Transcription
Quantum Physics NotesJ D CresserDepartment of PhysicsMacquarie University31st August 2011
Prefacehe world of our every-day experiences – the world of the not too big (compared to, say, agalaxy), and the not too small, (compared to something the size and mass of an atom), andwhere nothing moves too fast (compared to the speed of light) – is the world that is mostly directlyaccessible to our senses. This is the world usually more than adequately described by the theoriesof classical physics that dominated the nineteenth century: Newton’s laws of motion, includinghis law of gravitation, Maxwell’s equations for the electromagnetic field, and the three laws ofthermodynamics. These classical theories are characterized by, amongst other things, the notionthat 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 degreeof accuracy, limited only by our experimental ingenuity. According to this view, the universe isevolving in a way completely determined by these classical laws, so that if it were possible tomeasure the positions and momenta of all the constituent particles of the universe, and we knew allthe forces that acted between the particles, then we could in principle predict to what ever degreeof accuracy we desire, exactly how the universe (including ourselves) will evolve. Everythingis predetermined – there is no such thing as free will, there is no room for chance. Anythingapparently random only appears that way because of our ignorance of all the information that wewould need to have to be able to make precise predictions.TThis 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 physicaltheory, but a new set of fundamental principles that provides a framework into which all physicaltheories must fit: quantum mechanics. To a greater or lesser extent all natural phenomena appear tobe governed by the principles of quantum mechanics, so much so that this theory constitutes whatis undoubtedly the most successful theory of modern physics. One of the crucial consequencesof quantum mechanics was the realization that the world view implied by classical physics, asoutlined 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 stumbledon by Einstein, but never fully accepted by him. But more than that, quantum mechanics admits thepossibility of an interconnectedness or an ‘entanglement’ between physical systems, even thosepossibly separated by vast distances, that has no analogue in classical physics, and which playshavoc 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 throughits successes in describing the structure and properties of atoms and molecules – the chemicalproperties of matter – the structure of atomic nuclei and the properties of elementary particles. Butthis is true only insofar as the fact that peculiarly quantum effects are most readily observed at theatomic level. In the everyday world that we usually experience, where the classical laws of Newtonand Maxwell seem to be able to explain so much, it quickly becomes apparent that classical theoryis unable to explain many things e.g. why a solid is ‘solid’, or why a hot object has the colour thatit does. Beyond that, quantum mechanics is needed to explain radioactivity, how semiconductingdevices – the backbone of modern high technology – work, the origin of superconductivity, whatmakes a laser do what it does . . . . Even on the very large scale, quantum effects leave their markc J D Cresser 2011
Forewordiiin unexpected ways: the galaxies spread throughout the universe are believed to be macroscopicmanifestations of microscopic quantum-induced inhomogeneities present shortly after the birthof the universe, when the universe itself was tinier than an atomic nucleus and almost whollyquantum mechanical. Indeed, the marriage of quantum mechanics – the physics of the very small– with general relativity – the physics of the very large – is believed by some to be the crucial stepin formulating a general ‘theory of everything’ that will hopefully contain all the basic laws ofnature in one package.The impact of quantum mechanics on our view of the world and the natural laws that govern it,cannot be underestimated. But the subject is not entirely esoteric. Its consequences have beenexploited in many ways that have an immediate impact on the quality of our lives. The economicimpact of quantum mechanics cannot be ignored: it has been estimated that about 30% of the grossnational product of the United States is based on inventions made possible by quantum mechanics.If anyone aims to have anything like a broad understanding of the sciences that underpin moderntechnology, as well as obtaining some insight into the modern view of the character of the physicalworld, then some knowledge and understanding of quantum mechanics is essential. In the broadercommunity, the peculiar things that quantum mechanics says about the way the world works hasmeant that general interest books on quantum mechanics and related subjects continue to popularwith laypersons. This is clear evidence that the community at large and not just the scientific andtechnological community are very interested in what quantum mechanics has to say. Note thateven the term ‘quantum’ has entered the vernacular – it is the name of a car, a market researchcompany, and a dishwasher amongst other things!! The phrase ‘quantum jump’ or ‘quantum leap’is