Quantum Mechanics 6th Edition by Alastair Rae, Jim Napolitano 1138458333 9781138458338

Quantum Mechanics 6th Edition by Alastair Rae, Jim Napolitano – Ebook PDF Instant Download/Delivery: 1138458333, 9781138458338

Full download Quantum Mechanics 6th Edition after payment

Product details:

ISBN 10: 1138458333

ISBN 13: 9781138458338

Author: Alastair I. M. Rae; Jim Napolitano

A Thorough Update of One of the Most Highly Regarded Textbooks on Quantum MechanicsContinuing to offer an exceptionally clear, up-to-date treatment of the subject, Quantum Mechanics, Sixth Edition explains the concepts of quantum mechanics for undergraduate students in physics and related disciplines and provides the foundation necessary for other

Table of contents:

I Waves, Electromagnetism, and the Limits of Classical Physics

CHAPTER 1 The Physics and Mathematics of Waves

Figure 1.1 A simple mechanical system used to illustrate simple Harmonic Motion. A mass m is attached to a spring with stiffness constant k and the other end of the spring is attached to a fixed wall. The variable x measures the extension of the spring. That is, when x = 0, the spring is relaxed and the mass is at its equilibrium position.

1.1 A REVIEW OF SIMPLE HARMONIC MOTION

Euler’s formula

Worked Example 1.1

Figure 1.2 Section of a string under tension, slightly deformed, and variables used to derive its equation of motion. The vertical displacement of the string is the function y(x, t).

1.2 THE STRETCHED STRING EQUATION OF MOTION

Figure 1.3 A representation of the solution expressed by Equation 1.13. The solid line plots an arbitrary function y(x). The dotted line, and axis, plot the same curve, but displaced to the right by an amount vt. That is, the curve moves to the right with a speed v.

Principle of linear superposition

Worked Example 1.2

Figure 1.4 Graphic illustration of a solution to the wave equation, for a “bump” that is initially at rest. The shape splits in two with one half moving to the right, and half moving to the left.

Sinusoidal waves

1.3 STANDING WAVES, AND FOURIER SERIES

Figure 1.5 Graphic illustration of a solution to the wave equation which meets a boundary condition at x = 0, namely that the string remains motionless.

String fixed at both ends

Figure 1.6 Modes of a vibrating string, each with a fixed wavelength and frequency. The left shows the shape at t = 0 for the first three mode numbers, while the right shows the shape of the n = 2 mode at t = 0, at one eighth of a period, and at half a period.

Fourier series

Worked Example 1.3

Figure 1.7 On the left is the initial shape of a “plucked” stretched string. The plot on the right shows the shape of the string at different times, each a fraction of a full period.

1.4 THE FOURIER TRANSFORM

Worked Example 1.4

Figure 1.8 A square pulse f(x), left, and its Fourier transform a(k), right.

The Dirac δ-function

1.5 PROBLEMS

CHAPTER 2 Maxwell’s Equations and Electromagnetic Waves

Notation for working in three dimensions

On the question of units in electromagnetism

2.1 MAXWELL’S EQUATIONS AS INTEGRALS

Fields, forces, and flux

Application: the resistor

FIGURE 2.1 The resistor as used in electrical circuits. The upper symbol was completely standard, but has more recently been replaced by the lower one. They are used in circuit diagrams regardless of the specific type of resistor used. We can model a resistor as a long cylinder, made of some uniform, isotropic material, with wires attached to the flat faces. The model quite resembles a standard carbon resistor. The colored bands on the resistor are a code for its resistance value and precision standards.

2.2 SURFACE THEOREMS IN VECTOR CALCULUS

Figure 2.2 Geometric explanation of the derivation of the divergence theorem. (a) The volume integral over a region is the sum of small rectangles (or rectangular blocks in three dimensions). (b) Any quantity proportional to the area of a volume element cancels on adjacent cells, leaving only the edges on the boundary. (c) Coordinate system for analyzing the integral over one small cell.

