Quantum Physics Notes

Quantum Physics Notes 26
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QuantumPhysicsNotes J D Cresser Department of Physics Macquarie University st 31 August 2011Chapter 1 Introduction here are three fundamental theories on which modern physics is built: the theory of relativity, T statistical mechanics/thermodynamics, and quantum mechanics. Each one has forced upon us the need to consider the possibility that the character of the physical world, as we perceive it and understand it on a day to day basis, may be far di erent from what we take for granted. Already, the theory of special relativity, through the mere fact that nothing can ever be observed to travel faster than the speed of light, has forced us to reconsider the nature of space and time – that there is no absolute space, nor is time ‘like a uniformly flowing river’. The concept of ‘now’ or ‘the present’ is not absolute, something that everyone can agree on – each person has their own private ‘now’. The theory of general relativity then tells us that space and time are curved, that the universe ought to be expanding from an initial singularity (the big bang), and will possibly continue expanding until the sky, everywhere, is uniformly cold and dark. Statistical mechanics/thermodynamics gives us the concept of entropy and the second law: the entropy of a closed system can never decrease. First introduced in thermodynamics – the study of matter in bulk, and in equilibrium – it is an aid, amongst other things, in understanding the ‘direction in time’ in which natural processes happen. We remember the past, not the future, even though the laws of physics do not make a distinction between the two temporal directions ‘into the past’ and ‘into the future’. All physical processes have what we perceive as the ‘right’ way for them to occur – if we see something happening ‘the wrong way round’ it looks very odd indeed: eggs are often observed to break, but never seen to reassemble themselves. The sense of uni-directionality of events defines for us an ‘arrow of time’. But what is entropy? Statistical mechanics – which attempts to explain the properties of matter in bulk in terms of the aggregate behaviour of the vast numbers of atoms that make up matter – stepped in and told us that this quantity, entropy, is not a substance in any sense. Rather, it is a measure of the degree of disorder that a physical system can possess, and that the natural direction in which systems evolve is in the direction such that, overall, entropy never decreases. Amongst other things, this appears to have the consequence that the universe, as it ages, could evolve into a state of maximum disorder in which the universe is a cold, uniform, amorphous blob – the so-called heat death of the universe. So what does quantum mechanics do for us? What treasured view of the world is turned upside down by the edicts of this theory? It appears that quantum mechanics delivers to us a world view in which  There is a loss of certainty – unavoidable, unremovable randomness pervades the physical world. Einstein was very dissatisfied with this, as expressed in his well-known statement: “God does not play dice with the universe.” It even appears that the very process of making an observation can a ect the subject of this observation in an uncontrollably random way (even if no physical contact is made with the object under observation). c J D Cresser 2011Chapter 1 Introduction 2  Physical systems appear to behave as if they are doing a number of mutually exclusive things simultaneously. For instance an electron fired at a wall with two holes in it can appear to behave as if it goes through both holes simultaneously.  Widely separated physical systems can behave as if they are entangled by what Einstein termed some ‘spooky action at a distance’ so that they are correlated in ways that appear to defy either the laws of probability or the rules of special relativity. It is this last property of quantum mechanics that leads us to the conclusion that there are some aspects of the physical world that cannot be said to be objectively ‘real’. For instance, in the game known as The Shell Game, a pea is hidden under one of three cups, which have been shued around by the purveyor of the game, so that bystanders lose track of which cup the pea is under. Now suppose you are a bystander, and you are asked to guess which cup the pea is under. You might be lucky and guess which cup first time round, but you might have to have another attempt to find the cup under which the pea is hidden. But whatever happens, when you do find the pea, you implicitly believe that the pea was under that cup all along. But is it possible that the pea really wasn’t at any one of the possible positions at all, and the sheer process of looking to see which cup the pea is under, which amounts to measuring the position of the pea, ‘forces’ it to be in the position where it is ultimately observed to be? Was the pea ‘really’ there beforehand? Quantum mechanics says that, just maybe, it wasn’t there all along Einstein had a comment or two about this as well. He once asked a fellow physicist (Pascual Jordan): “Do you believe the moon exists only when you look at it?” The above three points are all clearly in defiance of our classical view of the world, based on the theories of classical physics, which goes had-in-hand with a particular view of the world some- times referred to as objective realism. 1.1 Classical Physics Before we look at what quantum mechanics has to say about how we are to understand the natural world, it is useful to have a look at what the classical physics perspective is on this. According to classical physics, by which we mean pre-quantum physics, it is essentially taken for granted that there is an ‘objectively real world’ out there, one whose properties, and whose very existence, is totally indi erent to whether or not we exist. These ideas of classical physics are not tied to any one person – it appears to be the world-view of Galileo, Newton, Laplace, Einstein and many other scientists and thinkers – and in all likelihood reflects an intuitive understanding of reality, at least in the Western world. This view of classical physics can be referred to as ‘objective reality’. The equations of the theories of classical physics, which include Newtonian mechan- ics, Maxwell’s theory of the electromag- netic field and Einstein’s theory of general relativity, are then presumed to describe what is ‘really happening’ with a physical sys- tem. For example, it is assumed that every particle has a definite position and velocity and that the solution to Newton’s equations Observed path Calculated path for a particle in motion is a perfect repre- Figure 1.1: Comparison of observed and calculated sentation of what the particle is ‘actually paths of a tennis ball according to classical physics doing’. Within this view of reality, we can speak about a particle moving through space, such as a tennis ball flying through the air, as if it has, at any time, a definite position and velocity. Moreover, it c J D Cresser 2011Chapter 1 Introduction 3 would have that definite position and velocity whether or not there was anyone or anything mon- itoring its behaviour. After all, these are properties of the tennis ball, not something attributable to our measurement e orts. Well, that is the classical way of looking at things. It is then up to us to decide whether or not we want to measure this pre-existing position and velocity. They both have definite values at any instant in time, but it is totally a function of our experimental ingenuity whether or not we can measure these values, and the level of precision to which we can measure them. There is an implicit belief that by refining our experiments — e.g. by measuring to to the 100th decimal place, then the 1000th, then the 10000th — we are getting closer and closer to the values of the position and velocity that the particle ‘really’ has. There is no law of physics, at least according to classical physics, that says that we definitely cannot determine these values to as many decimal places as we desire – the only limitation is, once again, our experimental inge- nuity. We can also, in principle, calculate, with unlimited accuracy, the future behaviour of any physical system by solving Newton’s equations, Maxwell’s equations and so on. In practice, there are limits to accuracy of measurement and/or calculation, but in principle there are no such limits. 