Theoretical Physics III Quantum Theory
Theoretical Physics IIIQuantum TheoryPeter E. BlöchlCaution! This version is not completed and may containerrorsInstitute of Theoretical Physics; Clausthal University of Technology;D-38678 Clausthal Zellerfeld; Germany;http://www.pt.tu-clausthal.de/atp/
2 Peter Blöchl, 2000-July 6, 2011Source: ission to make digital or hard copies of this work or portions thereof for personal or classroomuse is granted provided that copies are not made or distributed for profit or commercial advantage andthat copies bear this notice and the full citation. To copy otherwise requires prior specific permissionby the author.11 Tothe title page: What is the meaning of ΦSX? Firstly, it reads like “Physics”. Secondly the symbols stand forthe three main pillars of theoretical physics: “X” is the symbol for the coordinate of a particle and represents ClassicalMechanics. “Φ” is the symbol for the wave function and represents Quantum Mechanics and “S” is the symbol for theEntropy and represents Statistical Physics.
Foreword and OutlookAt the beginning of the 20th century physics was shaken by two big revolutions. One was the theoryof relativity, the other was quantum theory. The theory of relativity introduces a large but finiteparameter, the speed of light. The consequence is a unified description of space and time. Quantumtheory introduces a small but finite parameter, the Planck constant. The consequence of quantumtheory is a unified concept of particles and waves.Quantum theory describes the behavior of small things. Small things behave radically differentthan what we know from big things. Quantum theory becomes unavoidable to describe nature atsmall dimensions. Not only that, but the implications of quantum mechanics determine also how ourmacroscopic world looks like. For example, in a purely classical world all matter would essentiallycollapse into a point, leaving a lot of light around.Quantum mechanics is often considered difficult. It seems counterintuitive, because the pictureof the world provided in classical physics, which we have taken for granted, is inaccurate. Our mindnaturally goes into opposition, if observations conflict with the well proven views we make of ourworld and which work so successfully for our everyday experiences. Teaching, and learning, quantummechanics is therefore a considerable challenge. It is also one of the most exciting tasks, becauseit enters some fundamentally new aspects into our thinking. In contrast to many quotes, it is mystrong belief that one can “understand” quantum mechanics just as one can “understand” classicalmechanics2 . However, on the way, we have to let go of a number of prejudices that root deep in ourmind.Because quantum mechanics appears counterintuitive and even incomplete due to its probabilisticelements, a number of different interpretations of quantum mechanics have been developed. Theycorrespond to different mathematical representations of the same theory. I have chosen to startwith a description using fields which is due to Erwin Schrödinger, because I can borrow a number ofimportant concepts from the classical theory of electromagnetic radiation. The remaining mysteryof quantum mechanics is to introduce the particle concept into a field theory. However, otherformulations will be introduced later in the course.I have structured the lecture in the following way:In the first part, I will start from the experimental observations that contradict the so-calledclassical description of matter.I will then try to sketch how a theory can be constructed that captures those observations. Iwill not start from postulates, as it is often done, but demonstrate the process of constructing thetheory. I find this process most exciting, because it is what a scientist has to do whenever he is facedwith a new experimental facts that do not fit into an existing theory. Quantum mechanics comes intwo stages, called first and second quantization. In order to understand what quantum mechanicsis about, both stages are required, even though the second part is considered difficult and is oftentaught in a separate course. I will right from the beginning describe what both stages are about, inorder to clear up the concepts and in order to put the material into a proper context. However, inorder to keep the project tractable, I will demonstrate the concepts on a very simple, but specific2 This philosophical question requires however some deep thinking about what understanding means. Often theproblems understanding quantum mechanics are rooted in misunderstandings about what it means to understandclassical mechanics.3
4example and sacrifice mathematical rigor and generality. This is, however, not a course on secondquantization, and after the introduction, I will restrict the material to the first quantization until Icome to the last chapter.Once the basic elements of the theory have been developed, I will demonstrate the consequenceson some one-dimensional examples. Then it is time to prepare a rigorous mathematical formulation. I will discuss symmetry in some detail. In order to solve real world problems it is importantto become familiar with the most common approximation techniques. Now we are prepared to approach physical problems such as atoms and molecules. Finally we will close the circle by discussingrelativistic particles and many particle problems.There are a number of textbooks available. The following citations3 are not necessarily completeor refer to the most recent edition. Nevertheless, the information should allow to locate the mostrecent version of the book. Cohen-Tannoudji, Quantenmechanik[6]. In my opinion a very good and modern textbook withlots of interesting extensions. Schiff, Quantum Mechanics[1]. Classical textbook. Gasiorewicz, Quantenphysik[2]. Merzbacher, Quantum mechanics[3] Alonso and Finn, Quantenphysik[4], Atkins, Molecular Quantum Mechanics[7]. This book is a introductory text on quantum mechanics with focuses on applications to molecules and descriptions of spectroscopic techniques. Messiah, Quantum Mechanics[5]. A very good older text with extended explanations. Veryuseful are the mathematical appendices. W. Nolting, Grundkurs Theoretische Physik 5: Quantenmechanik[8]. Compact and detailedtext with a lot of problems and solutions. C. Kiefer Quantentheorie [9] Not a text book but easy reading for relaxation. [11]J.-L. Basdevant and J. Dalibard, Quantum mechanics. A very good course book from the Ecole Polytechnique, which links theory well upon modern themes. It is very recent (2002).3 Detailedcitations are compiled at the end of this booklet.