Figure 2.3 Geometric explanation of the derivation of Stokes’ theorem. (a) The surface integral over a region is the sum of small rectangles. (b) Any quantity proportional to a loop over the area element cancels on adjacent cells, leaving only the edges on the boundary. (c) Coordinate system for analyzing the integral over one small cell.

2.3 MAXWELL’S EQUATIONS AS DERIVATIVES

The electrostatic and magnetic vector potentials

Charge conservation and the continuity equation

Application: the betatron

Figure 2.4 A Betatron accelerates particles using the electric field from magnetic induction. The diagram shows the principle, and defines quantities that allow us to derive the so-called Betatron Condition. The photograph shows early Betatrons with inventor, Donald Kerst.

2.4 ELECTROMAGNETIC WAVES

2.5 ELECTROMAGNETIC RADIATION

Figure 2.5 A schematic representation of electric dipole radiation. The frames from left to right represent increasing time, while each frame represents the same portion of space. The lines are contours of constant electric field strength. The source is an infinitesimally small oscillating electric dipole at the origin, oriented in the vertical direction. The field strength is inversely proportional to distance from the origin, with the field lines forming closed loops that move away from the dipole.

Thomson scattering

2.6 PROBLEMS

CHAPTER 3 Particle Mechanics, Relativity, and Photons

3.1 NEWTON, MAXWELL, AND EINSTEIN

The principle of relativity

Figure 3.1 Two different inertial reference frames S and S′. The coordinates x and y locate a point in frame S, while coordinates x′ and y′ locate a point in frame S′. Frame S′ moves to the right, that is in the positive x-direction, with speed V relative to S.

Galilean relativity

The constancy of the speed of light

Figure 3.2 Schematic of a “clock” based on a light pulse that bounces between two mirrors. The mirrors are arranged vertically and are separated by a distance L. In (a) the clock is seen by a stationary observer, while in (b), the clock is moving horizontally with speed V.

3.2 SPACETIME IN SPECIAL RELATIVITY

Events and the invariant interval

Four-vectors

The Lorentz transformation

3.3 VELOCITY, MOMENTUM, AND ENERGY

Worked Example 3.1

Massless particles: photons

Figure 3.3 Scattering of a massless particle with energy E from a stationary particle with mass m. The massless particle moves off at an angle θ and with energy E′. The massive particle moves off with momentum p.

Application: invariant mass and the Higgs boson

Figure 3.4 Two very high energy protons collide in the CMS detector at CERN, producing hundreds of elementary particles. Two of these are identified as photons in one particular collision. Plotting the invariant mass of pairs of photons in many thousands of events, shows a “bump” in the spectrum at 125 GeV/c2. This is one observation of the Higgs boson. The diagram appears courtesy CERN. The plot is from the publication by V. Khachatryan, et al, EPJC, October 2014, 74:3076, courtesy Springer.

3.4 PROBLEMS

CHAPTER 4 The Early Development of Quantum Mechanics

4.1 THE PHOTOELECTRIC EFFECT

Worked Example 4.1

Application: photomultiplier tubes

Figure 4.1 A schematic of a photomultiplier tube (PMT), with a “box and grid” dynode structure, along with a photograph of a Hamamatsu R1306 PMT which uses an analogous construction. Materials courtesy of Hamamatsu Photonics K.K., reprinted with permission.

4.2 THE COMPTON EFFECT

Figure 4.2 In Compton scattering an x-ray photon of angular frequency ω and wave vector k collides with an electron initially at rest. After the collision the photon frequency and wave vector are changed to ω′ and k′, respectively, and the electron recoils with momentum pe.

Worked Example 4.2

4.3 LINE SPECTRA AND ATOMIC STRUCTURE

Worked Example 4.3

4.4 DE BROGLIE WAVES

Worked Example 4.4

Figure 4.3 In recent years, it has been possible to produce neutron beams with de Broglie wavelengths around 2 nm which can be detectably diffracted by double slits of separation about 0.1 mm. A typical experimental arrangement is shown in (a) and the slit arrangement is illustrated in (b). The number of neutrons recorded along a line perpendicular to the diffracted beam 5 m beyond the slits is shown in (c), along with the intensity calculated from diffraction theory, assuming a wave model for the neutron beam. The agreement is clearly excellent.

