every dynamical variable A in quantum mechanics, there corresponds an eigenvalue equation, usually written A ψˆ = a ψ (39)
The eigenvalues a represent the possible measured values of the variable A. The Schr¨odinger equation (37) is the best known instance of an eigenvalue equation, with its eigenvalues corresponding to the allowed energy levels of the quantum system.
The Wavefunction
For a single-particle system, the wavefunction Ψ(r, t), or ψ(r) for the time- independent case, represents the amplitude of the still vaguely defined mat- ter waves. The relationship between amplitude and intensity of electro- magnetic waves we developed for Eq (6) can be extended to matter waves.
The most commonly accepted interpretation of the wavefunction is due to Max Born (1926), according to which ρ(r), the square of the absolute value of ψ(r) is proportional to the probability density (probability per unit vol- ume) that the particle will be found at the position r. Probability density is the three-dimensional analog of the diffraction pattern that appears on the two-dimensional screen in the double-slit diffraction experiment for elec- trons described in the preceding Section. In the latter case we had the relative probability a scintillation would appear at a given point on the screen. The function ρ(r) becomes equal, rather than just proportional to, the probability density when the wavefunction is normalized, that is,
∫ |ψ(r)|^2 dτ = 1 (40)
This simply accounts for the fact that the total probability of finding the particle somewhere adds up to unity. The integration in (40) extends over all space and the symbol dτ designates the approrpiate volume element. For example, in cartesian coordinates, dτ = dx dy dz; in spherical polar coordinates, dτ = r^2 sin θ dr dθ dφ. The physical significance of the wavefunctions makes certain demands on its mathematical behavior. The wavefunction must be a single-valued function of all its coordinates, since the probability density ought to be uniquely determined at each point in space. Moreover, the wavefunction should be finite and continuous everywhere, since a physically-meaningful probability density must have the same attributes. The conditions that the wavefunction be single-valued, finite and continuous—in short, “well- behaved”—lead to restrictions on solutions of the Schr¨odinger equation such that only certain values of the energy and other dynamical variables are allowed. This is called quantization and is in fact the feature that gives quantum mechanics its name.