Contents1Waves? Particles? Particle waves!2Experiment: the double slit2.1 Macroscopic particles: playing golf2.2 Macroscopic waves: water waves .2.3 Microscopic waves: light . . . . .2.4 Microscopic particles: electrons . .2.5 Summary . . . . . . . . . . . . .3413.Towards a theory3.1 Particles: classical mechanics revisited . . . . . . . . . . . .3.1.1 Hamilton formalism . . . . . . . . . . . . . . . . . .3.2 Waves: the classical linear chain . . . . . . . . . . . . . . .3.2.1 Equations of motion . . . . . . . . . . . . . . . . .3.2.2 . and their solutions . . . . . . . . . . . . . . . . .3.3 Continuum limit: transition to a field theory . . . . . . . . .3.4 Differential operators, wave packets, and dispersion relations3.5 Breaking translational symmetry . . . . . . . . . . . . . . .3.6 Introducing mass into the linear chain (Home study) . . . .3.7 Measurements . . . . . . . . . . . . . . . . . . . . . . . . .3.8 Postulates of Quantum mechanics . . . . . . . . . . . . . .3.9 Is Quantum theory a complete theory? . . . . . . . . . . . .3.10 Planck constant . . . . . . . . . . . . . . . . . . . . . . . .3.11 Wavy waves are chunky waves: second quantization . . . . .3.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .Quantum effects4.1 Quasi-one-dimensional problems in the real world . . . . . .4.2 Time-independent Schrödinger equation . . . . . . . . . . .4.3 Method of separation of variables . . . . . . . . . . . . . .4.4 Free particles and spreading wave packets . . . . . . . . . .4.4.1 Wave packets delocalize with time . . . . . . . . . .4.4.2 Wick rotation . . . . . . . . . . . . . . . . . . . . .4.4.3 Galilei invariance (home study) . . . . . . . . . . . .4.5 Particle in a box, quantized energies and zero-point motion .4.5.1 Time-dependent solutions for the particle in the .47485051545557575862
6CONTENTS4.6Potential step and splitting of wave packets . . . . . . . . . . . . . . . . . .4.6.1 Kinetic energy larger than the potential step . . . . . . . . . . . . .4.6.2 Construct the other partial solution using symmetry transformations .4.6.3 Kinetic energy smaller than the potential step . . . . . . . . . . . . .4.6.4 Logarithmic derivative . . . . . . . . . . . . . . . . . . . . . . . . .4.7 Barrier and tunnel effect . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.8 Particle in a well and resonances . . . . . . . . . . . . . . . . . . . . . . . .4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .567Language of quantum mechanics5.1 Kets and the linear space of states . . . . . . . . . . . . . .5.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . .5.1.2 Bra’s and brackets . . . . . . . . . . . . . . . . . .5.1.3 Some vocabulary . . . . . . . . . . . . . . . . . . .5.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . .5.2.2 Some vocabulary . . . . . . . . . . . . . . . . . . .5.3 Analogy between states and vectors . . . . . . . . . . . . .5.4 The power of the Dirac notation . . . . . . . . . . . . . . .5.5 Extended Hilbert space . . . . . . . . . . . . . . . . . . . .5.6 Application: harmonic oscillator . . . . . . . . . . . . . . .5.6.1 Algebraic treatment of the one-dimensional harmonic5.6.2 Wave functions of the harmonic oscillator . . . . . .5.6.3 Multidimensional harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .oscillator. . . . . . . . . . Representations6.1 Unity operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2 Representation of a state . . . . . . . . . . . . . . . . . . . . . . . . . . .6.3 Representation of an operator . . . . . . . . . . . . . . . . . . . . . . . .6.4 Change of representations . . . . . . . . . . . . . . . . . . . . . . . . . .6.5 From bra’s and ket’s to wave functions . . . . . . . . . . . . . . . . . . .6.5.1 Real-space representation . . . . . . . . . . . . . . . . . . . . . .6.5.2 Momentum representation . . . . . . . . . . . . . . . . . . . . . .6.5.3 Orthonormality condition of momentum eigenstates (Home study)6.6 Application: Two-state system . . . . . . . . . . . . . . . . . . . . . . . .6.6.1 Pauli Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.6.2 Excursion: The Fermionic harmonic oscillator (Home study) . . . .1 Expectation values . . . . . . .7.2 Certain measurements . . . . .7.2.1 Schwarz’ inequality . . .7.3 Eigenstates . . . . . . . . . . .7.4 Heisenberg’s uncertainty relation7.5 Measurement process . . . . . .7.6 Kopenhagen interpretation . . .113113113115116118120120.