1.1.1 Classical Randomness and Ignorance of Information Of course, we recognize, for a macroscopic object, that we cannot hope to measure all the positions and velocities of all the particles making such an object. In the instance of a litre of air in a bottle at 26 room temperature, there are something like 10 particles whizzing around in the bottle, colliding with one another and with the walls of the bottle. There is no way of ever being able to measure the position and velocities of each one of these gas particles at some instant in time. But that does not stop us from believing that each particle does in fact possess a definite position and velocity at each instant. It is just too dicult to get at the information. Likewise, we are unable to predict the mo- tion of a pollen grain suspended in a liquid: Brownian motion (random walk) of pollen grain due to collisions with molecules of liquid. According to classical physics, the information is ‘really there’ – we just can’t get at it. Random behaviour only appears random be- cause we do not have enough information to describe it exactly. It is not really ran- dom because we believe that if we could re- peat an experiment under exactly identical conditions we ought to get the same result Figure 1.2: Random walk of a pollen grain suspended in every time, and hence the outcome of the a liquid. experiment would be perfectly predictable. In the end, we accept a certain level of ignorance about the possible information that we could, in principle, have about the gas. Because of this, we cannot hope to make accurate predictions about what the future behaviour of the gas is going to be. We compensate for this ignorance by using statistical methods to work out the chances of the gas particles behaving in various possible ways. For instance, it is possible to show that the chances of all the gas particles spontaneously rushing 26 10 to one end of the bottle is something like 1 in 10 – appallingly unlikely. The use of statistical methods to deal with a situation involving ignorance of complete information is reminiscent of what a punter betting on a horse race has to do. In the absence of complete information about each of the horses in the race, the state of mind of the jockeys, the state of the track, what the weather is going to do in the next half hour and any of a myriad other possible c J D Cresser 2011Chapter 1 Introduction 4 influences on the outcome of the race, the best that any punter can do is assign odds on each horse winning according to what information is at hand, and bet accordingly. If, on the other hand, the punter knew everything beforehand, the outcome of the race is totally foreordained in the mind of the punter, so (s)he could make a bet that was guaranteed to win. According to classical physics, the situation is the same when it comes to, for instance, the evolu- tion of the whole universe. If we knew at some instant all the positions and all the velocities of all the particles making up the universe, and all the forces that can act between these particles, then we ought to be able to calculate the entire future history of the universe. Even if we cannot carry out such a calculation, the sheer fact that, in principle, it could be done, tells us that the future of the universe is already ordained. This prospect was first proposed by the mathematical physicist Pierre-Simon Laplace (1749-1827) and is hence known as Laplacian determinism, and in some sense represents the classical view of the world taken to its most extreme limits. So there is no such thing, in classical physics, as true randomness. Any uncertainty we experience is purely a consequence of our ignorance – things only appear random because we do not have enough infor- mation to make precise predictions. Nevertheless, behind the scenes, everything is evolving in an entirely preordained way – everything is deterministic, there is no such thing as making a decision, free will is merely an illusion 1.2 Quantum Physics The classical world-view works fine at the everyday (macroscopic) level – much of modern en- gineering relies on this – but there are things at the macroscopic level that cannot be understood using classical physics, these including the colour of a heated object, the existence of solid objects . . . . So where does classical physics come unstuck? Non-classical behaviour is most readily observed for microscopic systems – atoms and molecules, but is in fact present at all scales. The sort of behaviour exhibited by microscopic systems that are indicators of a failure of classical physics are  Intrinsic Randomness  Interference phenomena (e.g. particles acting like waves)  Entanglement Intrinsic Randomness It is impossible to prepare any physical system in such a way that all its physical attributes are precisely specified at the same time – e.g. we cannot pin down both the position and the momentum of a particle at the same time. If we trap a particle in a tiny box, thereby giving us a precise idea of its position, and then measure its velocity, we find, after many repetitions of the experiment, that the velocity of the particle always varies in a random fashion from one measurement to the next. For instance, for an electron trapped in a box 1 micron 1 in size, the velocity of the electron can be measured to vary by at least50 ms . Refinement of the experiment cannot result in this randomness being reduced — it can never be removed, and making the box even tinier just makes the situation worse. More generally, it is found that for any experiment repeated under exactly identical conditions there will always be some physical quantity, some physical property of the systems making up the experiment, which, when measured, will always yield randomly varying results from one run of the experiment to the next. This is not because we do a lousy job when setting up the experiment or carrying out the measurement. The randomness is irreducible: it cannot be totally removed by improvement in experimental technique. What this is essentially telling us is that nature places limits on how much information we can gather about any physical system. We apparently cannot know with precision as much about c J D Cresser 2011Chapter 1 Introduction 5 a system as we thought we could according to classical physics. This tempts us to ask if this missing information is still there, but merely inaccessible to us for some reason. For instance, does a particle whose position is known also have a precise momentum (or velocity), but we simply cannot measure its value? It appears that in fact this information is not missing – it is not there in the first place. Thus the randomness that is seen to occur is not a reflection of our ignorance of some information. It is not randomness that can be resolved and made deterministic by digging deeper to get at missing information – it is apparently ‘uncaused’ random behaviour. Interference Microscopic physical systems can behave as if they are doing mutually exclusive things at the same time. The best known example of this is the famous two slit experiment in which electrons are fired, one at a time, at a screen in which there are two narrow slits. The electrons are observed to strike an observation screen placed beyond the screen with the slits. What is expected is that the electrons will strike this second screen in regions immediately opposite the two slits. What is observed is that the electrons arriving at this observation screen tend to arrive in preferred locations that are found to have all the characteristics of a wave-like interference pattern, i.e. the pattern formed as would be observed if it were waves (e.g. light waves) being directed towards the slits. Electrons strike screen at random.     