1 The Need for Quantum Mechanics

Transcription

1 MATH3385/5385. Quantum Mechanics. Handout # 2: Experimental Evidence; Wave-Particle Duality 1 The Need for Quantum Mechanics Towards the beginning of the century experimental evidence started to emerge that the classical theory did not give an accurate description of nature when dealing with phenomena on the atomic level. Among the experiments that demonstrated the inadequacy of the classical theory we mention the following: Blackbody radiation: The distribution of emitted heat by a blackbody (i.e. the hypothetical situation of an idealised body that absorbs all radiation that falls onto it 1 ) is a universal function E(λ, T), depending on temperature T and wavelength λ = c/ν, ν being the frequency. The energy density in the cavity u(ν,t) = 4E(λ,T)/ν 2 follows experimentally Wien s law (1894) u(ν,t) = ν 3 g(ν/t) in the high frequency (i.e. low wavelength) regime. However, Wien s law is not in accordance with certain principles of classical physics. In the low-frequency regime the classical theory predicts a behaviour of the energy density according to u(ν,t) = 8πν2 c 3 kt (where k = erg/kelvin is Boltzmann s constant) which is Rayleigh-Jeans law (1900). However, this formula does not make sense in the high-frequency regime since the total energy according to this law is infinite. It was Planck, in 1900, who proposed another formula u(ν,t) = 8πh c 3 ν 3 e hν/kt 1, Planck s Law that interpolates correctly and in accordance with experiment between the low- and high frequency regime. The new constant h is called Planck s constant and it turns out to be a constant of nature taking the value: h = erg-sec. In order to arrive at the formula, Planck had to assume that the energy associated with the electromagnetic waves increases with the frequency in a discrete manner, namely in steps equal to ǫ = hν, meaning that radiation is emitted in quanta. This was a revolutionary idea which could not be explained by the classical theory. 1 Such a body can be modelled by considering a closed box with a small hole in it, enclosing a cavity. No radiation entering through the hole can escape the cavity. 1

2 The Bohr Atom: The planetary model of the atom (consisting of a heavy positively charged nucleus surrounded by negatively charged electrons moving in orbits around the nucleus) was proposed in 1908 by E. Rutherford on the basis of experiments with α-particles carried out by H.W. Geiger and E. Marsden. However, by general principles this model is not in accordance with the classical theory. In fact, since an accelerating charge emits electromagnetic radiation, the electrons will constantly lose energy thereby spiralling towards the nucleus and the atom will eventually collapse. Thus, the classical theory cannot explain the stability of the atom. Niels Bohr in 1913 drew from this the consequences, and proposed some postulates to explain the structure of radiation spectra, however thereby breaking drastically with the classical theory. He postulated: 1. that the electrons move in orbits characterised by angular momentum restricted to integer multiples of = h/(2π), 2. that electrons can only make discontinuous transitions in energy, and hence can only emit radiation with frequency of given by ν = E/h. Based on these postulates, he was able to explain the correct (experimentally observed) structure of the emission/absorption-spectra of atoms which behave as with integers n 1, n 2. 1 λ 1 n n 2 2 The Compton effect: Experiments by A.H. Compton demonstrated that the scattering of X-ray radiation when sent through thin metallic foils is not in accordance with the classical theory. Whereas the classical theory would predict a variation of intensity of the scattered waves according to (1 + cos 2 θ), where θ is the scattering angle, and independent of the wavelength, the experiments showed that the intensity would exhibit a distinct peak with wavelength shifted from the orignal wavelength which would depend on the scattering angle proportional to (1 cos θ). An explanation of this phenomenon could be given in terms of a collision process of particles rather than by scattering of waves on atoms according to the classical theory. This led to the assumption that in certain circumstances monochromatic radiation would behave like a beam of particles, called photons, of mass zero and energy given by E = hν, where ν is the frequency of the wave. These particles, following relativistic kinematics, would have a momentum equal to p = hν/c moving with the speed of light c. Thus, the Compton effect seemed to indicate the dual wave/particle behaviour of radiation. Diffraction patterns: Prince Louis De Broglie in 1923 suggested that also matter would exhibit a dual particle/wave behaviour under certain circumstances. He, thus, associated with a particle of momentum p a wavelength given by λ = h/p. Experimental evidence to this suggestion was provided by electron diffraction experiments by C.J. Davisson and L.H. Germer, who investigated the scattering of electrons by a crystal surface. The interference patterns that were produced in these experiments indeed seemed to indicate that electron beams show distinct similarities with the diffraction 2