CONTENTS7.77.889Decoherence . . . . . . . . . . . . . . . . . . . . .Difference between a superposition and a statistical7.8.1 Mixture of states . . . . . . . . . . . . . .7.8.2 Superposition of states . . . . . . . . . . .7.8.3 Implications for the measurement process .Dynamics8.1 Dynamics of an expectation value . . . . . . . .8.2 Quantum numbers . . . . . . . . . . . . . . . .8.3 Ehrenfest Theorem . . . . . . . . . . . . . . . .8.4 Particle conservation and probability current . . .8.5 Schrödinger, Heisenberg and Interaction pictures8.5.1 Schrödinger picture . . . . . . . . . . . .8.5.2 Heisenberg picture . . . . . . . . . . . .8.5.3 Interaction picture . . . . . . . . . . . . . . . . . . . . .mixture of states. . . . . . . . . . . . . . . . . . . . . . . . . . . .Symmetry9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .9.1.1 Some properties of unitary operators . . . . . . .9.1.2 Symmetry groups . . . . . . . . . . . . . . . . .9.2 Finite symmetry groups . . . . . . . . . . . . . . . . . .9.3 Continuous symmetries . . . . . . . . . . . . . . . . . .9.3.1 Shift operator . . . . . . . . . . . . . . . . . . .9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .9.4.1 Translation operator from canonical 2134.13713714014114114214214414410 Specific Symmetries10.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.2 n-fold rotation about an axis . . . . . . . . . . . . . . . . .10.3 Exchange of two particles . . . . . . . . . . . . . . . . . . .10.3.1 Fermions . . . . . . . . . . . . . . . . . . . . . . .10.3.2 Bosons . . . . . . . . . . . . . . . . . . . . . . . .10.4 Continuous translations . . . . . . . . . . . . . . . . . . . .10.5 Discrete translations, crystals and Bloch theorem . . . . . .10.5.1 Real and reciprocal lattice: One-dimensional example10.5.2 Real and reciprocal lattice in three dimensions . . .10.5.3 Bloch Theorem . . . . . . . . . . . . . . . . . . . .10.5.4 Schrödinger equation in momentum representation .10.6 Some common Lattices . . . . . . . . . . . . . . . . . . . .10.6.1 Cubic lattices . . . . . . . . . . . . . . . . . . . . .10.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.7.1 Symmetrization of states . . . . . . . . . . . . . . .10.7.2 Dirac comb . . . . . . . . . . . . . . . . . . . . . 64.11 Rotations, Angular Momentum and Spin17111.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17111.1.1 Derivation of the Angular momentum . . . . . . . . . . . . . . . . . . . . . 172
8CONTENTS11.211.311.411.511.611.711.811.1.2 Another derivation of the angular momentum (HomeCommutator Relations . . . . . . . . . . . . . . . . . . . .Algebraic derivation of the eigenvalue spectrum . . . . . . .Eigenstates of angular momentum: spherical harmonics . . .Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Addition of angular momenta . . . . . . . . . . . . . . . . .Products of spherical harmonics . . . . . . . . . . . . . . .Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .study). . . . . . . . . . . . . . . . . . . . . .17217317418018418618818912 Atoms19112.1 Radial Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19112.2 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19513 Approximation techniques13.1 Perturbation theory . . . . . . . . . . . . . . . . . .13.2 General principle of perturbation theory . . . . . . .13.3 Time-independent perturbation theory . . . . . . . .13.3.1 Degenerate case . . . . . . . . . . . . . . .13.3.2 First order perturbation theory . . . . . . . .13.3.3 Second order perturbation theory . . . . . .13.4 Time-dependent perturbation theory . . . . . . . . .13.4.1 Transition probabilities . . . . . . . . . . . .13.5 Variational or Rayleigh-Ritz principle . . . . . . . . .13.6 WKB-Approximation . . . . . . . . . . . . . . . . .13.6.1 Classical turning points . . . . . . . . . . . .13.7 Numerical integration of one-dimensional problems .13.7.1 Stability . . . . . . . . . . . . . . . . . . . .13.7.2 Natural boundary conditions . . . . . . . . .13.7.3 Radial wave functions for atoms . . . . . . .13.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . .13.8.1 A model for optical absorption . . . . . . . .13.8.2 Gamow’s theory of alpha decay . . . . . . .13.8.3 Transmission coefficient of a tunneling barrier13.8.4 Fowler-Nordheim tunneling . . . . . . . . . .14 Relativistic particles14.1 A brief review of theory of relativity . .14.2 Relativistic Electrons . . . . . . . . . .14.3 Electron in the electromagnetic field . .14.4 Down-folding the positron component .14.5 The non-relativistic limit: Pauli equation14.6 Relativistic corrections . . . . . . . . .14.6.1 Spin-Orbit coupling . . . . . . .14.6.2 Darwin term . . . . . . . . . .14.6.3 Mass-velocity term . . . . . . .15 Many 8238239