Application: electron microscopy

Figure 4.4 The left shows a general schematic of a Transmission Electron Microscope (TEM). Electrons are accelerated and emitted by the gun at the top, producing a downward going beam. (This schematic is courtesy of Wikipedia). The right image is of a human immunodeficiency virus (HIV) organism, next to a human T-Cell. The effective magnification shown here is ×145000. Note the scale in the image, a size not resolvable with optical microscopy. Image courtesy of the Dartmouth College Electron Microscope Facility.

4.5 WAVE-PARTICLE DUALITY

Figure 4.5 A measurement of the position of a particle by a microscope is accompanied by a corresponding uncertainty in our knowledge of the particle momentum.

The uncertainty principle

4.6 THE REST OF THIS BOOK

4.7 PROBLEMS

II Elementary Wave Mechanics

CHAPTER 5 The One-dimensional Schrödinger Equations

5.1 THE TIME-DEPENDENT SCHRÖDINGER EQUATION

The wave function

5.2 THE TIME-INDEPENDENT SCHRÖDINGER EQUATION

5.3 BOUNDARY CONDITIONS

5.4 THE INFINITE SQUARE WELL

Figure 5.1 a) Shows the potential V(x) as a function of x for an infinite square well, along with the energy levels, En, of the four lowest energy states. The wave functions, un, and position probability distributions, |un|2, corresponding to energy states with n = 1, 2, 3, and 8 are shown in (b) and (c) respectively.

Worked Example 5.1

Worked Example 5.2

5.5 THE FINITE SQUARE WELL

Figure 5.2 The circular quadrant represents κa and the other lines are graphs of tan ka and cot ka. We note that k and κ are both positive by definition. Intersections between the quadrant and the other curves correspond to the solutions of equation (5.44).

Figure 5.3 a) Shows the potential V as a function of x for a finite square well in the case where V0 = 25ħ2/2ma2, along with the energies of the four bound states. The wave functions and position probability distributions for these states are shown in (b) and (c) respectively.

TABLE 5.1 Values of the quantities ka, κa and E that are consistent with the boundary conditions for a potential well whose sides are of height V0 in the case where V0 = 25ħ2/2ma2. The energies of the corresponding states in the case where V0 is infinite are represented by E∞.

Worked Example 5.3

5.6 QUANTUM MECHANICAL TUNNELLING

Figure 5.4 A beam of particles represented by a plane wave is incident on a potential barrier. Most particles are reflected, but some are transmitted by quantum-mechanical tunnelling.

Alpha decay

Figure 5.5 a) Shows the potential energy of interaction between an alpha particle and a nucleus as a function of its distance from the centre of the nucleus while (b) shows the potential energy of an electron near the surface of a metal with and without (broken line) an applied electric field. In each case the particles can pass through the potential barrier by quantum-mechanical tunnelling.

Cold electron emission

Application: scanning tunnelling microscopy

Figure 5.6 An image of the (111) surface of silicon obtained by scanning tunnelling microscopy. The bright peaks correspond to silicon atoms. The hexagonal symmetry is a characteristic feature of this surface. (Supplied by P. A. Sloan and R. E. Palmer of the Nanoscale Physics Research Laboratory at the University of Birmingham.)

Worked Example 5.4

Figure 5.7 Particles in states with energy between V1 and V2 can escape from the potential well illustrated in (a) by quantum-mechanical tunnelling. (b) shows the real part of the wave function of such a state.

5.7 THE HARMONIC OSCILLATOR

Figure 5.8 The wave functions corresponding to the four lowest energy states of the harmonic oscillator are shown in (a), while (b) depicts the position probability distribution in the case where n = 10, compared with that calculated classically (broken curve). The limits of the classical oscillation are indicated by vertical lines in each case. Note that (a) and (b) are on different scales.