now in common usage, and incorrectly too: a quantum jump is usually understood to representa substantial change whereas a quantum jump in its physics context is usually something that isvery small.The successes of quantum mechanics have been extraordinary. Following the principles of quantum mechanics, it is possible to provide an explanation of everything from the state of the universeimmediately after the big bang, to the structure of DNA, to the colour of your socks. Yet for allof that, and in spite of the fact that the theory is now roughly 100 years old, if Planck’s theory ofblack body radiation is taken as being the birth of quantum mechanics, it as true now as it wasthen that no one truly understands the theory, though in recent times, a greater awareness has developed of what quantum mechanics is all about: as well as being a physical theory, it is also atheory of information, that is, it is a theory concerning what information we can gain about theworld about us – nature places limitations on what we can ‘know’ about the physical world, butit also gives us greater freedoms concerning what we can do with this ‘quantum information’ (ascompared to what we could expect classically), as realized by recent developments in quantumcomputation, quantum teleportation, quantum cryptography and so on. For instance, hundreds ofmillions of dollars are being invested world-wide on research into quantum computing. Amongstother things, if quantum computing ever becomes realizable, then all security protocols used bybanks, defense, and businesses can be cracked on the time scale on the order of months, or maybea few years, a task that would take a modern classical computer 1010 years to achieve! On theother hand, quantum cryptography, an already functioning technology, offers us perfect security.It presents a means by which it is always possible to know if there is an eavesdropper listeningin on what is supposed to be a secure communication channel. But even if the goal of building aquantum computer is never reached, trying to achieve it has meant an explosion in our understanding of the quantum information aspects of quantum mechanics, and which may perhaps one dayfinally lead us to a full understanding of quantum mechanics itself.The Language of Quantum MechanicsAs mentioned above, quantum mechanics provides a framework into which all physical theoriesmust fit. Thus any of the theories of physics, such as Maxwell’s theory of the electromagnetic field,c J D Cresser 2011
Forewordiiior Newton’s description of the mechanical properties of matter, or Einstein’s general relativistictheory of gravity, or any other conceivable theory, must be constructed in a way that respectsthe edicts of quantum mechanics. This is clearly a very general task, and as such it is clear thatquantum 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 physicalsystem. Of course, the theories of classical physics are built on the information gained about thephysical world, but the difference here is that quantum mechanics provides a set of rules regardingthe information that can be gained about the state of any physical system and how this informationcan 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 propertiesof 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 required as compared to that of classical mechanics. At its heart quantum mechanics is a mathematically abstract subject expressed in terms of the language of complex linear vector spaces — inother words, linear algebra. In fact, it was in this form that quantum mechanics was first workedout, by Werner Heisenberg, in the 1920s who showed how to represent the physically observableproperties of systems in terms of matrices. But not long after, a second version of quantum mechanics appeared, that due to Erwin Schrödinger. Instead of being expressed in terms of matricesand vectors, it was written down in the terms of waves propagating through space and time (atleast for a single particle system). These waves were represented by the so-called wave functionΨ(x, t), and the equation that determined the wave function in any given circumstance was knownas the Schrödinger equation.This version of the quantum theory was, and still is, called ‘wave mechanics’. It is fully equivalentto Heisenberg’s version, but because it is expressed in terms of the then more familiar mathematicallanguage of functions and wave equations, and as it was usually far easier to solve Schrödinger’sequation than it was to work with (and understand) Heisenberg’s version, it rapidly became ‘theway’ 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 asgiving the probability of finding a particle in some region in space, is to describing the structureof matter at the atomic level where the positions of the particles is important, such as in thedistribution in space of electrons and nuclei in atomic, molecular and solid state physics. Butquantum mechanics is much more than the mechanics of the wave function, and its applicabilitygoes way beyond atomic, molecular or solid state theory. There is an underlying, more generaltheory of which wave mechanics is but one mathematical manifestation or representation. In asense wave mechanics is one step removed from this deeper theory in that the latter highlights theinformational interpretation of quantum mechanics. The language of this more general theory isthe language of vector spaces, of state vectors and linear superpositions of states, of Hermiteanoperators 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, andwith the development of the ‘quantum information’ interpretation of quantum mechanics, moreand more the tendency is to move away from wave mechanics to the more abstract linear algebraversion, chiefly expressed in the notation due to Dirac. It is this more general view of quantummechanics 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 summarizingthe essence of the wave mechanical point of view. A study is then made of the one experimentthat is supposed to embody all of the mystery of quantum mechanics – the double slit interferenceexperiment. 