CONTENTS15.1 Bosons . . . . . . . . . . . . . . . . . . . . . . . . .15.1.1 Quantum mechanics as classical wave theory15.1.2 Second quantization . . . . . . . . . . . . .15.1.3 Coordinate representation . . . . . . . . . .15.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . .15.3 Final remarks . . . . . . . . . . . . . . . . . . . . .I.Appendix9239239240242243243245A Galilei invariance247B Spreading wave packet of a free particle249B.1 Probability distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251C The one-dimensional rectangular barrier253C.1 E V , the barrier can be classically surmounted . . . . . . . . . . . . . . . . . . . 254C.2 Tunneling effect E V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256D Particle scattering at a one-dimensional square well259D.1 Square well E 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259D.2 Square well E 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259E Alternative proof of Heisenberg’s uncertainty principle263F Spherical harmonics267F.1 Spherical harmonics addition theorem . . . . . . . . . . . . . . . . . . . . . . . . . 267F.2 Unsöld’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267F.3 Condon-Shortley phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267G Random Phase approximationG.1 Repeated random phase approximation .G.2 Transition matrix element . . . . . . . .G.3 Rate equation . . . . . . . . . . . . . .G.4 Off-diagonal elements . . . . . . . . . .269271272273273H Propagator for time dependent Hamiltonian275IMatching of WKB solution at the classical turning point277JPositrons281J.1 The action of quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 283K Method of separation of variables287L Trigonometric functions289M Gauss’ theorem291
10CONTENTSN Fourier transformN.1 General transformations . . . . . . . . .N.2 Fourier transform in an finite interval . .N.3 Fourier transform on an infinite intervalN.4 Table of Fourier transforms . . . . . . .N.5 Dirac’s δ-function . . . . . . . . . . . .293293293294295295O Pauli Matrices297P Linear AlgebraP.1 Half Group . .P.2 Vector space .P.3 Banach spaceP.4 Hilbert space.299299299300300Q Matrix identitiesQ.1 Notation . . . . . . . . . . . . . . . . . . . . . . .Q.2 Identities related to the trace . . . . . . . . . . . .Q.2.1 Definition . . . . . . . . . . . . . . . . . .Q.2.2 Invariance under commutation of a productQ.2.3 Invariance under cyclic permutation . . . .Q.2.4 Invariance under unitary transformation . .Q.3 Identities related to the determinant . . . . . . . .Q.3.1 Definition . . . . . . . . . . . . . . . . . .Q.3.2 Product rule . . . . . . . . . . . . . . . . .Q.3.3 Permutation of a product . . . . . . . . . .Q.3.4 1 . . . . . . . . . . . . . . . . . . . . . . .301301301301301302302302302302303304R Special FunctionsR.1 Bessel and Hankel functionsR.2 Hermite Polynomials . . . .R.3 Legendre Polynomials . . . .R.4 Laguerre Polynomials . . . .305305305305305.S Principle of least action for fields309T ℓ-degeneracy of the hydrogen atomT.1 Laplace-Runge-Lenz Vector . . . . . . . . . . . . . . . . . . . .T.1.1 Commutator relations of Runge-Lenz vector and angularT.1.2 Rescale to obtain a closed algebra . . . . . . . . . . . .T.2 SO(4) symmetry . . . . . . . . . . . . . . . . . . . . . . . . .T.3 ℓ-Degeneracy of the hydrogen atom . . . . . . . . . . . . . . .T.4 Derivation of commutator relations used in this chapter . . . . . ˆ 2 . . . . . . . . . . . . . . . . . . . . .T.4.1 Calculation of M ˆ . . . . . . . . . . . . . . . . ˆ and HT.4.2 Commutator of M ˆ and L ˆ . . . . . . . . . . . . . . . .T.4.3 Commutators of Mˆ(1)T.4.4 Commutators of J and J ˆ(2) . . . . . . . . . . . . . . . . . . . .momentum. . . . . . . . . . . . . . . . . . . . . . . . .311311312312312313314.316319321323.
