De Broglie's postulate - wave-like properties of particles

Transcription

1 3 De Broglie's postulate - wave-like properties of particles Wave-particle duality Electromagnetic radiation can exhibit both particle and wave properties. It appears dicult to reconcile these facts. The full paradox of particle-wave duality is illustrated by the following two types of experiments. The resolution is achieved by a probabilistic interpretation of wave mechanics. Photon absorption experiment Electromagnetic waves have polarization. Polarizer is a material which is transparent to light of only one particular polarization. Consider two slabs of a polarizer material with a preferred axis. A light beam shined on the rst slab gets ltered, so that only a linearly polarized beam along the preferred axis direction comes out. This beam falls on the second slab, called analyzer. If the preferred axis of the second slab is parallel to that of the rst slab, the polarized light will pass through without any intensity loss. If the two preferred axes are at a right angle, no light will pass through the analyzer. If the two axes are at an angle θ, then the intensity of the light ltered through the analyzer will be I = I 0 cos θ, where I 0 E 0 is the incident light intensity on the analyzer. This can be understood by decomposing the linearly polarized electric eld vector in two components, one parallel and one perpendicular to the analyzer axis. The parallel component has magnitude E 0 cos θ and passes through. The above relationship follows from I E. This behavior is experimentally veried (and can be easily demonstrated using 3D glasses). But, what happens when the incident light intensity is lowered so much that only one photon passes through the system at a time? The properties of blackbody radiation and photoelectric eect could be explained only by assuming that photons are absorbed and emitted in their entirety. There is no way to absorb or emit a part of a photon. Therefore, the analyzer must either let a photon through, or completely absorb it. How can one reproduce the experimental nding I = I 0 cos θ for an arbitrary angle θ when the light intensity is carried by a single photon at a time? Since 0 cos θ 1 we cannot allow the same thing to happen to every photon that arrives at the analyzer. Some photons must pass through, and some must be absorbed. The most fair (general) situation, given that photons are identical, is that absorption be a random process: the probability that a photon will pass through the analyzer is cos θ. The microscopic mechanism responsible for such a probabilistic dynamics is not known. However, we have no choice but to accept the experimentally observed reality and seek the best possible way to mathematically describe it, after which we can make new physical predictions. Double-slit experiment Consider a beam of monochromatic light incident onto a plate with two parallel slits. Light can pass through the slits onto a screen placed behind the plate, where its intensity is measured as a function of position. The separation d between the slits is comparable with the wavelength of the light λ. Classically, light propagates as a wave. The two slits act as point-like sources of coherent light, so an interference pattern will be observable on the screen. Let us set up a coordinate system so that the incident light beam is parallel to y-axis, the plate is as y = 0, the screen is at y = l and the slits are at locations x = ±d/, y = 0. The electric eld at the screen is a superposition of elds which propagate radially (in the xy-plane) from the slits: E(x, l) = E 0 cos π λ ( l + x d ) + cos π l λ + ( x + d )

2 4 We can simplify this by assuming l x, d and by using 1 + x = 1 + x/ + O(x ): ( πl E(x, l) = E 0 [cos λ + π ( x d ) ) ( πl + cos λl λ + π ( x + d ) )] λl ( πl = E 0 cos λ + π λl )) (x + d cos 4 ( ) πd λl x ( ) α β. The oscillatory dependence on x of the ( ) where we also used cos α + cos β = cos α+β cos measured light intensity I E on the screen is a manifestation of wave interference. Now, reduce the incident light intensity until a single photon at a time passes through the slits. If a photon were merely a point-like particle, it would either fall on the plate and get absorbed, or pass through one of the slits and land on the screen. We might not be able to predict which slit a successful photon will pass through, but we could assume that the slits are identical so that a photon has an equal probability of 1/ to pass through either slit. The most natural expectation is that the photon would then land on the screen in close proximity to the slit it passed through. Therefore, if we counted over a long period of time how often photons land at dierent positions on the screen, we should nd a probability prole with two bright maximums at x = ±d/ just behind the slits, and the rest of the plate should cast a shadow. Clearly, this scenario contradicts the experimentally observed oscillatory interference pattern for large light intensities when a large number of photons impede per unit time. But, since photons do not interact among themselves, it must not matter what the light intensity is. Each photon must behave the same and independently from other photons contribute to the interference pattern. Therefore, when a single photon passes through the slits, it must land at a random location on the screen but with the probability which reects the interference pattern. This is the same logic as in the polarizer/analyzer experiment. Therefore, the naive interpretation of particle-like motion of photons is incorrect. We can reconcile the existence of photons with interference only if we give up the notion that the photon passes through only one slit at a time. Even a single photon must be regarded as a wave when it goes through the slits. But, we know that photons can behave like particles. So, let us place a photon detector at each slit. We shall imagine that these detectors are designed to not block a slit for light passage. However, since a measurable physical interaction of light and matter must involve photon absorption, a detector must absorb an incoming photon and then re-emit it after registering the event. Then, if a photon were found to pass through the slit at x = +d/, the upper detector would generate a signal, while the lower detector would not. This is necessary because a photon cannot be partially absorbed, say a half on each detector. Analogous would be true for a photon detected to pass through the slit at x = d/. Now notice that the random photon detection events trigger re-emission of photons by either one detector, but not both at the same time. The re-emitted photons are collected at the screen in a manner consistent with particle dynamics, and there can be no interference patterns. The above experiment is the most general manner in which one could detect a position of a photon. We see that by learning about a photon position we destroy its wave-like properties. Alternatively, in order to see interference we must not have information about photon's particle properties, such as position. Matter waves Planck's and Einstein's postulates were extremely successful in explaining the properties of electromagnetic radiation which could not be understood within the classical physics framework. These postulates introduce a quantization of electromagnetic eld: electromagnetic eld is emitted, absorbed and transmitted in lumps of energy, called photons.

