Quantumphysicsnotes, Thesis of Quantum Mechanics

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Quantum Physics Notes

J D Cresser Department of Physics Macquarie University

31 st^ August 2011

Preface

T

he world of our every-day experiences – the world of the not too big (compared to, say, a galaxy), and the not too small, (compared to something the size and mass of an atom), and where nothing moves too fast (compared to the speed of light) – is the world that is mostly directly accessible to our senses. This is the world usually more than adequately described by the theories of classical physics that dominated the nineteenth century: Newton’s laws of motion, including his law of gravitation, Maxwell’s equations for the electromagnetic field, and the three laws of thermodynamics. These classical theories are characterized by, amongst other things, the notion that there is a ‘real’ world out there, one that has an existence independent of ourselves, in which, for instance, objects have a definite position and momentum which we could measure to any degree of accuracy, limited only by our experimental ingenuity. According to this view, the universe is evolving in a way completely determined by these classical laws, so that if it were possible to measure the positions and momenta of all the constituent particles of the universe, and we knew all the forces that acted between the particles, then we could in principle predict to what ever degree of accuracy we desire, exactly how the universe (including ourselves) will evolve. Everything is predetermined – there is no such thing as free will, there is no room for chance. Anything apparently random only appears that way because of our ignorance of all the information that we would need to have to be able to make precise predictions.

This rather gloomy view of the nature of our world did not survive long into the twentieth century. It was the beginning of that century which saw the formulation of, not so much a new physical theory, but a new set of fundamental principles that provides a framework into which all physical theories must fit: quantum mechanics. To a greater or lesser extent all natural phenomena appear to be governed by the principles of quantum mechanics, so much so that this theory constitutes what is undoubtedly the most successful theory of modern physics. One of the crucial consequences of quantum mechanics was the realization that the world view implied by classical physics, as outlined above, was no longer tenable. Irreducible randomness was built into the laws of nature. The world is inherently probabilistic in that events can happen without a cause, a fact first stumbled on by Einstein, but never fully accepted by him. But more than that, quantum mechanics admits the possibility of an interconnectedness or an ‘entanglement’ between physical systems, even those possibly separated by vast distances, that has no analogue in classical physics, and which plays havoc with our strongly held presumptions that there is an objectively real world ‘out there’.

Quantum mechanics is often thought of as being the physics of the very small as seen through its successes in describing the structure and properties of atoms and molecules – the chemical properties of matter – the structure of atomic nuclei and the properties of elementary particles. But this is true only insofar as the fact that peculiarly quantum effects are most readily observed at the atomic level. In the everyday world that we usually experience, where the classical laws of Newton and Maxwell seem to be able to explain so much, it quickly becomes apparent that classical theory is unable to explain many things e.g. why a solid is ‘solid’, or why a hot object has the colour that it does. Beyond that, quantum mechanics is needed to explain radioactivity, how semiconducting devices – the backbone of modern high technology – work, the origin of superconductivity, what makes a laser do what it does.... Even on the very large scale, quantum effects leave their mark

Foreword iii

or Newton’s description of the mechanical properties of matter, or Einstein’s general relativistic theory of gravity, or any other conceivable theory, must be constructed in a way that respects the edicts of quantum mechanics. This is clearly a very general task, and as such it is clear that quantum mechanics must refer to some deeply fundamental, common feature of all these theories. This common feature is the information that can be known about the physical state of a physical system. Of course, the theories of classical physics are built on the information gained about the physical world, but the difference here is that quantum mechanics provides a set of rules regarding the information that can be gained about the state of any physical system and how this information can be processed, that are quite distinct from those implicit in classical physics. These rules tell us, amongst other things, that it is possible to have exact information about some physical properties of a system, but everything else is subject to the laws of probability.

To describe the quantum properties of any physical system, a new mathematical language is re- quired as compared to that of classical mechanics. At its heart quantum mechanics is a mathemat- ically abstract subject expressed in terms of the language of complex linear vector spaces — in other words, linear algebra. In fact, it was in this form that quantum mechanics was first worked out, by Werner Heisenberg, in the 1920s who showed how to represent the physically observable properties of systems in terms of matrices. But not long after, a second version of quantum me- chanics appeared, that due to Erwin Schr¨odinger. Instead of being expressed in terms of matrices and vectors, it was written down in the terms of waves propagating through space and time (at least for a single particle system). These waves were represented by the so-called wave function Ψ(x, t), and the equation that determined the wave function in any given circumstance was known as the Schr¨odinger equation.