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. Subsequently, working with this general way of presenting quantum mechanics, the physical content ofc J D Cresser 2011
Forewordivthe theory is developed.The overall approach adopted here is one of inductive reasoning, that is the subject is developedby a process of trying to see what might work, or what meaning might be given to a certainmathematical or physical result or observation, and then testing the proposal against the scientificevidence. The procedure is not a totally logical one, but the result is a logical edifice that is onlylogical after the fact, i.e. the justification of what is proposed is based purely on its ability to agreewith what is known about the physical world.c J D Cresser 2011
ContentsPrefacei1 Introduction11.1 Classical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21.1.1 Classical Randomness and Ignorance of Information . . . . . . . . . . .31.2 Quantum Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.3 Observation, Information and the Theories of Physics62 The Early History of Quantum Mechanics3 The Wave Function. . . . . . . . . . . . . .9143.1 The Harmonic Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 The Heisenberg Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . . 183.3.1 The Heisenberg microscope: the effect of measurement . . . . . . . . . 203.3.2 The Size of an Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.3 The Minimum Energy of a Simple Harmonic Oscillator . . . . . . . . . . 254 The Two Slit Experiment274.1 An Experiment with Bullets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 An Experiment with Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 An Experiment with Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3.1 Monitoring the slits: the Feynman microscope . . . . . . . . . . . . . . . 324.3.2 The Role of Information: The Quantum Eraser. . . . . . . . . . . . . . 334.3.3 Wave-particle duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Probability Amplitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5 The Fundamental Nature of Quantum Probability . . . . . . . . . . . . . . . . . 37c J D Cresser 2011
CONTENTSvi5 Wave Mechanics385.1 The Probability Interpretation of the Wave Function . . . . . . . . . . . . . . . . 385.1.1 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Expectation Values and Uncertainties . . . . . . . . . . . . . . . . . . . . . . . 435.3 Particle in an Infinite Potential Well. . . . . . . . . . . . . . . . . . . . . . . . 455.3.1 Some Properties of Infinite Well Wave Functions . . . . . . . . . . . . . 485.4 The Schrödinger Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 545.4.1 The Time Dependent Schrödinger Wave Equation . . . . . . . . . . . . 545.4.2 The Time Independent Schrödinger Equation . . . . . . . . . . . . . . . 555.4.3 Boundary Conditions and the Quantization of Energy . . . . . . . . . . . 565.4.4 Continuity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4.5 Bound States and Scattering States . . . . . . . . . . . . . . . . . . . . 585.5 Solving the Time Independent Schrödinger Equation . . . . . . . . . . . . . . . 585.5.1 The Infinite Potential Well Revisited . . . . . . . . . . . . . . . . . . . . 585.5.2 The Finite Potential Well. . . . . . . . . . . . . . . . . . . . . . . . . . 615.5.3 Scattering from a Potential Barrier . . . . . . . . . . . . . . . . . . . . . 665.6 Expectation Value of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 705.7 Is the wave function all that is needed?. . . . . . . . . . . . . . . . . . . . . . 726 Particle Spin and the Stern-Gerlach Experiment736.1 Classical Spin Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Quantum Spin Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 746.3 The Stern-Gerlach Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.4 Quantum Properties of Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.4.1 Spin Preparation and Measurement . . . . . . . . . . . . . . . . . . . . 786.4.2 Repeated spin measurements . . . . . . . . . . . . . . . . . . . . . . . 786.4.3 Quantum randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.4.4 Probabilities for Spin. . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.5 Quantum Interference for Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 Probability Amplitudes7.1 The State of a System89. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897.1.1 Limits to knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927.2 The Two Slit Experiment Revisited . . . . . . . . . . . . . . . . . . . . . . . . . 927.2.1 Sum of Amplitudes in Bra(c)ket Notation . . . . . . . . . . . . . . . . . . 93c J D Cresser 2011