5.8 PROBLEMS

CHAPTER 6 The Three-dimensional Schrödinger Equations

6.1 THE WAVE EQUATIONS

6.2 SEPARATION IN CARTESIAN COORDINATES

The three-dimensional “box”

Figure 6.1 The top two diagrams show the shape of the wave functions corresponding to the states n1 = 2, n2 = 1, n3 = 1, and n1 = 1, n2 = 2, n3 = 1, as calculated in the plane z = 0 for the case of a particle in a rectangular box with a = b. The lower diagram shows the position probability distribution calculated as the average of the squares of the two wave functions.

Worked Example 6.1 The three-dimensional harmonic oscillator

Figure 6.2 The geometrical relationship between the spherical polar coordinates r, θ, ϕ and the Cartesian axes x, y, z.

6.3 SEPARATION IN SPHERICAL POLAR COORDINATES

Figure 6.3 Representations of the shapes of the spherical harmonics with quantum numbers l, m, where l ≤ 2 and the z axis is vertical. In the case where m = 0, the dark and light regions are of opposite sign; when m ≠ 0, the function is complex and its phase changes by 2mπ during a complete circuit of the z axis.

The radial equation

Worked Example 6.2 The spherical infinite square well.

Worked Example 6.3 The isotropic harmonic oscillator.

6.4 THE HYDROGENIC ATOM

The hydrogenic atom wave functions

Figure 6.4 The radial parts, Rn,l, of the wave functions corresponding to some of the energy states of the hydrogen atom (a). The corresponding radial probability distributions, |χn,l|2 = r2|Rn,l|2, are displayed in (b).

Worked Example 6.4

6.5 PROBLEMS

III Formal Foundations

CHAPTER 7 The Basic Postulates of Quantum Mechanics

7.1 THE WAVE FUNCTION

Notation

7.2 THE DYNAMICAL VARIABLES

Orthonormality

Worked Example 7.1

7.3 PROBABILITY DISTRIBUTIONS

Figure 7.1 An example of quantum measurement theory. The left-hand part of the diagram shows how the wave function before a position measurement can be expressed as a sum of parts, each of which corresponds to one of the seven possible outcomes. After the measurement the wave function “collapses” at random into one of the functions shown on the right. (NB: The vertical scale of the right-hand peaks is reduced: they should all be the same size with their squares mutiplied by Δ normalized to unity.)

Worked Example 7.2

Continuous eigenvalues

Summary

Expectation values

Worked Example 7.3

7.4 COMMUTATION RELATIONS

Worked Example 7.4

Compatibility

7.5 THE UNCERTAINTY PRINCIPLE

Figure 7.2 Single-slit diffraction. Light reaching the right-hand slit has also passed through the left-hand slit and is therefore travelling parallel to the horizontal axis. When it emerges from the second slit, it spreads out to form a diffraction pattern The graph at the right shows the intensity of the diffraction pattern as a function of θ.

Single-slit diffraction

Worked Example 7.5

7.6 THE TIME DEPENDENCE OF THE WAVE FUNCTION

7.7 DEGENERACY

Worked Example 7.6

7.8 THE HARMONIC OSCILLATOR AGAIN

Worked Example 7.7

7.9 THE MEASUREMENT OF MOMENTUM BY COMPTON SCATTERING

Figure 7.3 A comparison of the theoretical and experimental intensities of Compton scattering from helium. The continuous line represents the momentum calculated in the manner described in the text and the points represent measured intensities—after J. W. M. Dumond and H. A. Kirkpatrick, Physical Review, vol. 52, pp. 419-436, 1937.

7.10 PROBLEMS

CHAPTER 8 Angular Momentum I

8.1 THE ANGULAR-MOMENTUM OPERATORS

8.2 THE ANGULAR MOMENTUM EIGENVALUES AND EIGENFUNCTIONS

Figure 8.1 The possible orientations of the angular momentum vector relative to the z axis in the case where l = 2. Note that the orientation within the xy plane is unknown if the z component has a definite value, so the angular momentum vector joins the origin to the edge of a cone whose axis is z.

Worked Example 8.1

Worked Example 8.2

8.3 THE EXPERIMENTAL MEASUREMENT OF ANGULAR MOMENTUM

The Stern-Gerlach experiment

Figure 8.2 (a) A beam of spin-half atoms is split by an inhomogeneous magnetic field. The form of the field is shown in projection down the beam direction in (b).