This version of the quantum theory was, and still is, called ‘wave mechanics’. It is fully equivalent to Heisenberg’s version, but because it is expressed in terms of the then more familiar mathematical language of functions and wave equations, and as it was usually far easier to solve Schr¨odinger’s equation than it was to work with (and understand) Heisenberg’s version, it rapidly became ‘the way’ of doing quantum mechanics, and stayed that way for most of the rest of the 20th century. Its most usual application, built around the wave function Ψ and the interpretation of |Ψ|^2 as giving the probability of finding a particle in some region in space, is to describing the structure of matter at the atomic level where the positions of the particles is important, such as in the distribution in space of electrons and nuclei in atomic, molecular and solid state physics. But quantum mechanics is much more than the mechanics of the wave function, and its applicability goes way beyond atomic, molecular or solid state theory. There is an underlying, more general theory of which wave mechanics is but one mathematical manifestation or representation. In a sense wave mechanics is one step removed from this deeper theory in that the latter highlights the informational interpretation of quantum mechanics. The language of this more general theory is the language of vector spaces, of state vectors and linear superpositions of states, of Hermitean operators and observables, of eigenvalues and eigenvectors, of time evolution operators, and so on. As the subject has matured in the latter decades of the 20th century and into the 21st century, and with the development of the ‘quantum information’ interpretation of quantum mechanics, more and more the tendency is to move away from wave mechanics to the more abstract linear algebra version, chiefly expressed in the notation due to Dirac. It is this more general view of quantum mechanics that is presented in these notes.

The starting point is a look at what distinguishes quantum mechanics from classical mechanics, followed by a quick review of the history of quantum mechanics, with the aim of summarizing the essence of the wave mechanical point of view. A study is then made of the one experiment that is supposed to embody all of the mystery of quantum mechanics – the double slit interference experiment. A closer analysis of this experiment also leads to the introduction of a new notation – the Dirac notation – along with a new interpretation in terms of vectors in a Hilbert space. Subse- quently, working with this general way of presenting quantum mechanics, the physical content of

Foreword iv

the theory is developed.

The overall approach adopted here is one of inductive reasoning, that is the subject is developed by a process of trying to see what might work, or what meaning might be given to a certain mathematical or physical result or observation, and then testing the proposal against the scientific evidence. The procedure is not a totally logical one, but the result is a logical edifice that is only logical after the fact, i.e. the justification of what is proposed is based purely on its ability to agree with what is known about the physical world.