Electron Two slit gun interference pattern The detailed nature of the interference pattern is determined by the separation of the slits: increas- ing this separation produces a finer interference pattern. This seems to suggest that an electron, which, being a particle, can only go through one slit or the other, somehow has ‘knowledge’ of the position of the other slit. If it did not have that information, then it is hard to see how the electron could arrive on the observation screen in such a manner as to produce a pattern whose features are directly determined by the slit separation And yet, if the slit through which each electron passes is observed in some fashion, the interference pattern disappears – the electrons strike the screen at positions directly opposite the slits The uncomfortable conclusion that is forced on us is that if the path of the electron is not observed then, in some sense, it passes through both slits much as waves do, and ultimately falls on the observation screen in such a way as to produce an interference pattern, once again, much as waves do. This propensity for quantum system to behave as if they can be two places at once, or more generally in di erent states at the same time, is termed ‘the superposition of states’ and is a singular property of quantum systems that leads to the formulation of a mathematical description based on the ideas of vector spaces. Entanglement Suppose for reasons known only to yourself that while sitting in a hotel room in Sydney looking at a pair of shoes that you really regret buying, you decided to send one of the pair to a friend in Brisbane, and the other to a friend in Melbourne, without observing which shoe went where. It would not come as a surprise to hear that if the friend in Melbourne discovered that the shoe they received was a left shoe, then the shoe that made it to Brisbane was a right shoe, c J D Cresser 2011Chapter 1 Introduction 6 and vice versa. If this strange habit of splitting up perfectly good pairs of shoes and sending one at random to Brisbane and the other to Melbourne were repeated many times, then while it is not possible to predict for sure what the friend in, say Brisbane, will observe on receipt of a shoe, it is nevertheless always the case that the results observed in Brisbane and Melbourne were always perfectly correlated – a left shoe paired o with a right shoe. Similar experiments can be undertaken with atomic particles, though it is the spins of pairs of particles that are paired o : each is spinning in exactly the opposite fashion to the other, so that the total angular momentum is zero. Measurements are then made of the spin of each particle when it arrives in Brisbane, or in Melbourne. Here it is not so simple as measuring whether or not the spins are equal and opposite, i.e. it goes beyond the simple example of left or right shoe, but the idea is nevertheless to measure the correlations between the spins of the particles. As was shown by John Bell, it is possible for the spinning particles to be prepared in states for which the correlation between these measured spin values is greater than what classical physics permits. The systems are in an ‘entangled state’, a quantum state that has no classical analogue. This is a conclusion that is experimentally testable via Bell’s inequalities, and has been overwhelmingly confirmed. Amongst other things it seems to suggest the two systems are ‘communicating’ instantaneously, i.e. faster than the speed of light which is inconsistent with Einstein’s theory of relativity. As it turns out, it can be shown that there is no faster-than-light communication at play here. But it can be argued that this result forces us to the conclusion that physical systems acquire some (maybe all?) properties only through the act of observation, e.g. a particle does not ‘really’ have a specific position until it is measured. The sorts of quantum mechanical behaviour seen in the three instances discussed above are be- lieved to be common to all physical systems. So what is quantum mechanics? It is saying some- thing about all physical systems. Quantum mechanics is not a physical theory specific to a limited range of physical systems i.e. it is not a theory that applies only to atoms and molecules and the like. It is a meta-theory. At its heart, quantum mechanics is a set of fundamental principles that constrain the form of physical theories themselves, whether it be a theory describing the mechan- ical properties of matter as given by Newton’s laws of motion, or describing the properties of the electromagnetic field, as contained in Maxwell’s equations or any other conceivable theory. Another example of a meta-theory is relativity — both special and general — which places strict conditions on the properties of space and time. In other words, space and time must be treated in all (fundamental) physical theories in a way that is consistent with the edicts of relativity. To what aspect of all physical theories do the principles of quantum mechanics apply? The princi- ples must apply to theories as diverse as Newton’s Laws describing the mechanical properties of matter, Maxwell’s equations describing the electromagnetic field, the laws of thermodynamics – what is the common feature? The answer lies in noting how a theory in physics is formulated. 1.3 Observation, Information and the Theories of Physics Modern physical theories are not arrived at by pure thought (except, maybe, general relativity). The common feature of all physical theories is that they deal with the information that we can ob- tain about physical systems through experiment, or observation. For instance, Maxwell’s equations for the electromagnetic field are little more than a succinct summary of the observed properties of electric and magnetic fields and any associated charges and currents. These equations were ab- stracted from the results of innumerable experiments performed over centuries, along with some clever interpolation on the part of Maxwell. Similar comments could be made about Newton’s laws of motion, or thermodynamics. Data is collected, either by casual observation or controlled experiment on, for instance the motion of physical objects, or on the temperature, pressure, vol- ume of solids, liquids, or gases and so on. Within this data, regularities are observed which are c J D Cresser 2011Chapter 1 Introduction 7 best summarized as equations: F = ma — Newton’s second law; B rE = — One of Maxwell’s equations (Faraday’s law); t PV = NkT — Ideal gas law (not really a fundamental law) What these equations represent are relationships between information gained by observation of various physical systems and as such are a succinct way of summarizing the relationship between the data, or the information, collected about a physical system. The laws are expressed in a manner consistent with how we understand the world from the view point of classical physics in that the symbols replace precisely known or knowable values of the physical quantities they represent. There is no uncertainty