8.4 A GENERAL SOLUTION TO THE ANGULAR-MOMENTUM EIGENVALUE PROBLEM

Notation

Worked Example 8.3

Worked Example 8.4

Normalization

Worked Example 8.5

8.5 PROBLEMS

CHAPTER 9 Angular Momentum II

9.1 MATRIX REPRESENTATIONS

9.2 PAULI SPIN MATRICES

TABLE 9.1 The eigenvalues and eigenvectors of the matrices representing the angular momentum components of a spin-half particle. NB: The overall phase of an eigenvector, like that of a wave function, is arbitrary.

Worked Example 9.1

9.3 SPIN AND THE QUANTUM THEORY OF MEASUREMENT

Figure 9.1 Successive measurements of the angular momentum components of spin-half particles. The boxes represent sets of Stern-Gerlach apparatus which direct a particle into the upper or lower output channel depending on whether the appropriate spin component is found to be positive or negative. The y direction is from left to right.

Worked Example 9.2

9.4 DIRAC NOTATION

9.5 SPIN-ORBIT COUPLING AND THE ZEEMAN EFFECT

Spin-orbit coupling

Figure 9.2 The maximum (minimum) possible values of j are obtained when l and s are as near parallel (antiparallel) as possible, given that the squared magnitudes of j, l and s must be proportional to j(j + 1), l(l + 1) and s(s + 1) respectively, where j, l and s are integers or half-integers. The diagram shows the case where l = 1, and or

Worked Example 9.3

Figure 9.3 The effect of spin-orbit coupling on the energy levels of a one-electron atom.

The strong-field Zeeman effect

The weak-field Zeeman effect

Figure 9.4 The orbital and spin magnetic moments are parallel to the respective angular momenta, but the constants of proportionality are different, so that the total magnetic moment, μ, is not parallel to j. The vectors representing l, s, j, μl, μs and μ are shown, assuming that they all lie in the plane of the page. However, for a given j, l and s lie on the surfaces of the cones and, as a result, μ also lies on a cone whose axis is j. In weak magnetic fields, the average value of μ is therefore its component parallel to j.

Worked Example 9.4

9.6 A MORE GENERAL TREATMENT OF THE COUPLING OF ANGULAR MOMENTA

Figure 9.5 The splitting ΔE of the states with l = 1 in a one-electron atom as a function of applied magnetic field. Energy is in units of E0 = ħ2 and the field is in units of E0/μB.

TABLE 9.2 Clebsch-Gordan coefficients for the case l = 1 and Eigenfunctions of the total angular momentum with given values of (j, mj) are constructed as linear combinations of products of the angular momentum eigenfunctions denoted by the values of (ml, ms).

Worked Example 9.5

TABLE 9.3 The matrix representing in the case where l = 1 and E0 = 〈f(r)〉ħ2 and μB = eħ/2me.

9.7 PROBLEMS

IV Extensions and Approximation Schemes

CHAPTER 10 Time-independent Perturbation Theory and the Variational Principle

10.1 PERTURBATION THEORY FOR NONDEGENERATE ENERGY LEVELS

Worked Example 10.1 The anharmonic oscillator.

TABLE 10.1 The ground-state energy of an anharmonic oscillator calculated using first-order perturbation theory (E01) and by numerical methods (E0)

The atomic polarizability of hydrogen

Worked Example 10.2

10.2 PERTURBATION THEORY FOR DEGENERATE ENERGY LEVELS

Worked Example 10.3

The Stark effect in hydrogen

Figure 10.1 A section at y = 0 through the position probability distribution corresponding to the n = 2 energy state of a hydrogen atom subject to an electric field in the z direction. (A number of contours have been omitted in the high peaks represented by the cross-hatched areas.)

Nearly degenerate systems

Electrons in a one-dimensional solid

Figure 10.2 The energy of an electron in a one-dimensional metal as a function of wave number k, compared with that of a free electron (broken line), showing the energy gaps at k = ±π/a.