CONTENTS11U A small Dictionary325V Greek Alphabet329W Philosophy of the ΦSX Series331X About the Author333
12CONTENTS
Chapter 1Waves? Particles? Particle waves!Quantum mechanics is about unifying two concepts that we use todescribe nature, namely that of particles and that of waves. Waves are used to describe continuous distributions such as water waves, sound waves, electromagnetic radiation such as lightor radio waves. Even highway traffic has properties of waves withregions of dense traffic and regions with low traffic. The characteristic properties of waves are that they are delocalized andcontinuous. On the other hand we use the concept of particles to describe themotion of balls, bullets, atoms, atomic nuclei, electrons, or in theexample of highway traffic the behavior of individual cars. Thecharacteristic properties of particles are that they are localizedand discrete. They are discrete because we cannot imagine halfa particle. (We may divide a bigger junk into smaller pieces, butFig. 1.1:Louis Victorthen we would say that the larger junk consisted out of severalde Broglie, 1892-1987.particles.)French physicist. Nobelprice in Physics 1929 forHowever, if we look a little closer, we find similarities between thesethe de Broglie relationsconcepts:E ω and p k,linking energy to frequency A wave can be localized in a small region of space. For example,and momentum to inversea single water droplet may be considered a particle, because itwavelength, which he posis localized, and it changes its size only slowly. Waves can alsotulated 1924 in his PhDbe discrete, in the case of bound states of a wave equation: Athesis.[1]violin string (g: Violinensaite) can vibrate with its fixed naturalvibrational frequency, the pitch, or with its first, second, etc. harmonic. The first overtonehas two times the frequency of the pitch (g:Grundton), the second overtone has three timesthe frequency of the pitch and so on. Unless the artist changes the length of the vibratingpart of the chord, the frequencies of these waves are fixed and discrete. We might considerthe number of the overtone as analogous to number of particles, which changes in discrete,equi-spaced steps. This view of particles as overtones of some vibrating object is very close toparticle concept in quantum field theory. If we consider many particles, such as water molecules, we preferto look at them as a single object, such as a water wave. Even ifwe consider a single particle, and we lack some information aboutit, we can use a probability distribution to describe its whereabouts13
141 WAVES? PARTICLES? PARTICLE WAVES!at least in an approximate manner. This is the realm of statisticalmechanics.Quantum mechanics shows that particles and waves are actually two aspects of more generalobjects, which I will call particle waves. In our everyday life, the two aspects of this generalizedobject are well separated. The common features become evident only at small length and timescales. There is a small parameter, namely the Planck constant h, that determines when the twoaspects of particle waves are well separated and when they begin to blur1 . In practice, the reduceddef hPlanck constant “hbar”, 2π, is used2 . Compared to our length and time scales, is so tinythat our understanding of the world has sofar been based on the assumption that 0. However,as one approached very small length and time scales, however, it became evident that 0, and afew well established concepts, such as those of particles and waves, went over board.A similar case, where a new finite parameter had to be introduced causing a lot of philosophicalturmoil is the theory of relativity. The speed of light, c, is so much larger than the speed of thefastest object we can conceive, that we can safely regard this quantity as infinite in our everydaylife. More accurate experiments, however, have found that c . As a result, two very differentquantities, space and time, had to be unified.Interestingly, the concept of a maximum velocity such as the speed of light, which underlies thetheory of relativity, is a natural (even though not necessary) consequence of a wave theory, such asquantum mechanics. Thus we may argue that the theory of relativity is actually a consequence ofquantum mechanics.1 “toblur” means in german “verschwimmen”Planck constant is defined as h. This has been a somewhat unfortunate choice, because h appears nearlyalways in combination with 1/(2π). Therefore, the new symbol h/(2π), denoted reduced Planck constant hasbeen introduced. The value of is 10 34 Js to within 6%.2 The