or randomness as a consequence of our ignorance of information about a system implicit in any of these equations. Moreover, classical physics says that this information is a faithful representation of what is ‘really’ going on in the physical world. These might be called the ‘classical laws of information’ implicit in classical physics. What these pre-quantum experimenters were not to know was that the information they were gathering was not refined enough to show that there were fundamental limitations to the accuracy with which they could measure physical properties. Moreover, there was some information that they might have taken for granted as being accessible, simply by trying hard enough, but which we now know could not have been obtained at all There was in operation unsuspected laws of nature that placed constraints on the information that could be obtained about any physical system. In the absence in the data of any evidence of these laws of nature, the information that was gathered was ultimately organised into mathematical statements that constituted classical laws of physics: Maxwell’s equations, or Newton’s laws of motion. But in the late nineteenth century and on into the twentieth century, experimental evidence began to accrue that suggested that there was something seriously amiss with the classical laws of physics: the data could no longer be fitted to the equations, or, in other words, the theory could not explain the observed experimental results. The choice was clear: either modify the existing theories, or formulate new ones. It was the latter approach that succeeded. Ultimately, what was formulated was a new set of laws of nature, the laws of quantum mechanics, which were essentially a set of laws concerning the information that could be gained about the physical world. These are not the same laws as implicit in classical physics. For instance, there are limits on the information that can be gained about a physical system. For instance, if in an experiment we 1 measure the position x of a particle with an accuracy of x, and then measure the momentum p of the particle we find that the result for p randomly varies from one run of the experiment to the next, spread over a range p. But there is still law here. Quantum mechanics tells us that 1 xp — the Heisenberg Uncertainty Relation 2 Quantum mechanics also tells us how this information is processed e.g. as a system evolves in time (the Schrodinger ¨ equation) or what results might be obtained in in a randomly varying way in a measurement. Quantum mechanics is a theory of information, quantum information theory. What are the consequences? First, it seems that we lose the apparent certainty and determinism of classical physics, this being replaced by uncertainty and randomness. This randomness is not due to our inadequacies as experimenters — it is built into the very fabric of the physical world. But on the positive side, these quantum laws mean that physical systems can do so much more within these restrictions. A particle with position or momentum uncertain by amounts x and p means we do not quite know where it is, or how fast it is going, and we can never know this. But 1 Accuracy indicates closeness to the true value, precision is the repeatability or reproducibility of the measurement. c J D Cresser 2011Chapter 2 The Early History of Quantum Mechanics n the early years of the twentieth century, Max Planck, Albert Einstein, Louis de Broglie, Neils I Bohr, Werner Heisenberg, Erwin Schrodinger ¨ , Max Born, Paul Dirac and others created the theory now known as quantum mechanics. The theory was not developed in a strictly logical way – rather, a series of guesses inspired by profound physical insight and a thorough command of new mathematical methods was sewn together to create a theoretical edifice whose predictive power is such that quantum mechanics is considered the most successful theoretical physics construct of the human mind. Roughly speaking the history is as follows: Planck’s Black Body Theory (1900) One of the major challenges of theoretical physics towards the end of the nineteenth century was to derive an expression for the spectrum of the electromag- netic energy emitted by an object in thermal equilibrium at some temperature T. Such an object is known as a black body, so named because it absorbs light of any frequency falling on it. A black body also emits electromagnetic radiation, this being known as black body radiation, and it was a formula for the spectrum of this radiation that was being sort for. One popular candidate for the formula was Wein’s law: 3 f=T S( f; T) = f e (2.1) The quantityS( f; T), otherwise known as the spec- S( f; T) 6 Rayleigh-Jeans tral distribution function, is such thatS( f; T)d f is the energy contained in unit volume of electromagnetic Planck radiation in thermal equilibrium at an absolute tem- perature T due to waves of frequency between f and f + d f . The above expression for S was not so much Wein derived from a more fundamental theory as quite sim- ply guessed. It was a formula that worked well at high frequencies, but was found to fail when improved ex- perimental techniques made it possible to measureS at lower (infrared) frequencies. There was another - candidate forS which was derived using arguments f from classical physics which lead to a formula for Figure 2.1: Rayleigh-Jeans (classical), S( f; T) known as the Rayleigh-Jeans formula: Wein, and Planck spectral distributions. 2 8 f S( f; T) = k T (2.2) B 3 c where k is a constant known as Boltzmann’s constant. This formula worked well at low fre- B quencies, but su ered from a serious problem – it clearly increases without limit with increasing frequency – there is more and more energy in the electromagnetic field at higher and higher fre- quencies. This amounts to saying that an object at any temperature would radiate an infinite c J D Cresser 2011Chapter 2 The Early History of Quantum Mechanics 10 amount of energy at infinitely high frequencies. This result, ultimately to become known as the ‘ultra-violet catastrophe’, is obviously incorrect, and indicates a deep flaw in classical physics. In an attempt to understand the form of the spectrum of the electromagnetic radiation emitted by a black body, Planck proposed a formula which he obtained by looking for a formula that fitted Wein’s law at high frequencies, and also fitted the new low frequency experimental results (which happen to be given by the Rayleigh-Jeans formula, though Planck was not aware of this). It was when he tried to provide a deeper explanation for the origin of this formula that he made an important discovery whose significance even he did not fully appreciate. In this derivation, Planck proposed that the atoms making up the black body object absorbed and emitted light of frequency f in multiples of a fundamental unit of energy, or quantum of energy, E = h f . On the basis of this assumption, he was able to rederive the formula he had earlier guessed: 3 8h f 1 S( f; T) = : (2.3) 3 c exp(h f=kT) 1 This curve did not diverge at high frequencies – there was no ultraviolet catastrophe. Moreover, by fitting this formula to experimental results, he was able to determine the value of the constant 34 h, that is, h = 6:6218 10 Joule-sec. This constant was soon recognized as a new fundamental constant of nature, and is now known as Planck’s constant. In later years, as quantum mechanics evolved, it was found that the ratio h=2 arose time and again. As a consequence, Dirac introduced a new quantity = h=2, pronounced ‘h-bar’, which is now the constant most commonly encountered. In terms of, Planck’s formula for the quantum of energy becomes E = h f = (h=2) 2 f = (2.4) where is the angular frequency of the light wave. Einstein’s Light