10.3 THE VARIATIONAL PRINCIPLE

Worked Example 10.4 The harmonic oscillator.

The atomic polarizability of hydrogen

Figure 10.3 The wave function corresponding to the ground state of a one-dimensional harmonic oscillator (broken line) compared with that for an infinite-sided well whose parameters are chosen using the variational principle (continuous line).

10.4 PROBLEMS

CHAPTER 11 Time Dependence

11.1 TIME-INDEPENDENT HAMILTONIANS

Worked Example 11.1 The harmonic oscillator.

Figure 11.1 The time evolution of the position probability density corresponding to a particle in a harmonic oscillator potential whose wave function at t = 0 is the normalized sum of the eigenfunctions corresponding to the two lowest energy states.

Spin-half particle in a magnetic field

Figure 11.2 Neutrons pass through the lower slit and then enter a region where there is a magnetic field directed either vertically upwards (on the left) or downwards (on the right). The resulting interference pattern has a minimum in the centre if the relative rotation of the spins passing along the two paths is an odd number multiplied by 2π.

Worked Example 11.2 The spreading wave packet.

11.2 THE SUDDEN APPROXIMATION

Worked Example 11.3

11.3 TIME-DEPENDENT PERTURBATION THEORY

Periodic perturbations

Figure 11.3 The transition probability |cm1|2 as a function of (ωmn − ω) in the case of a system that has been subject to a periodic perturbation of angular frequency ω for a time t.

Application: magnetic resonance imaging

Figure 11.4 A magnetic resonance image (MRI) of the brain of a patient suffering from Parkinson’s disease. The gray scale of each image represents the proton density distribution. The different images each focus on different areas of the brain. Graham, et al, Brain (2000) 123, 2423-2431 with permission.

11.4 TRANSITIONS BETWEEN ATOMIC ENERGY LEVELS

Spontaneous emission

Selection rules

Worked Example 11.4

11.5 THE EHRENFEST THEOREM

11.6 THE AMMONIA MASER

Figure 11.5 The geometrical configuration of the ammonia molecule in states where the nitrogen atom is above (continuous lines) and below (broken lines) the plane of the three hydrogens.

Figure 11.6 The ammonia maser. Ammonia molecules leave the container C and pass through a region of nonuniform electric field ℰ. Excited molecules are deviated towards the lower field region and directed into a microwave cavity M.

11.7 PROBLEMS

CHAPTER 12 Scattering

12.1 SCATTERING IN ONE DIMENSION

Particle flux

Figure 12.1 The potential V and the real parts of the incident, reflected (broken line), and transmitted waves in the case of the scattering of particles by a potential step.

Scattering by a potential step

Worked Example 12.1

Worked Example 12.2

12.2 SCATTERING IN THREE DIMENSIONS

Cross section

Centre-of-mass frame

12.3 THE BORN APPROXIMATION

Figure 12.2 The relationship between the scattering angle (θ), the scattering vector (K), and the wave vectors of the incident (k0) and scattered (k) waves.

Worked Example 12.3 Scattering by a spherical potential well or step.

Application: the size of atomic nuclei

Figure 12.3 Data on elastic electron scattering from 40Ca at three different beam energies, taken from B. B. P. Sinha, et al, Phys. Rev. C7(1973)1930. For each beam energy, the cross section shows a sharp minimum at K = 1.15, 1.95, and 3.10 fm−1.

Worked Example 12.4 Rutherford scattering.

12.4 PARTIAL WAVE ANALYSIS

Figure 12.4 Particles passing through the collimator C are represented by a plane wave that is scattered by the scatterer S. Only the scattered wave enters the detector D, but there is a region around the scatterer, much larger than the scatterer itself, over which the wave function is a linear combination of the incident and scattered waves.

Worked Example 12.5 Scattering by hard spheres.