viiCONTENTS7.2.2 Superposition of States for Two Slit Experiment . . . . . . . . . . . . . . 947.3 The Stern-Gerlach Experiment Revisited. . . . . . . . . . . . . . . . . . . . . 957.3.1 Probability amplitudes for particle spin . . . . . . . . . . . . . . . . . . . 957.3.2 Superposition of States for Spin Half . . . . . . . . . . . . . . . . . . . . 967.3.3 A derivation of sum-over-probability-amplitudes for spin half . . . . . . . 977.4 The General Case of Many Intermediate States . . . . . . . . . . . . . . . . . . 1017.5 Probabilities vs probability amplitudes . . . . . . . . . . . . . . . . . . . . . . . 1038 Vector Spaces in Quantum Mechanics8.1 Vectors in Two Dimensional Space105. . . . . . . . . . . . . . . . . . . . . . . . 1058.1.1 Linear Combinations of Vectors – Vector Addition . . . . . . . . . . . . . 1058.1.2 Inner or Scalar Products . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.2 Generalization to higher dimensions and complex vectors . . . . . . . . . . . . 1078.3 Spin Half Quantum States as Vectors . . . . . . . . . . . . . . . . . . . . . . . 1098.3.1 The Normalization Condition . . . . . . . . . . . . . . . . . . . . . . . . 1128.3.2 The General Spin Half State . . . . . . . . . . . . . . . . . . . . . . . . 1128.3.3 Is every linear combination a state of the system? . . . . . . . . . . . . . 1148.4 Constructing State Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.4.1 A General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1188.4.2 Further Examples of State Spaces. . . . . . . . . . . . . . . . . . . . 1208.4.3 States with multiple labels . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.5 States of Macroscopic Systems — the role of decoherence9 General Mathematical Description of a Quantum System. . . . . . . . . . . 1231269.1 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269.2 Probability Amplitudes and the Inner Product of State Vectors . . . . . . . . . . 1279.2.1 Bra Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12910 State Spaces of Infinite Dimension13110.1 Examples of state spaces of infinite dimension . . . . . . . . . . . . . . . . . . 13110.2 Some Mathematical Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13310.2.1 States of Infinite Norm. . . . . . . . . . . . . . . . . . . . . . . . . . . 13310.2.2 Continuous Basis States . . . . . . . . . . . . . . . . . . . . . . . . . . 13410.2.3 The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . 13510.2.4 Separable State Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 138c J D Cresser 2011
CONTENTSviii11 Operations on States13911.1 Definition and Properties of Operators . . . . . . . . . . . . . . . . . . . . . . . 13911.1.1 Definition of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 13911.1.2 Linear and Antilinear Operators. . . . . . . . . . . . . . . . . . . . . . 14111.1.3 Properties of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 14211.2 Action of Operators on Bra Vectors. . . . . . . . . . . . . . . . . . . . . . . . 14711.3 The Hermitean Adjoint of an Operator . . . . . . . . . . . . . . . . . . . . . . . 15111.3.1 Hermitean and Unitary Operators. . . . . . . . . . . . . . . . . . . . . 15411.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 15611.4.1 Eigenkets and Eigenbras . . . . . . . . . . . . . . . . . . . . . . . . . . 15911.4.2 Eigenstates and Eigenvalues of Hermitean Operators11.4.3 Continuous Eigenvalues. . . . . . . . . . 159. . . . . . . . . . . . . . . . . . . . . . . . . . 16111.5 Dirac Notation for Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16212 Matrix Representations of State Vectors and Operators16812.1 Representation of Vectors In Euclidean Space as Column and Row Vectors. . 16812.1.1 Column Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16812.1.2 Row Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17012.2 Representations of State Vectors and Operators . . . . . . . . . . . . . . . . . 17012.2.1 Row and Column Vector Representations for Spin Half State Vectors . . 17112.2.2 Representation of Ket and Bra Vectors . . . . . . . . . . . . . . . . . . . 17112.2.3 Representation of Operators . . . . . . . . . . . . . . . . . . . . . . . . 17312.2.4 Properties of Matrix Representations of Operators . . . . . . . . . . . . 17512.2.5 Eigenvectors and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 17812.2.6 Hermitean Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17913 Observables and Measurements in Quantum Mechanics18113.1 Measurements in Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . 18113.2 Observables and Hermitean Operators. . . . . . . . . . . . . . . . . . . . . . 18213.3 Observables with Discrete Values . . . . . . . . . . . . . . . . . . . . . . . . . 18413.3.1 The Von Neumann Measurement Postulate . . . . . . . . . . . . . . . . 18613.4 The Collapse of the State Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 18713.4.1 Sequences of measurements . . . . . . . . . . . . . . . . . . . . . . . . 18713.5 Examples of Discrete Valued Observables . . . . . . . . . . . . . . . . . . . . . 18813.5.1 Position of a particle (in one dimension) . . . . . . . . . . . . . . . . . . 188c J D Cresser 2011