3 In classical physics a wave is characterized by its frequency ν, wavelength λ, and phase velocity c = λν, but it does not have the properties of instantaneous position and velocity. A (point) particle is characterized by its position and velocity, but clearly not by frequency, etc. The actual experimentally veried truth about electromagnetic radiation is that it has both wave-like and particle-like properties in dierent circumstances. Its wave-like properties are best illustrated by interference and diraction, while the photoelectric eect and Compton scattering illustrate its particle-like properties. Classical physics regards matter as something that has particle-like properties, and radiation (elds) as something that has wave-like properties. The true properties of radiation blurred this distinction. Also, Einstein's theory of relativity blurs the distinction between particles and radiation by stating that energy carried by the mass of particles can be converted into radiation (E = mc ), which found a dramatic conrmation in the discovery of nuclear reactions. De Broglie's postulate Radiation can behave like matter. Shouldn't matter be able to behave like radiation? De Broglie's generalization of Einstein's postulate to matter assigns a property of frequency and wavelength to matter. A particle with energy E and momentum p should have frequency ν = E/h and wavelength λ = h/p, just like a photon. Matter should behave like a wave in certain situations. This postulate was experimentally conrmed by observing diraction of electrons, slow neutrons, atoms and even molecules. The rst such observation was electron diraction on crystals. Our everyday experience does not give us any reason to think that matter has wave properties. But, the same is true regarding light in geometrical optics. A collection of lenses, mirrors, etc. will generally deect and scatter light as if a light beam were nothing but a stream of particle-like photons. In order to see wave-like phenomena it is necessary to present obstacles to light whose dimensions (size) are comparable with the wavelength. For example, diraction on a grating will be noticeable only when the wavelength becomes comparable with the size of openings in the grating. Momentum carried by matter particles or objects is typically much larger than that carried by radiation, so even sub-atomic distances are much larger than the de Broglie wavelength and do not give rise to diraction. Probabilistic interpretation of wave dynamics De Broglie's matter waves are postulated to have exactly the same kind of particle-wave duality as radiation. The matter wave dynamics is mathematically captured by a wavefunction ψ(r, t) which plays the same role as electric eld in electrodynamics. However, the wavefunction is a complex scalar rather than a real vector like electromagnetic eld, because matter does not have polarization. Choosing complex numbers for ψ is a matter of mathematical convenience. We do not have any physical interpretation for ψ, it is not a measurable quantity. We only use it to calculate superposition of matter waves. We assume that the observed matter particles are quanta of de Broglie waves, just like photons are quanta of electromagnetic waves. We have interpreted the intensity of light I(r, t) E(r, t) as the number of photons passing through a beam cross-section per unit time. We saw that in order to dene a continuously varying intensity in terms of discrete photons we had to redene intensity as the number of photons times the probability that a photon participates in the beam. The number of photons is a discrete (integer) number, but probability is not. We must resort to the same trick for matter waves. We assign the following physical meaning to the wavefunction: ψ(r, t) is the probability density that a particle would be detected at location r at time t. Note the analogy between this denition and I E of electrodynamics in the limit of a single photon. The wavefunction is meant to describe a single particle.