• 1 Introduction Preface i
  • 1.1 Classical Physics
    • 1.1.1 Classical Randomness and Ignorance of Information
  • 1.2 Quantum Physics
  • 1.3 Observation, Information and the Theories of Physics
• 2 The Early History of Quantum Mechanics
• 3 The Wave Function
  • 3.1 The Harmonic Wave Function
  • 3.2 Wave Packets
  • 3.3 The Heisenberg Uncertainty Relation
    • 3.3.1 The Heisenberg microscope: the effect of measurement
    • 3.3.2 The Size of an Atom
    • 3.3.3 The Minimum Energy of a Simple Harmonic Oscillator
• 4 The Two Slit Experiment
  • 4.1 An Experiment with Bullets
  • 4.2 An Experiment with Waves
  • 4.3 An Experiment with Electrons
    • 4.3.1 Monitoring the slits: the Feynman microscope
    • 4.3.2 The Role of Information: The Quantum Eraser
    • 4.3.3 Wave-particle duality
  • 4.4 Probability Amplitudes
  • 4.5 The Fundamental Nature of Quantum Probability
• 5 Wave Mechanics CONTENTS vi
  • 5.1 The Probability Interpretation of the Wave Function
    • 5.1.1 Normalization
  • 5.2 Expectation Values and Uncertainties
  • 5.3 Particle in an Infinite Potential Well
    • 5.3.1 Some Properties of Infinite Well Wave Functions
  • 5.4 The Schr ¨odinger Wave Equation
    • 5.4.1 The Time Dependent Schr ¨odinger Wave Equation
    • 5.4.2 The Time Independent Schr ¨odinger Equation
    • 5.4.3 Boundary Conditions and the Quantization of Energy
    • 5.4.4 Continuity Conditions
    • 5.4.5 Bound States and Scattering States
  • 5.5 Solving the Time Independent Schr ¨odinger Equation
    • 5.5.1 The Infinite Potential Well Revisited
    • 5.5.2 The Finite Potential Well
    • 5.5.3 Scattering from a Potential Barrier
  • 5.6 Expectation Value of Momentum
  • 5.7 Is the wave function all that is needed?
• 6 Particle Spin and the Stern-Gerlach Experiment
  • 6.1 Classical Spin Angular Momentum
  • 6.2 Quantum Spin Angular Momentum
  • 6.3 The Stern-Gerlach Experiment
  • 6.4 Quantum Properties of Spin
    • 6.4.1 Spin Preparation and Measurement
    • 6.4.2 Repeated spin measurements
    • 6.4.3 Quantum randomness
    • 6.4.4 Probabilities for Spin
  • 6.5 Quantum Interference for Spin
• 7 Probability Amplitudes
  • 7.1 The State of a System
    • 7.1.1 Limits to knowledge
  • 7.2 The Two Slit Experiment Revisited
    • 7.2.1 Sum of Amplitudes in Bra(c)ket Notation
    • 7.2.2 Superposition of States for Two Slit Experiment CONTENTS vii
  • 7.3 The Stern-Gerlach Experiment Revisited
    • 7.3.1 Probability amplitudes for particle spin
    • 7.3.2 Superposition of States for Spin Half
    • 7.3.3 A derivation of sum-over-probability-amplitudes for spin half
  • 7.4 The General Case of Many Intermediate States
  • 7.5 Probabilities vs probability amplitudes
• 8 Vector Spaces in Quantum Mechanics
  • 8.1 Vectors in Two Dimensional Space
    • 8.1.1 Linear Combinations of Vectors – Vector Addition
    • 8.1.2 Inner or Scalar Products
  • 8.2 Generalization to higher dimensions and complex vectors
  • 8.3 Spin Half Quantum States as Vectors
    • 8.3.1 The Normalization Condition
    • 8.3.2 The General Spin Half State
    • 8.3.3 Is every linear combination a state of the system?
  • 8.4 Constructing State Spaces
    • 8.4.1 A General Formulation
    • 8.4.2 Further Examples of State Spaces
    • 8.4.3 States with multiple labels
  • 8.5 States of Macroscopic Systems — the role of decoherence
• 9 General Mathematical Description of a Quantum System
  • 9.1 State Space
  • 9.2 Probability Amplitudes and the Inner Product of State Vectors
    • 9.2.1 Bra Vectors
• 10 State Spaces of Infinite Dimension
  • 10.1 Examples of state spaces of infinite dimension
  • 10.2 Some Mathematical Issues
    • 10.2.1 States of Infinite Norm
    • 10.2.2 Continuous Basis States
    • 10.2.3 The Dirac Delta Function
    • 10.2.4 Separable State Spaces
• 11 Operations on States CONTENTS viii
  • 11.1 Definition and Properties of Operators
    • 11.1.1 Definition of an Operator
    • 11.1.2 Linear and Antilinear Operators
    • 11.1.3 Properties of Operators
  • 11.2 Action of Operators on Bra Vectors
  • 11.3 The Hermitean Adjoint of an Operator
    • 11.3.1 Hermitean and Unitary Operators
  • 11.4 Eigenvalues and Eigenvectors
    • 11.4.1 Eigenkets and Eigenbras
    • 11.4.2 Eigenstates and Eigenvalues of Hermitean Operators
    • 11.4.3 Continuous Eigenvalues
  • 11.5 Dirac Notation for Operators
• 12 Matrix Representations of State Vectors and Operators
  • 12.1 Representation of Vectors In Euclidean Space as Column and Row Vectors
    • 12.1.1 Column Vectors
    • 12.1.2 Row Vectors
  • 12.2 Representations of State Vectors and Operators
    • 12.2.1 Row and Column Vector Representations for Spin Half State Vectors
    • 12.2.2 Representation of Ket and Bra Vectors
    • 12.2.3 Representation of Operators
    • 12.2.4 Properties of Matrix Representations of Operators
    • 12.2.5 Eigenvectors and Eigenvalues
    • 12.2.6 Hermitean Operators
• 13 Observables and Measurements in Quantum Mechanics
  • 13.1 Measurements in Quantum Mechanics
  • 13.2 Observables and Hermitean Operators
  • 13.3 Observables with Discrete Values
    • 13.3.1 The Von Neumann Measurement Postulate
  • 13.4 The Collapse of the State Vector
    • 13.4.1 Sequences of measurements
  • 13.5 Examples of Discrete Valued Observables
    • 13.5.1 Position of a particle (in one dimension)
    • 13.5.2 Momentum of a particle (in one dimension) CONTENTS ix
    • 13.5.3 Energy of a Particle (in one dimension)
    • 13.5.4 The O− 2 Ion: An Example of a Two-State System
    • 13.5.5 Observables for a Single Mode EM Field
  • 13.6 Observables with Continuous Values
    • 13.6.1 Measurement of Particle Position
    • 13.6.2 General Postulates for Continuous Valued Observables
  • 13.7 Examples of Continuous Valued Observables
    • 13.7.1 Position and momentum of a particle (in one dimension)
    • 13.7.2 Field operators for a single mode cavity
• 14 Probability, Expectation Value and Uncertainty
  • 14.1 Observables with Discrete Values
    • 14.1.1 Probability
    • 14.1.2 Expectation Value
    • 14.1.3 Uncertainty
  • 14.2 Observables with Continuous Values
    • 14.2.1 Probability
    • 14.2.2 Expectation Values
    • 14.2.3 Uncertainty
  • 14.3 The Heisenberg Uncertainty Relation
  • 14.4 Compatible and Incompatible Observables
    • 14.4.1 Degeneracy
• 15 Time Evolution in Quantum Mechanics
  • 15.1 Stationary States
  • 15.2 The Schr ¨odinger Equation – a ‘Derivation’.
    • 15.2.1 Solving the Schr ¨odinger equation: An illustrative example
    • 15.2.2 The physical interpretation of the O− 2 Hamiltonian
  • 15.3 The Time Evolution Operator
• 16 Displacements in Space
• 17 Rotations in Space
• 18 Symmetry and Conservation Laws