Scattering by a potential well or step

12.5 PROBLEMS

CHAPTER 13 Many-particle Systems

13.1 GENERAL CONSIDERATIONS

13.2 ISOLATED SYSTEMS

Worked Example 13.1

13.3 NONINTERACTING PARTICLES

13.4 INDISTINGUISHABLE PARTICLES

Worked Example 13.2

Noninteracting indistinguishable particles

13.5 MANY-PARTICLE SYSTEMS

Bosons

Figure 13.1 The distribution of atomic velocities in a collection of rubidium atoms held in a trap for three temperatures that span the Bose condensation temperature of 170 nK. Zero velocity corresponds to the centre of the peak in each case. The condensate is absent at 200 nK, well established at 100 nK and very nearly complete at 20 nK.

Fermions

Worked Example 13.3

Superconductivity

Application: Superconducting magnets

Figure 13.2 The left-hand photo (courtesy GE) shows a commercial magnetic resonance imaging system, including patient table, for medical diagnostics. On the right we show cross sections of two types of superconducting wire, manufactured by Oxford Instruments. The superconducting NbTi alloy is in the form of thin filaments within a ~ 1 mm diameter copper matrix. This gives a copper-to-superconductor ratio of 1.35:1, but embedding the wire in a solid copper channel, as in the top photo, increases the ratio to ~ 50:1 and gives superior performance for some applications.

13.6 THE HELIUM ATOM

Interelectronic interactions

Figure 13.3 The energies of the singlet and triplet states of the helium atom compared with the energy levels of the hydrogen atom. The number beside each energy level is the principal quantum number n. Each triplet state is actually three closely spaced levels split by the spin-orbit interaction. Zero energy corresponds to a singly ionized He atom.

13.7 SCATTERING OF IDENTICAL PARTICLES

13.8 PROBLEMS

V Advanced Topics

CHAPTER 14 Relativity and Quantum Mechanics

14.1 BASIC RESULTS IN SPECIAL RELATIVITY

14.2 THE DIRAC EQUATION

A free particle

Worked Example 14.1

A particle in an electromagnetic field

A particle in a magnetic field

A particle subject to a scalar potential

Worked Example 14.2

14.3 ANTIPARTICLES

Figure 14.1 An electron can be excited from one of the filled negative energy states to create a free electron of positive energy and a vacancy. The properties of the negative-energy sea containing a vacancy are the same as those of a positron.

14.4 OTHER WAVE EQUATIONS

14.5 QUANTUM FIELD THEORY AND THE SPIN-STATISTICS THEOREM

14.6 PROBLEMS

CHAPTER 15 Quantum Information

15.1 QUANTUM CRYPTOGRAPHY

Figure 15.1 Alice sends a series of qubits to Bob, which are αz/βz or αx/βx at random. Bob makes a similar set of randomly selected x or z measurements on the qubits he receives. They then exchange information on the measurement orientations they have chosen and use the results for the subset where these are the same, to define the key to be used. In the absence of Eve’s intervention, Alice and Bob have the same key, but this no longer holds if Eve has attempted to make a measurement.

TABLE 15.1 The properties of a typical set of 12 qubits analyzed by the apparatus shown in Figure 15.1. The asterisks in the final column mark cases where Eve’s intervention has resulted in Bob having the wrong result. (O: apparatus orientation; S: state of sent qubit; M: message; R: state of received qubit; a/r: accepted or rejected; K agreed common key; K′: key corrupted by Eve)

Worked Example 15.1

15.2 ENTANGLEMENT

15.3 CLONING AND TELEPORTATION

Teleportation

Figure 15.2 Quantum teleportation. SEP represents a source of entangled pairs of particles. One member of a pair is sent to Bob while the other is sent to Alice, who allows it to interact with the object qubit and uses the Bell measurement apparatus (BMA) to project their state into one of the four Bell states. Alice then uses a classical communication channel to tell Bob which of these results she obtained. Bob can then transform the state of the teleported qubit into one identical to the initial state of the object qubit.

Figure 15.3 Two photons (a and b) are simultaneously incident on a semitransparent mirror where each is reflected and/or transmitted with equal probability, and then detected by one of the detectors, D. If photons are simultaneously detected in both detectors, this constitutes a Bell-state measurement.