3 patterns that one would expect from interference phenomena of radiation. A simplified description of such phenomena is the Young double-slit experiment which is a so-called Gedankenexperiment (thought experiment) where one imagines a particle or radiation beam falling on a screen with two slits. In both cases an interference pattern emerges. The interference pattern in the case of the electron beam is not a consequence of possible interactions between different particles in the beam, because the pattern still arises if the beam is diluted to such an extent that only individual particles can move through the slits at a time. However, the diffraction pattern of the particle beam is not the superposition of the patterns corresponding to the situation that either one of the slits is closed, since in the latter case the diffraction pattern of the two-slit situation is destroyed. Thus, it seems as if determining through which slit the particle is going influences in a nontrivial way the outcome of the experiment!, in view of the observation that the single-slit patterns correspond to the case that one knows through which one of the slits the particle has moved. The only viable explanation is that particles behave in these experiments in precisely the same way as a beam of light, which when passing through both slits produces a different superposition pattern than the one obtained when only one slit at a time is open. The conclusions of this Gedanken-experiment are in full accordance with the interpretations of diffraction patterns in realistic experiments. The two main new insights that resulted from these, and other experimental data 2, are the following: Some fundamental phenomena in nature are quantised, i.e. they take place through discontinuous transitions, and the measure of this discreteness is given by a new constant of nature: the Planck s constant h = 6, Joules-sec. The wave/particle distinction between matter and radiation in the classical theory can no longer hold true and must make way for a particle/wave duality that would be fundamental to a correct description of nature. In the early stages of the development of the quantum theory the various interpretations of the observed phenomena led to a number of rather ad hoc principles, e.g. the Bohr- Sommerfeld quantisation rules. The theory developed in these early years (i.e. roughly between 1913 and 1925) is sometimes referred to as the Old Quantum Theory. Due to the contributions of various people, such as N. Bohr, P. Dirac, W. Heisenberg, E. Schrödinger, and others the quantum theory developed eventually into a coherent theory whose internal consistency has been proved. Nonetheless, the conceptual changes that the quantum theory has brought in our understanding of nature are far-reaching and often counter-intuitive. It requires that we have to set aside some preconceived notions that we are brought up with and which stem from of our experience of the macroscopic world. Since the effects of the quantum theory are measured in terms of the very small constant h we usually only see the effects of the quantum theory on the level of the microscopic 2 For a fuller account of the various experimental evidence, please consult some of the textbooks (e.g. S. Gasiorowicz: Quantum Physics, John Wiley & Sons Inc., 1996). 3

4 world 3 This leads us to postulate the correspondence principle: if we consider h to be a (variable) parameter of the theory, we should recover from the quantum theory the results of the classical theory if we would perform the mathematical limit h 0. Obviously, h being a constant of nature, this can only be done as a mathematical exercise, and we call this limit: the classical limit of the theory. Information on the history of QM and its personalities can be found on the website: 2 Quantum Mechanics and Wave Packets When we talk about quantum mechanics we mean the theory that was developed in the late 1920 s, early 1930 s, i.e. the quantum theory of nonrelativistic particles. In this theory we usually don t include the quantum theory of electromagnetic radiation which would necessitate the incorporation of relativistic effects (since photons move at the speed of light). Thus, in quantum mechanics we still do not treat radiation and matter on an equal footing, which is not entirely satisfactory. The theory is, therefore, not yet fully applicable and has to be expanded in a more extensive theory. The latter theory is usually called quantum theory of radiation, or quantum field theory, and it came to full bloom well after the second world war. In this module we can only deal with the more restrictive theory which is the quantum mechanics of particles 4. Our first aim is to arrive at a wave description of particles. In order to do that we have to develop some rough first ideas. What we need is a description that captures both aspects of a particle: its particle -nature (i.e. its behaviour as a localised object in space) and its wave -nature (i.e. the possibility that it expands throughout space and behaves as wave trains). This leads us to the notion of a wave packet: a superposition of waves whose wave envelope is more or less localised depending on the frequencies and wave lengths of the wave components that make up the wave packet. A (complex) plane wave is described by the exponent: e i(k r ωt) where k = (k x,k y,k z ) is the wave vector, the wave length being given by λ = 2π/ k, and where ω = 2πν is the angular frequency. A superposition of waves is some combination of the form g n e i(kn r ωnt) n 3 In recent years it has been recognised that on an intermediate level, between the microscopic and the macroscopic world which is sometimes called the mesoscopic level, some quantum effects already play an important role. This has very practical consequences: in the development of nano-devices, i.e. the ultra-small devices that are being nowadays developed in applications all over the world, the effects of the quantum theory is no longer negligeable. 4 This means that if we would like to incorporate the effects of electromagnetic radiation we can only treat it as an external field interacting with the particles. We will avoid this problem altogether and not deal with the effects of electromagnetism at all. 4