4 6 If we attempt to measure the position of a particle, we might nd a dierent result every time we measure, just like in the case of light in the double-slit experiment. We can only determine the probability of measurement outcomes by calculating ψ, but we cannot make a precise prediction of where a particle is located (unless the wavefunction is zero everywhere except at a single point in space). The wavefunction must be normalized, implying that a particle must be somewhere at any given time: d 3 r ψ(r, t) = 1 The fundamental dynamics of matter must describe wave propagation of de Broglie waves. We will construct the equation of motion later. Here, we will focus on the kinematics of de Broglie waves. Heisenberg uncertainty principle A plane matter wave is given by the wavefunction ψ(r, t) = ψ 0 e i(kr ωt), where k is a wavevector ( k = π/λ) and ω = πν angular frequency. This wavefunction describes a particle with completely undetermined position, since ψ(r, t) = ψ 0 = const. However, its momentum is well dened, being equal to p = h/λ, and its energy is E = hν. A localized particle is described by the wavefunction ψ(r, t) = δ(r r 0 ), where δ(r) is the Dirac deltafunction which vanishes when its argument is not equal to zero. If we Fourier transform this function we nd: d 3 k δ(r) = (π) 3 eikr Therefore, a fully localized wavefunction is an equal-amplitude superposition of all possible plane waves. Since a wavevector is related to particle momentum, we see that a localized particle has a completely undetermined momentum. We see that wave kinematics together with the relationships p = h/λ and E = hν has the ingredients necessary to describe the phenomenon seen in the double-slit experiment: it is not possible to simultaneously observe the wave-like and particle-like behavior. Observing interference prohibits knowing through which slit a particle goes through, while measurements of the latter destroy the interference pattern. In classical physics one assumes that both position and momentum can be measured at the same time with arbitrary accuracy. This is not possible in quantum mechanics. One can be measured accurately only at the expense of not knowing the other one accurately. Wavepackets How does a wave which minimizes the uncertainty of both momentum and position look like? Such a wave is called a wavepacket. Let us for now ignore time dependence and consider a one-dimensional wavepacket given by the wavefunction ψ(x). This wavefunction is localized mostly within a nite distance x around the centerof-mass point x 0. Its Fourier transform ψ(k) is also localized in k-space within a nite distance k around the wavevector k 0. The Fourier and inverse-fourier transforms can be dened as: dk ψ(x) = ψ(k)e ikx π ψ(k) = dx π ψ(x)e ikx These transformations look practically the same, there is a certain symmetry between real-space coordinates x and momentum coordinates k.

5 7 Consider a function ψ(x) localized within x 0 x x x 0 + x. For example: { ψ(x) = 1 1, x 0 x x x 0 + x x 0, otherwise Its Fourier transform is: ψ(k) = = x0+ x 1 dx e ikx = ie ikx0 x π k ( e ik x e ik x) π x x 0 x x sin(k x) e ikx0 π k x The function sin(α)/α is oscillatory under an envelope which decays as 1/α as α. Therefore, ψ(k) is localized in momentum space. The localization interval in momentum space can be estimated as the value of k at which α becomes large enough, say α = 1 (this is an arbitrary choice). Hence we nd that k x 1. This is a very general conclusion, independent of the precise choice of ψ(x). It also works in the opposite direction: we could have assumed localization in momentum space and deduced localization in real space from the inverse Fourier transform. The uncertainty in real space is inverse-proportional to the uncertainty in momentum space. We could write this as x k = α. What is the smallest possible value for α? The answer reveals a fundamental limit for the accuracy of simultaneous measurements of position and momentum. It is not hard to guess that the symmetry between Fourier and inverse-fourier transforms implies that the smallest α occurs when ψ(x) and ψ(k) have the same dependence on their arguments. What function ψ(x) has the Fourier transform which looks the same as ψ(x)? Such a function is Gaussian (we set x 0 = 0 for simplicity, without loss of generality, and temporarily give up proper normalization in order to achieve a complete formal symmetry between the function and its Fourier transform): Its Fourier transform is: ψ(k) = ψ(x) = 1 πδx 1 πδx e x δx dx π e ikx e x δx One calculates a Gaussian integral by completing the square in the exponent: and by using the formula: Fe nd: 1 δx x + ikx = 1 δx (x ikδx ) k δx ψ(k) = dxe ax = π a 1 1 πδx e k δx 1 = e k δk πδx π πδk where we have written δx = 1/δk. The Fourier-transform has exactly the same form as the original function, but in terms of k instead of x. This assures us that the obtained δkδx = 1 is the best possible. As a nal touch, we note the relationship between δx and the root-mean-square x dened as: dx (x x 0 ) ψ(x) x = (x x 0 ) = dx ψ(x)