Quanta (1905) Although Planck believed that the rule for the absorption and emission of light in quanta applied only to black body radiation, and was a property of the atoms, rather than the radiation, Einstein saw it as a property of electromagnetic radiation, whether it was black body radiation or of any other origin. In particular, in his work on the photoelectric e ect, he proposed that light of frequency was made up of quanta or ‘packets’ of energy which could be only absorbed or emitted in their entirety. Bohr’s Model of the Hydrogen Atom (1913) Bohr then made use of Einstein’s ideas in an at- tempt to understand why hydrogen atoms do not self destruct, as they should according to the laws of classical electromagnetic theory. As implied by the Rutherford scattering experiments, a hydro- gen atom consists of a positively charged nucleus (a proton) around which circulates a very light (relative to the proton mass) negatively charged particle, an electron. Classical electromagnetism says that as the electron is accelerating in its circular path, it should be radiating away energy in 12 the form of electromagnetic waves, and do so on a time scale of 10 seconds, during which time the electron would spiral into the proton and the hydrogen atom would cease to exist. This obviously does not occur. Bohr’s solution was to propose that provided the electron circulates in orbits whose radii r satisfy an ad hoc rule, now known as a quantization condition, applied to the angular momentum L of the electron L = mvr = n (2.5) where v is the speed of the electron and m its mass, and n a positive integer (now referred to as a quantum number), then these orbits would be stable – the hydrogen atom was said to be in a stationary state. He could give no physical reason why this should be the case, but on the basis of c J D Cresser 2011Chapter 2 The Early History of Quantum Mechanics 11 this proposal he was able to show that the hydrogen atom could only have energies given by the formula 2 ke 1 E = (2.6) n 2 2a n 0 where k = 1=4 and 0 2 4 0 a = = 0:0529 nm (2.7) 0 2 me is known as the Bohr radius, and roughly speaking gives an indication of the size of an atom as determined by the rules of quantum mechanics. Later we shall see how an argument based on the uncertainty principle gives a similar result. The tie-in with Einstein’s work came with the further proposal that the hydrogen atom emits or absorbs light quanta by ‘jumping’ between the energy levels, such that the frequency f of the photon emitted in a downward transition from the stationary state with quantum number n to i another of lower energy with quantum number n would be f " 2 E E n n ke 1 1 i f f = = : (2.8) 2 2 h 2a h 0 n n f i Einstein used these ideas of Bohr to rederive the black body spectrum result of Planck. In doing so, he set up the theory of emission and absorption of light quanta, including spontaneous (i.e. ‘uncaused’ emission) – the first intimation that there were processes occurring at the atomic level that were intrinsically probabilistic. This work also lead him to the conclusion that the light quanta were more than packets of energy, but carried momentum in a particular direction – the light quanta were, in fact, particles, subsequently named photons by the chemist Gilbert Lewis. There was some success in extracting a general method, now known as the ‘old’ quantum theory, from Bohr’s model of the hydrogen atom. But this theory, while quite successful for the hydrogen atom, was an utter failure when applied to even the next most complex atom, the helium atom. The ad hoc character of the assumptions on which it was based gave little clue to the nature of the underlying physics, nor was it a theory that could describe a dynamical system, i.e. one that was evolving in time. Its role seems to have been one of ‘breaking the ice’, freeing up the attitudes of researchers at that time to old paradigms, and opening up new ways of looking at the physics of the atomic world. De Broglie’s Hypothesis (1924) Inspired by Einstein’s picture of light, a form of wave motion, as also behaving in some circumstances as if it was made up of particles, and inspired also by the success of the Bohr model of the hydrogen atom, de Broglie was lead, by purely aesthetic arguments to make a radical proposal: if light waves can behave under some circumstances like particles, then by symmetry it is reasonable to suppose that particles such as an electron (or a planet?) can behave like waves. More precisely, if light waves of frequency can behave like a collection of particles of energy E = , then by symmetry, a massive particle of energy E, an electron say, should behave under some circumstances like a wave of frequency = E=. But assigning a frequency to these waves is not the end of the story. A wave is also characterised by its wavelength, so it is also necessary to assign a wavelength to these ‘matter waves’. For a particle of light, a photon, the wavelength of the associated wave is = c= f where f = =2. So what is it for a massive particle? A possible formula for this wavelength can be obtained by looking a little further at the case of the photon. In Einstein’s theory of relativity, a photon is recognized as a particle of zero rest mass, and as such the energy of a photon (moving freely in empty space) is related to its momentum p by E = pc. From this it follows that E = = 2c= = pc (2.9) c J D Cresser 2011Chapter 2 The Early History of Quantum Mechanics 12 so that, since = h=2 p = h=: (2.10) This equation then gave the wavelength of the photon in terms of its momentum, but it is also an expression that contains nothing that is specific to a photon. So de Broglie assumed that this relationship applied to all free particles, whether they were photons or electrons or anything else, and so arrived at the pair of equations f = E=h  = h=p (2.11) which gave the frequency and wavelength of the waves that were to be associated with a free particle of kinetic energy E and momentum p. Strictly speaking, the relativistic expressions for the momentum and energy of a particle of non-zero rest mass ought to be used in these formula, as these above formulae were derived by making use of results of special relativity. However, here we will be concerned solely with the non-relativistic limit, and so the non-relativistic expressions, 1 2 1 E = mv and p = mv will suce . 2 This work constituted de Broglie’s PhD thesis. It was a pretty thin a air, a few pages long, and while it was looked upon with some scepticism by the thesis examiners, the power and elegance of his ideas and his results were immediately appreciated by Einstein, more reluctantly by others, and lead ultimately to the discovery of the wave equation by Schrodinger ¨ , and the development of wave mechanics as a theory describing the atomic world. Experimentally, the first evidence of the wave nature of massive particles was seen by Davisson and Germer in 1926 when they fired a beam of electrons of known energy at a nickel crystal in which the nickel atoms are arranged in a regular array. Much to the surprise of the experimenters (who were not looking for any evidence of wave properties of electrons), the electrons reflected o the surface of the crystal to form an interference pattern. The characteristics of this pattern were entirely consistent with the electrons behaving as waves, with a wavelength given by the de Broglie formula, that were reflected by the periodic array of atoms in the crystal (which acted much like slits in a di raction grating). An immediate success of de Broglie’s hypothesis was that it gave an explanation, of sorts, of the quantization condition L = n. If the electron circulating around  the nucleus is associated with a wave of wavelength r , then for the wave not to