15.4 QUANTUM COMPUTING

Worked Example 15.2

The quantum Fourier transform

Figure 15.4 Quantum Fourier transform. The boxes labelled H represent Hadamard gates that perform a π/2 rotation of spin about the y axis. The boxes labelled R2 and R3 rotate the spin about the z axis so as to introduce phase changes of π/2 and π/4, respectively if the relevant control bit equals 1. The three boxes labelled C are CNOT gates, whose combined effect is to swap the states of qubits 1 and 3.

15.5 PROBLEMS

CHAPTER 16 The Conceptual Problems of Quantum Mechanics

16.1 THE CONCEPTUAL PROBLEMS

Determinism

Reality

16.2 HIDDEN-VARIABLE THEORIES

de Broglie-Bohm theory

Figure 16.1 The particle trajectories calculated for double-slit diffraction using the de Broglie-Bohm hidden-variable theory. The trajectories cluster around the diffraction maxima in the expected way, as indicated by the Fraunhofer intensity pattern on the right.

Worked Example 16.1

Problems with DBB

16.3 NONLOCALITY

Figure 16.2 An example of nonlocality. A pair of spin-half particles is created in a state with zero total spin and the spin of each is analyzed by a Stern-Gerlach magnet. Quantum mechanics says that a measurement on one of the particles collapses the whole wave function.

Bell’s theorem

Figure 16.3 a) The measurement directions and the angles between them. (b) The left-hand side of (16.23) as a function of θ. The fact that it is negative for θ > 60° proves Bell’s theorem.

Worked Example 16.2 A “real-spin” hidden-variable theory

Figure 16.4 According to the “real-spin” hidden-variable theory, the number of particle pairs where the z component of the spin of the first particle and that of the second particle in the direction defined by θ are both positive is proportional to the shaded area.

Experiments

Figure 16.5 The Aspect experiment to compare the predictions of quantum mechanics with those of Bell’s theorem for pairs of polarized photons. The photons can be switched into either channel on each side, so that measurements can be made with four different pairs of polarizations. The switches marked S are changed at a rate faster than the time required to send a signal from one side of the apparatus to the other at the speed of light.

Figure 16.6 (a) A pair of wavepackets (open circles) containing Bohm particles (filled circles) in their upper halves are split by Stern-Gerlach magnets oriented to measure the z component of spin. The figure shows the positions of the packets and particles at several stages of evolution. (b) The unbroken lines illustrate the same process in the z1/z2 plane of the two-particle configuration space, while the broken lines show what would happen if the right-hand magnet were the closer to the source than that on the left.

16.4 THE QUANTUM MEASUREMENT PROBLEM

Figure 16.7 Spin-half particles with positive x component of spin pass through an SGZ apparatus and are then directed along a common path into an SGX without any record of their z component having been made. A further measurement of their x component invariably produces a positive result.

Schrödinger’s cat

Decoherence

Many worlds

Figure 16.8 Water flowing along a stream divides into two channels. If small floating objects are scattered randomly onto the water above the splitting point, they will be more probably found in the wider channel. However, it is meaningless to say that the water itself is more probably in one channel or the other as it is actually in both. By analogy, probabilities can be straightforwardly assigned to particle properties in DBB, but not to “worlds” in many-worlds theory.

Spontaneous collapse

Figure 16.9 According to GRW theory, there is a finite probability of a particle collapsing into a localized state at any time. A macroscopic body is delocalized (picture on left) when one of the atoms collapses so that one of its possible positions is left empty, while the other is “over-full” (central picture). To avoid the inevitable increase in energy, the whole object must collapse around the atom (picture on right).

Gravity

The Copenhagen interpretation

Subjectivism

Consistent histories

16.5 THE ONTOLOGICAL PROBLEM

16.6 PROBLEMS

Back Matter

Bibliography

BACKGROUND

COMPLEMENTARY

ADVANCED

Addition

THE CONCEPTUAL PROBLEMS OF QUANTUM MECHANICS

Additions

APPLICATIONS

People also search:

6th postulate of quantum mechanics

quantum mechanics in layman’s terms

quantum mechanics explained

quantum mechanics salary

quantum mechanics simplified

Tags: Alastair Rae, Jim Napolitano, Quantum, Mechanics