chapter 2 - waves and particles, Exams of Quantum Mechanics

Download chapter 2 - waves and particles and more Exams Quantum Mechanics in PDF only on Docsity! 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 1 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. 2 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= ρ1 + ρ2. Rather it is the wave amplitudes which add: Ψ = Ψ1 + Ψ2 (7) with the intensity given by the square of the amplitude: ρ = Ψ2 = Ψ2 1 + Ψ2 2 + 2Ψ1Ψ2 (8) The cross term 2Ψ1Ψ2 is responsible for the constructive and destructive interference. Where Ψ1 and Ψ2 have the same sign, constructive interference makes the total intensity greater than the the sum of ρ1 and ρ2. Where Ψ1 and Ψ2 have opposite signs, there is destructive interference. If, in fact, Ψ1 = −Ψ2 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. 5 Figure 4. Scintillations observed after dimming laser intensity by several orders of magnitude. These are evidently caused by individual photons! But wonders do not cease even here. Even though the individual scintilla- tions appear at random positions on the screen, their statistical behavior reproduces the original high-intensity diffraction pattern. Evidently the sta- tistical behavior of the photons follows a predictable pattern, even though the behavior of individual photons is unpredictable. This impies that each individual photon, even though it behaves mostly like a particle, somehow carry with it a “knowledge” of the entire wavelike diffraction pattern. In some sense, a single photon must be able to go through both slits at the same time. This is what is known as the wave-particle duality for light: under appropriate circumstances light can behave as a wave or as a particle. Planck’s resolution of the problem of blackbody radiation and Ein- stein’s explanation of the photoelectric effect can be summarized by a rela- tion between the energy of a photon to its frequency: E = hν (8) where h = 6.626 × 10−34 J sec, known as Planck’s constant. Much later, the Compton effect was discovered, wherein an x-ray or gamma ray photon ejects an electron from an atom, as shown in Fig. 5. Assuming conservation 6 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 = mc2 (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). 7 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 [ 2πi (x λ − ν t )] (18) 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)/h̄] (19) where h̄ ≡ h 2π (20) 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/h̄) × exp[i(px − Et)/h̄] (21) 10 Thus ih̄ ∂Ψ ∂t = EΨ (22) Analogously −ih̄ ∂Ψ ∂x = pΨ (23) and −h̄2 ∂2Ψ ∂x2 = p2Ψ (24) The energy and momentum for a nonrelativistic free particle are related by E = 1 2 mv2 = p2 2m (25) Thus Ψ(x, t) satisfies the partial differential equation ih̄ ∂Ψ ∂t = − h̄2 2m ∂2Ψ ∂x2 (26) For a particle with a potential energy V (x), E = p2 2m + V (x) (27) we postulate that the equation for matter waves generalizes to ih̄ ∂Ψ ∂t = { − h̄2 2m ∂2 ∂x2 + V (x) } Ψ (28) For waves in three dimensions should then have ih̄ ∂ ∂t Ψ(r, t) = { − h̄2 2m ∇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 Schrödinger equation for the amplitude Ψ(r, t) of the matter waves associated with the particle. Its formulation in 1926 represents the 11 starting point of modern quantum mechanics. (Heisenberg in 1925 proposed another version known as matrix mechanics.) For conservative systems, in which the energy is a constant, we can separate out the time-dependent factor from (19) and write Ψ(r, t) = ψ(r) e−iEt/h̄ (30) where ψ(r) is a wavefunction dependent only on space coordinates. Putting (30) into (29) and cancelling the exponential factors, we obtain the time- independent Schrödinger equation: { − h̄2 2m ∇2 + V (r) } ψ(r) = Eψ(r) (31) Most of our applications of quantum mechanics to chemistry will be based on this equation. The bracketed object in Eq. (31) is called an operator. An operator is a generalization of the concept of a function. Whereas a function is a rule for turning one number into another, an operator is a rule for turning one function in another. The Laplacian is an example of an operator. We usually indicate that an object is an operator by placing a ‘hat’ over it, eg, Â. The action of an operator that turns the function f into the function g is represented by  f = g (32) Eq (23) implies that the operator for the x-component of momentum can be written p̂x = −ih̄ ∂ ∂x (33) and by analogy, we must have p̂y = −ih̄ ∂ ∂y , p̂z = −ih̄ ∂ ∂z (34) The energy, as in Eq (27), expressed as a function of position and momen- tum is known in classical mechanics as the Hamiltonian. Generalizing to three dimensions, H = p2 2m + V (r) = 1 2m (p2 x + p2 y + p2 z) + V (x, y, z) (35) 12