destructively interfere with itself, there must be a whole number of waves (see Fig. (2.2)) fitting into one circumference of the orbit, i.e. n = 2r: (2.12) Using the de Broglie relation = h=p then gives L = pr = n which is just Bohr’s quantization condition. Figure 2.2: De Broglie wave for which four But now, given that particles can exhibit wave like wavelengths fit into a circle of radius r. properties, the natural question that arises is: what is doing the ‘waving’? Further, as wave motion is usually describable in terms of some kind of wave equation, it is then also natural to ask what the wave equation is for these de Broglie waves. The latter question turned out to be much easier to answer than the first – these waves satisfy the famous Schrodinger ¨ wave equation. But what these waves are is still, largely speaking, an incom- pletely answered question: are they ‘real’ waves, as Schrodinger ¨ believed, in the sense that they represent some kind of physical vibration in the same way as water or sound or light waves, or are 1 For a particle moving in the presence of a spatially varying potential, momentum is not constant so the wavelength of the waves will also be spatially dependent – much like the way the wavelength of light waves varies as the wave moves through a medium with a spatially dependent refractive index. In that case, the de Broglie recipe is insucient, and a more general approach is needed – Schrodinger’ ¨ s equation. c J D Cresser 2011Chapter 2 The Early History of Quantum Mechanics 13 they something more abstract, waves carrying information, as Einstein seemed to be the first to in- timate. The latter is an interpretation that has been gaining in favour in recent times, a perspective that we can support somewhat by looking at what we can learn about a particle by studying the properties of these waves. It is this topic to which we now turn. c J D Cresser 2011Chapter 3 The Wave Function n the basis of the assumption that the de Broglie relations give the frequency and wavelength O of some kind of wave to be associated with a particle, plus the assumption that it makes sense to add together waves of di erent frequencies, it is possible to learn a considerable amount about these waves without actually knowing beforehand what they represent. But studying di er- ent examples does provide some insight into what the ultimate interpretation is, the so-called Born interpretation, which is that these waves are ‘probability waves’ in the sense that the amplitude squared of the waves gives the probability of observing (or detecting, or finding – a number of di erent terms are used) the particle in some region in space. Hand-in-hand with this interpreta- tion is the Heisenberg uncertainty principle which, historically, preceded the formulation of the probability interpretation. From this principle, it is possible to obtain a number of fundamental results even before the full machinery of wave mechanics is in place. In this Chapter, some of the consequences of de Broglie’s hypothesis of associating waves with particles are explored, leading to the concept of the wave function, and its probability interpreta- tion. 3.1 The Harmonic Wave Function On the basis of de Broglie’s hypothesis, there is associated with a particle of energy E and mo- mentum p, a wave of frequency f and wavelength given by the de Broglie relations Eq. (2.11). It is more usual to work in terms of the angular frequency = 2 f and wave number k = 2= so that the de Broglie relations become = E= k = p=: (3.1) With this in mind, and making use of what we already know about what the mathematical form is for a wave, we are in a position to make a reasonable guess at a mathematical expression for the wave associated with the particle. The possibilities include (in one dimension) i(kxt) (x; t) = A sin(kxt); A cos(kxt); Ae ; ::: (3.2) At this stage, we have no idea what the quantity (x; t) represents physically. It is given the name the wave function, and in this particular case we will use the term harmonic wave function to describe any trigonometric wave function of the kind listed above. As we will see later, in general it can take much more complicated forms than a simple single frequency wave, and is almost always a complex valued function. In fact, it turns out that the third possibility listed above is the appropriate wave function to associate with a free particle, but for the present we will work with real wave functions, if only because it gives us the possibility of visualizing their form while discussing their properties. c J D Cresser 2011Chapter 3 The Wave Function 15 In order to gain an understanding of what a wave function might represent, we will turn things around briefly and look at what we can learn about a particle if we know what its wave function is. We are implicitly bypassing here any consideration of whether we can understand a wave function as being a physical wave in the same way that a sound wave, a water wave, or a light wave are physical waves, i.e. waves made of some kind of physical ‘stu ’. Instead, we are going to look on a wave function as something that gives us information on the particle it is associated with. To this end, we will suppose that the particle has a wave function given by (x; t) = A cos(kxt). Then, given that the wave has angular frequency and wave number k, it is straightforward to calculate the wave velocity, that is, the phase velocity v of the wave, which is just the velocity of the wave p crests. This phase velocity is given by 1 2 mv E 2 1 v = = = = = v: (3.3) p 2 k k p mv Thus, given the frequency and wave number of a wave function, we can determine the speed of the particle from the phase velocity of its wave function, v = 2v . We could also try to learn p from the wave function the position of the particle. However, the wave function above tells us nothing about where the particle is to be found in space. We can make this statement because this wave function is more or less the same everywhere. For sure, the wave function is not exactly the same everywhere, but any feature that we might decide as being an indicator of the position of the particle, say where the wave function is a maximum, or zero, will not do: the wave function is periodic, so any feature, such as where the wave function vanishes, reoccurs an infinite number of times, and there is no way to distinguish any one of these repetitions from any other, see Fig. (3.1). (x; t) x Figure 3.1: A wave function of constant amplitude and wavelength. The wave is the same everywhere and so there is no distinguishing feature that could indicate one possible position of the particle from any other. Thus, this particular wave function gives no information on the whereabouts of the particle with which it is associated. So from a harmonic wave function it is possible to learn how fast a particle is moving, but not what the position is of the particle. 