5 where n labels the components of the wave, g n being the amplitudes. If the number of components becomes infinite we may replace the sum by an integral leading to: g(k)e i(k r ω(k)t) dk, dk = dk x dk y dk z where the integral is a three-fold integral over the various components of the wave vector. The dependence of the angular frequency ω(k) on the wave vector is called the dispersion relation. To study how localisation can occur in such integrals, let us investigate the following example. Example: We take for simplicity a one-dimensional situation (i.e. taking into account only the x component of the position vector and the corresponding component k of the wave vector), and consider the function f(x) = g(k)e ikx dk Let us take a Gaussian shape for the function g(k), i.e. g(k) = exp( α(k k 0 ) 2 x), where α is some positive real constant. Gaussian curve 5

6 Using the integral: we can evaluate f(x), namely: f(x) = = e ik 0x e ak2 dk = π a g(k)e i(k k 0)x e ik 0x dk = e ik 0x Thus, the absolute value of f(x) is given by e α(k ix/(2α)) 2 e x2 /(4α) dk = f(x) 2 = π α e x2 /(2α) e αk 2 e ik x dk π a eik 0x e x2 /(4α) whose graph is a simple Gaussian function (see the figure). Comparing the graphs of f(x) 2 and g(k) we can estimate the spreading of the wave packet by the region in the x- resp. k-variable where the graph drops down by a factor 1/e say from its peak value. In the x-value we thus get a measure of spreading x = 2 2α whereas in the variable k we get a spreading k = 2/ α. The product of these measures of uncertainty is x k = 4 2. What is important is not the precise number on the right-hand side of this formula, (which depends on how we measure the spreading of the wave packet), but the fact that however we measure it we get something of the order of or greater than 1, which we write as: x k 1 independent of the parameter α! We note that α changes the spreading of the individual graphs of f(x) 2 and g(k)), but the greater the spreading in x the smaller the spreading in k and vice versa. Note that as α or α 0 one of the graphs gets infinitely sharped peaked (with a an infinitely high peak) whereas the other gets infinitely spread out. If we regard these wave-packets as somehow representing a physical particle and the spreadings in x as a measure for the uncertainty in determining its position (due to the fuzziness of the particle), whereas using the De Broglie relation p = k the spreading in k measuring the uncertainty in determining its momentum, we are lead to the uncertainty relation : p x. Since is very small this fundamental uncertainty in determining both p as well as x to arbitrary precision which follows as a consequence of this wave description, it is negligeable on the macroscopic level, but becomes important on the microscopic (i.e. atomic) level. Let us now study how the wave packets representing particles might propagate in time, which means that we have to take into account the dispersion factor exp( iω(k)t in the superposition formula. Thus, in the one-dimensional case we get the integral f(x,t) = g(k)e i(kx ω(k)t) dk If we are dealing with a one-dimensional free particle, (i.e. there are no external forces), then its kinetic energy is related to the Planck s energy leading to: E = p2 2m = ω 6

7 On the other hand the velocity of the particle is expected to correspond to the group velocity of the wave packet, i.e. v g = p m = dω dk Thus, expressing ω in terms of p and using the De Broglie relation p = k we get ω = p2 2m = k2 2m v g = k m as expected. Thus, the wave packet can be rewritten in the form ψ(x,t) = 1 2π φ(p)e i(px Et)/ dp, E = p2 2m after changing somewhat our notation (the reason for the prefactor will be explained later). We have now used the in Quantum Mechanics more commonly used symbol ψ for the socalled wave function representing the particle. From this integral representation one can easily derive an equation for ψ, i.e. a formula not involving the function φ(p), namely the following partial differential equation: i ψ t = 2 2 ψ 2m x 2 which is the so-called Schrödinger equation (SE) for a one-dimensional free particle. The derivation of the SE from the integral is simple: take the derivatives inside the integral (assuming this is allowed) and check that when acting on the exponent the left-hand side equals the right-hand side. 7