ixCONTENTS13.5.2 Momentum of a particle (in one dimension) . . . . . . . . . . . . . . . . 18913.5.3 Energy of a Particle (in one dimension) . . . . . . . . . . . . . . . . . . 18913.5.4 The O 2 Ion: An Example of a Two-State System . . . . . . . . . . . . . 19113.5.5 Observables for a Single Mode EM Field. . . . . . . . . . . . . . . . . 19313.6 Observables with Continuous Values . . . . . . . . . . . . . . . . . . . . . . . . 19413.6.1 Measurement of Particle Position . . . . . . . . . . . . . . . . . . . . . . 19413.6.2 General Postulates for Continuous Valued Observables. . . . . . . . . 19813.7 Examples of Continuous Valued Observables . . . . . . . . . . . . . . . . . . . 19913.7.1 Position and momentum of a particle (in one dimension) . . . . . . . . . 19913.7.2 Field operators for a single mode cavity . . . . . . . . . . . . . . . . . . 20514 Probability, Expectation Value and Uncertainty20914.1 Observables with Discrete Values . . . . . . . . . . . . . . . . . . . . . . . . . 20914.1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20914.1.2 Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21014.1.3 Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21114.2 Observables with Continuous Values . . . . . . . . . . . . . . . . . . . . . . . . 21514.2.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21514.2.2 Expectation Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21614.2.3 Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21714.3 The Heisenberg Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . . 21714.4 Compatible and Incompatible Observables. . . . . . . . . . . . . . . . . . . . 21714.4.1 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21715 Time Evolution in Quantum Mechanics21815.1 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21815.2 The Schrödinger Equation – a ‘Derivation’. . . . . . . . . . . . . . . . . . . . 22015.2.1 Solving the Schrödinger equation: An illustrative example . . . . . . . . 22115.2.2 The physical interpretation of the O 2 Hamiltonian . . . . . . . . . . . . . 22415.3 The Time Evolution Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22516 Displacements in Space22617 Rotations in Space22718 Symmetry and Conservation Laws228c J D Cresser 2011
Chapter 1Introductionhere are three fundamental theories on which modern physics is built: the theory of relativity,statistical mechanics/thermodynamics, and quantum mechanics. Each one has forced uponus the need to consider the possibility that the character of the physical world, as we perceive itand understand it on a day to day basis, may be far different from what we take for granted.TAlready, the theory of special relativity, through the mere fact that nothing can ever be observedto 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 ownprivate ‘now’. The theory of general relativity then tells us that space and time are curved, thatthe universe ought to be expanding from an initial singularity (the big bang), and will possiblycontinue expanding until the sky, everywhere, is uniformly cold and dark.Statistical mechanics/thermodynamics gives us the concept of entropy and the second law: theentropy of a closed system can never decrease. First introduced in thermodynamics – the studyof 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 oddindeed: eggs are often observed to break, but never seen to reassemble themselves. The senseof uni-directionality of events defines for us an ‘arrow of time’. But what is entropy? Statisticalmechanics – which attempts to explain the properties of matter in bulk in terms of the aggregatebehaviour of the vast numbers of atoms that make up matter – stepped in and told us that thisquantity, entropy, is not a substance in any sense. Rather, it is a measure of the degree of disorderthat a physical system can possess, and that the natural direction in which systems evolve is in thedirection such that, overall, entropy never decreases. Amongst other things, this appears to havethe consequence that the universe, as it ages, could evolve into a state of maximum disorder inwhich 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 upsidedown by the edicts of this theory? It appears that quantum mechanics delivers to us a world viewin which There is a loss of certainty – unavoidable, unremovable randomness pervades the physicalworld. 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 makingan 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!).c J D Cresser 2011
Chapter 1Introduction2 Physical systems appear to behave as if they are doing a number of mutually exclusive thingssimultaneously. For instance an electron fired at a wall with two holes in it can appear tobehave as if it goes through both holes simultaneously. Widely separated physical systems can behave as if they are entangled by what Einsteintermed some ‘spooky action at a distance’ so that they are correlated in ways that appear todefy 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 someaspects of the physical world that cannot be said to be objectively ‘real’. For instance, in the gameknown as The Shell Game, a pea is hid
quantum mechanical. Indeed, the marriage of quantum mechanics – the physics of the very small – with general relativity – the physics of the very large – is believed by some to be the crucial step in formulating a general ‘theory of everything’ that will hopefully contain all the basic laws of nature in one package.