3.2 Wave Packets From what was said above, a wave function constant throughout all space cannot give information on the position of the particle. This suggests that a wave function that did not have the same amplitude throughout all space might be a candidate for a giving such information. In fact, since what we mean by a particle is a physical object that is confined to a highly localized region in space, ideally a point, it would be intuitively appealing to be able to devise a wave function that is zero or nearly so everywhere in space except for one localized region. It is in fact possible to construct, from the harmonic wave functions, a wave function which has this property. To show how this is done, we first consider what happens if we combine together two harmonic waves whose wave numbers are very close together. The result is well-known: a ‘beat note’ is produced, i.e. periodically in space the waves add together in phase to produce a local maximum, while c J D Cresser 2011Chapter 3 The Wave Function 16 midway in between the waves will be totally out of phase and hence will destructively interfere. This is illustrated in Fig. 3.2(a) where we have added together two waves cos(5x) + cos(5:25x). (a) (b) (c) (d) Figure 3.2: (a) Beat notes produced by adding together two cos waves: cos(5x) + cos(5:25x). (b) Combining five cos waves: cos(4:75x) + cos(4:875x) + cos(5x) + cos(5:125x) + cos(5:25x). (c) Combining seven cos waves: cos(4:8125x) + cos(4:875x) + cos(4:9375x) + cos(5x) + cos(5:0625x) + cos(5:125x) + cos(5:1875x). (d) An integral over a continuous range of wave numbers produces a single wave packet. Now suppose we add five such waves together, as in Fig. 3.2(b). The result is that some beats turn out to be much stronger than the others. If we repeat this process by adding seven waves together, but now make them closer in wave number, we get Fig. 3.2(c), we find that most of the beat notes tend to become very small, with the strong beat notes occurring increasingly far apart. Mathematically, what we are doing here is taking a limit of a sum, and turning this sum into an integral. In the limit, we find that there is only one beat note – in e ect, all the other beat notes become infinitely far away. This single isolated beat note is usually referred to as a wave packet. We need to look at this in a little more mathematical detail, so suppose we add together a large number of harmonic waves with wave numbers k ; k ; k ;::: all lying in the range: 1 2 3 k k. k . k + k (3.4) n around a value k, i.e. (x; t) =A(k ) cos(k x t) + A(k ) cos(k x t) +::: 1 1 1 2 2 2 X = A(k ) cos(k x t) (3.5) n n n n where A(k) is a function peaked about the value k with a full width at half maximum of 2k. (There is no significance to be attached to the use of cos functions here – the idea is simply to illustrate a c J D Cresser 2011Chapter 3 The Wave Function 17 point. We could equally well have used a sin function or indeed a complex exponential.) What is found is that in the limit in which the sum becomes an integral: Z +1 (x; t) = A(k) cos(kxt) dk (3.6) 1 all the waves interfere constructively to produce only a single beat note as illustrated in Fig. 3.2(d) 1 above . The wave function or wave packet so constructed is found to have essentially zero ampli- tude everywhere except for a single localized region in space, over a region of width 2x, i.e. the wave function (x; t) in this case takes the form of a single wave packet, see Fig. (3.3). 2k 8" A(k) 2x (x; t) 1'-1" k x k (a) (b) Figure 3.3: (a) The distribution of wave numbers k of harmonic waves contributing to the wave function (x; t). This distribution is peaked about k with a width of 2k. (b) The wave packet (x; t) of width 2x resulting from the addition of the waves with distribution A(k). The oscillatory part of the wave packet (the ‘carrier wave’) has wave number k. This wave packet is clearly particle-like in that its region of significant magnitude is confined to a localized region in space. Moreover, this wave packet is constructed out of a group of waves with an average wave number k, and so these waves could be associated in some sense with a particle of momentum p = k. If this were true, then the wave packet would be expected to move with a velocity of p=m. This is in fact found to be the case, as the following calculation shows. Because a wave packet is made up of individual waves which themselves are moving, though not with the same speed, the wave packet itself will move (and spread as well). The speed with which the wave packet moves is given by its group velocity v : g d v = : (3.7) g dk k=k This is the speed of the maximum of the wave packet i.e. it is the speed of the point on the wave packet where all the waves are in phase. Calculating the group velocity requires determining the relationship between to k, known as a dispersion relation. This dispersion relation is obtained from 2 p 1 2 E = mv = : (3.8) 2 2m 1 In Fig. 3.2(d), the wave packet is formed from the integral Z +1 1 2 ((k5)=4) (x; 0) = e cos(kx) dk: p 4  1 c J D Cresser 2011Chapter 3 The Wave Function 18 Substituting in the de Broglie relations Eq. (2.11) gives 2 2 k = (3.9) 2m from which follows the dispersion relation 2 k = : (3.10) 2m The group velocity of the wave packet is then d k v = = : (3.11) g dk m k=k Substituting p = k, this becomes v = p=m. i.e. the packet is indeed moving with the velocity g of a particle of momentum p, as suspected. This is a result of some significance, i.e. we have constructed a wave function of the form of a wave packet which is particle-like in nature. But unfortunately this is done at a cost. We had to combine together harmonic wave functions cos(kx t) with a range of k values 2k to produce a wave packet which has a spread in space of size 2x. The two ranges of k and x are not unrelated – their connection is embodied in an important result known as the Heisenberg Uncertainty Relation. 3.3 The Heisenberg Uncertainty Relation The wave packet constructed in the previous section obviously has properties that are reminiscent of a particle, but it is not entirely particle-like — the wave function is non-zero over a region in space of size 2x. In the absence of any better way of relating the wave function to the position of the atom, it is intuitively appealing to suppose that where (x; t) has its greatest amplitude is where the particle is most likely to be found, i.e. the particle is to be found somewhere in a region of size 2x. More than that, however, we have seen that to construct this wavepacket, harmonic waves having k values in the range (k k; k + k) were adding together. These ranges x and k are related by the bandwidth theorem, which applies when adding together harmonic waves, which tell us that xk& 1: (3.12) Using p = k, we have p = k so that xp& : (3.13) A closer look at this result is warranted. A wave packet that has a significant amplitude within a region of size 2x was constructed from harmonic wave functions which represent a range of momenta p p to p + p. We can say then say that the particle is likely to be found somewhere in the region 2x, and given that wave functions representing a range of possible momenta were used to form this wave packet, we could also say that the momentum of the particle will have 2 a value in the range p p to p + p . The quantities x and p are known as uncertainties, and the relation above Eq. (3.14) is known as the Heisenberg uncertainty relation for position and momentum. All this is rather abstract. We do not actually ‘see’ a wave function accompanying its particle, so how are we to know how ‘wide’ the wave packet is, and hence what the uncertainty in position and momentum might be for a given particle, say an electron orbiting in an atomic nucleus, or the 2 In fact, we can look on A(k) as a wave function for k or, since k = p= as e ectively a wave function for momentum analogous to (x; t) being a wave function for position. c J D Cresser 2011Chapter 3 The Wave Function 19 nucleus itself, or an electron in a metal or . . . ? The answer to this question is intimately linked with what has been suggested by the use above of such phrases as ‘where the particle is most likely to be found’ and so on, words that are hinting at the fundamental role of randomness as an intrinsic property of quantum systems, and role of probability in providing a meaning for the wave function. To get a flavour of what is meant here, we can suppose that we have a truly vast number of identical 25 particles, say 10 , all prepared one at a time in some experiment so that they all have associated with them the same wave packet. For half these