Chapter 1

Introduction

T

here are three fundamental theories on which modern physics is built: the theory of relativity, 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 different 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 affect the subject of this observation in an uncontrollably random way (even if no physical contact is made with the object under observation!).

Chapter 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 efforts. 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 room temperature, there are something like 10^26 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 difficult to get at the information.

Figure 1.2: Random walk of a pollen grain suspended in a liquid.

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 every time, and hence the outcome of the 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 to one end of the bottle is something like 1 in 10^10 26

• 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

Chapter 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 in size, the velocity of the electron can be measured to vary by at least ±50 ms−^1. 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

Chapter 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 off with a right shoe.

Similar experiments can be undertaken with atomic particles, though it is the spins of pairs of particles that are paired off: 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

Chapter 1 Introduction 7

best summarized as equations:

F = ma — Newton’s second law;

∇ × E = −

∂B

∂t — One of Maxwell’s equations (Faraday’s law); 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 measure the position x of a particle with an accuracy^1 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

∆x∆p ≥ 12 ℏ — the Heisenberg Uncertainty Relation

Quantum mechanics also tells us how this information is processed e.g. as a system evolves in time (the Schr¨odinger 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.

Chapter 2

The Early History of Quantum Mechanics

I

n the early years of the twentieth century, Max Planck, Albert Einstein, Louis de Broglie, Neils Bohr, Werner Heisenberg, Erwin Schr¨odinger, 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: S( f, T ) = α f 3 e−β^ f^ /T^ (2.1)

f

S( f, T ) Rayleigh-Jeans

Planck

Wein

6

Figure 2.1: Rayleigh-Jeans (classical), Wein, and Planck spectral distributions.

The quantity S( f, T ), otherwise known as the spec- tral distribution function, is such that S( f, T )d f is the energy contained in unit volume of electromagnetic 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 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 measure S at lower (infrared) frequencies. There was another candidate for S which was derived using arguments from classical physics which lead to a formula for S( f, T ) known as the Rayleigh-Jeans formula:

S( f, T ) =

8 π f 2 c^3 kBT (2.2)

where kB is a constant known as Boltzmann’s constant. This formula worked well at low fre- quencies, but suffered 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

Chapter 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:

S( f, T ) = 8 πh f 3 c^3

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 h, that is, h = 6. 6218 × 10 −^34 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 π arose time and again. As a consequence, Dirac introduced a new quantity ℏ = h/ 2 π, pronounced ‘h-bar’, which is now the constant most commonly encountered. In terms of ℏ, Planck’s formula for the quantum of energy becomes E = h f = (h/ 2 π) 2π f = ℏω (2.4)

where ω 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 effect, he proposed that light of frequency ω was made up of quanta or ‘packets’ of energy ℏω 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 the form of electromagnetic waves, and do so on a time scale of ∼ 10 −^12 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 = nℏ (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