6 8 x is the variance of the probability distribution given by ψ(x), written in the form which does not require the proper normalization of ψ(x) - the denominator automatically xes the correct normalization. For the Gaussian with x 0 = 0 used above we have: x = π dx ψ(x) = 1 πδx x δx x dx πδx e = π δx π dx e x δx = 1 π dx x e x dx e x = 1 π = π δx π π = δx (after integration-by-parts the integral in the second line is reduced to the integral of e x ). It is customary in literature to express uncertainty relationships in terms of the root-mean-square x = δx/ and k = δk/. Then, δkδx = 1 turns into x k = 1/. The uncertainty principle We want a wavepacket to describe a particle with energy E and momentum p which moves along a classical trajectory r 0 (t) = vt with velocity v. We must have E = p /m = ω and p = mv = k 0, where we have introduced Dirac's constant = h/π. We found that the uncertainty of position x and the uncertainty of wavevector k are at best constrained by x k 1/. We now multiply k by in order to obtain the uncertainty of momentum p. We discover that: x p This is known as the Heisenberg uncertainty principle. It is possible to measure the position of a particle, or momentum of a particle with an arbitrary accuracy. However, both cannot be measured with arbitrary accuracy at the same time. An analogous relationship follows by considering time dependence of a wavefunction: t ω 1/. Since E = ω we nd: t E This relationship gives as a lower bound on the amount of time t needed to complete a measurement of particle energy with accuracy E. Matter wave kinematics A Plane wave ψ(x, t) = ψ 0 e i(kx ωt) is characterized by the phase velocity v p = λν = (π/k)(ω/π) = ω/k. This is the velocity v p = dx/dt at which a point of xed phase θ = kx ωt = const. moves as the wave oscillates. For electromagnetic radiation v p = c is the speed of light. For relativistic matter waves we have: v p = h E p h = E p = p c + (mc ) = c p Using the relativistic expression for the momentum p = mv 1 v c 1 + ( ) mc p we nd: v p = c v

7 9 where v is the particle velocity. We see that the phase velocity is dierent from particle velocity, unless the particle moves at the speed of light, like a photon. Even worse, v p > c which is forbidden by the theory of relativity. The above problem is related to the notion that the wavefunction in its own right does not have a physical meaning. Only ψ is physical, but it does not betray the phase velocity ( ψ = ψ 0 = const.) How do we obtain the physical velocity v of a particle from its wavefunction? Consider a wavepacket dk dω dk ψ(x, t) = π π ψ(k, ω)ei(kx ωt) π ψ(k)ei(kx ω(k)t) Since momentum and energy are related by an equation of motion, there is a denite relationship between frequency and wavevector, ω = ω(k). The most likely position of the particle is calculated by regarding ψ as the probability density: x(t) = dx ψ(x, t) x We now Fourier transform both the ψ and ψ in the this integral: dk dk x(t) = dx π π ψ (k)e i(kx ω(k)t) ψ(k )e i(k x ω(k )t) x The only dependence on x in this integral comes from the phase factors. We can integrate out x by using Dirac delta-functions: Therefore, x(t) = = dxe ikx x = i d dk dxe ikx = πδ(k) dk dk π π ψ (k)ψ(k )e i(ω(k dk dk π π ψ (k)ψ(k )e i(ω(k dxe ikx = πiδ (k) ) ω(k))t dxe i(k k)x x ) ω(k))t ( πi)δ (k k) The Dirac delta-function has the following property: dxf(x)δ (x) = dxf (x)δ(x) = f (0) where f(x) is an arbitrary function. This follows from integration by parts and the denition of the delta-function, dxδ(x) = 1, and δ(x) = 0 ( x 0). Substituting in the integral for v we obtain (k = k): dk x(t) = πi π dk π dk π ψ (k)ψ(k) dω dk t d dk (ψ (k)ψ(k)e i(ω(k ) ω(k))t ) k =k where we assumed that the dominant contributions to this integral come from amplitudes ψ(k) which exhibit little dependence on k. Lastly, we assume that dω/dk is practically constant in the range of k which gives the main contribution to the integral. The normalization condition on the wavefunctions leaves us with: x(t) = dω dk t

8 30 and the particle velocity is: v = dx dt = dω dk In wave dynamics this is called group velocity. For a relativistic particle we have: v = d(e/ ) d(p/ ) = de dp = d p c dp + (mc ) = pc E = c v p which is in accord with the expression for the phase velocity obtained above. It is satisfying to see that the measurable predictions of wave dynamics are consistent with the theory of relativity, since one always nds v c.