particles, we measure their position at the same time, i.e. at, say, 10 sec after they emerge from the apparatus, and for the other half we measure their momentum. What we find is that the results for the position are not all the same: they are spread out randomly around some average value, and the range over which they are spread is most conveniently measured by the usual tool of statistics: the standard deviation. This standard deviation in position turns out to be just the uncertainty x we introduced above in a non-rigorous manner. Similarly, the results for the measurement of momentum for the other half are randomly scattered around some average value, and the spread around the average is given by the standard deviation once again. This standard deviation in momentum we identify with the uncertainty p introduced above. With uncertainties defined as standard deviations of random results, it is possible to give a more precise statement of the uncertainty relation, which is: 1 xp (3.14) 2 but we will mostly use the result Eq. (3.13). The detailed analysis is left to much later (See Chapter 1). The Heisenberg relation has an immediate interpretation. It tells us that we cannot determine, from knowledge of the wave function alone, the exact position and momentum of a particle at the same time. In the extreme case that x = 0, then the position uncertainty is zero, but Eq. (3.14) tells us that the uncertainty on the momentum is infinite, i.e. the momentum is entirely unknown. A similar statement applies if p = 0. In fact, this last possibility is the case for the example of a single harmonic wave function considered in Section 3.1. However, the uncertainty relation does not say that we cannot measure the position and the momentum at the same time. We certainly can, but we have to live with the fact that each time we repeat this simultaneous measurement of position and momentum on a collection of electrons all prepared such as to be associated with the same wave packet, the results that are obtained will vary randomly from one measurement to the next, i.e. the measurement results continue to carry with them uncertainty by virtue of the uncertainty relation. This conclusion that it is impossible for a particle to have zero uncertainty in both position and momentum at the same time flies in the face of our intuition, namely our belief that a particle moving through space will at any instant have a definite position and momentum which we can, in principle, measure to arbitrary accuracy. We could then feel quite justified in arguing that our wave function idea is all very interesting, but that it is not a valid description of the physical world, or perhaps it is a perfectly fine concept but that it is incomplete, that there is information missing from the wave function. Perhaps there is a prescription still to be found that will enable us to complete the picture: retain the wave function but add something further that will then not forbid our being able to measure the position and the momentum of the particle precisely and at the same time. This, of course, amounts to saying that the wave function by itself does not give complete information on the state of the particle. Einstein fought vigorously for this position i.e. that the wave function was not a complete description of ‘reality’, and that there was somewhere, in some sense, a repository of missing information that will remove the incompleteness of the wave function — so-called ‘hidden variables’. Unfortunately (for those who hold to his point c J D Cresser 2011Chapter 3 The Wave Function 20 of view) evidence has mounted, particularly in the past few decades, that the wave function (or its analogues in the more general formulation of quantum mechanics) does indeed represent the full picture — the most that can ever be known about a particle (or more generally any system) is what can be learned from its wave function. This means that the diculty encountered above concerning not being able to pinpoint the position and the momentum of a particle from knowledge of its wave function is not a reflection of any inadequacy on the part of experimentalists trying to measure these quantities, but is an irreducible property of the natural world. Nevertheless, at the macroscopic level the uncertainties mentioned above become so small as to be experimentally unmeasurable, so at this level the uncertainty relation has no apparent e ect. The limitations implied by the uncertainty relation as compared to classical physics may give the impression that something has been lost, that nature has prevented us, to an extent quantified by the uncertainty principle, from having complete information about the physical world. To someone wedded to the classical deterministic view of the the physical world (and Einstein would have to be counted as one such person), it appears to be the case that there is information that is hidden from us. This may then be seen as a cause for concern because it implies that we cannot, even in principle, make exact predictions about the behaviour of any physical system. However, the view can be taken that the opposite is true, that the uncertainty principle is an indicator of greater freedom. In a sense, the uncertainty relation means it is possible for a physical system to have a much broader range of possible physical properties consistent with the smaller amount of information that is available about its properties. This leads to a greater richness in the properties of the physical world than could ever be found within classical physics. 3.3.1 The Heisenberg microscope: the effect of measurement The Heisenberg Uncertainty Relation is enormously general. It applies without saying anything whatsoever about the nature of the particle, how it is prepared in an experiment, what it is doing, what it might be interacting with . . . . It is clearly a profoundly significant physical result. But at its heart it is simply a mathematical statement about the properties of waves that flows from the assumed wave properties of matter plus some assumptions about the physical interpretation of these waves. There is little indication of what the physics might be that underlies it. One way to uncover what physics might be present is to study what takes place if we attempt to measure the position or the momentum of a particle. This is in fact the problem initially addressed by Heisenberg, and leads to a result that is superficially the same as 3.14, but, from a physics point of view, there is in fact a subtle di erence between what Heisenberg was doing, and what Eq. (3.14) is saying. Heisenberg’s analysis was based on a thought experiment, i.e. an experiment that was not actually performed, but was instead analysed as a mental construct. From this experiment, it is possible to show, by taking account of the quantum nature of light and matter, that measuring the position of an electron results in an unavoidable, unpredictable change in its momentum. More than that, it is possible to show that if the particle’s position were measured with ever increasing precision, the result was an ever greater disturbance of the particle’s momentum. This is an outcome that is summarized mathematically by a formula essentially the same as Eq. (3.14). In his thought experiment, Heisenberg considered what was involved in attempting to measure the position of a particle, an electron say, by shining light on the electron and observing the scattered light through a microscope. To analyse this measurement process, arguments are used which are a curious mixture of ideas from classical optics (the wave theory of light) and from the quantum theory of light (that light is made up of particles). c J D Cresser 2011

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