Chapter 2Experiment: the double slitThe essence of quantum mechanics becomes evident from a singleexperiment[2, 3, 4, 5]: the double-slit experiment. The experimentis simple: We need a source of particles or waves, such as golf balls,electrons, water waves or light. We need an absorbing wall with twoholes in it and behind the wall, at a distance, an array of detectors ora single detector that can be moved parallel to the wall with the holes.We will now investigate what the detector sees.2.1Macroscopic particles: playing golfLet us first investigate the double-slit experiment for a simple case,namely macroscopic particles such as golf balls.Fig. 2.1: Thomas Young,1773-1829. English physician and physicist.Established a wave theory oflight with the help of thedouble-slit experiment.Fig. 2.2: Double-slit experiment for particles. Particles are shot randomly against a wall, whichcontains two slits. Some particles pass through one or the other slit in the wall and proceed to thenext wall. The number of particles arriving at the second wall is counted as function of their position,yielding a probability distribution as function of their position. This probability distribution, shown asfull line at the very right, is the superposition of the probability distributions, shown as dashed lines,which one obtains if one or the other slit is closed.We take a golfer as a source of golf-balls. We build a wall with two slits. As detector we simplyuse little baskets that collect the golf balls that pass through the holes. By counting the numberof balls in each basket that arrive per unit time, we obtain a distribution P12 (y ) as function of thebasket position. We observe a distribution with two maxima, one behind each hole. If the two slits15
162 EXPERIMENT: THE DOUBLE SLITare close together the two maxima may also merge into on
Merzbacher, Quantum mechanics[3] Alonso and Finn, Quantenphysik[4], Atkins, Molecular Quantum Mechanics[7]. This book is a introductory text on quantum me-chanics with focuses on applications to molecules and descriptions of spectroscopic techniques. Messiah, Quantum Mechanics[5]. A very good older text with extended .
Chapter 2 - Quantum Theory At the end of this chapter – the class will: Have basic concepts of quantum physical phenomena and a rudimentary working knowledge of quantum physics Have some familiarity with quantum mechanics and its application to atomic theory Quantization of energy; energy levels Quantum states, quantum number Implication on band theory
terpretation of quantum physics. It gives new foundations that connect all of quantum physics (including quantum mechanics, statistical mechanics, quantum field theory and their applications) to experiment. Quantum physics, as it is used in practice, does much more than predicting probabili
The Quantum Nanoscience Laboratory (QNL) bridges the gap between fundamental quantum physics and the engineering approaches needed to scale quantum devices into quantum machines. The team focuses on the quantum-classical interface and the scale-up of quantum technology. The QNL also applies quantum technology in biomedicine by pioneering new
For example, quantum cryptography is a direct application of quantum uncertainty and both quantum teleportation and quantum computation are direct applications of quantum entanglement, the con-cept underlying quantum nonlocality (Schro dinger, 1935). I will discuss a number of fundamental concepts in quantum physics with direct reference to .
Quantum Field Theory Quantum field theory is the natural language of physics: Particle physics Condensed matter Cosmology String theory/quantum gravity Applications in mathematics especially in geometry and topology Quantum field theory is the modern calculus Natural language for describing diverse phenomena
the quantum operations which form basic building blocks of quantum circuits are known as quantum gates. Quantum algorithms typically describe a quantum circuit de ning the evolution of multiple qubits using basic quantum gates. Compiler Implications: This theoretical background guides the design of an e ective quantum compiler. Some of
Rae, Alastair I. M. Quantum physics: illusion or reality? 1. Quantum theory I. Title 530.1’2 QC174.12 Library of Congress Cataloguing in Publication data Rae, Alastair I. M. Quantum physics: illusion or reality? Bibliography Includes index. 1. Quantum theory. 2. Physics – Philosophy. I. Title. QC174.12.R335 1985 530.1’2 85 – 13256
According to the quantum model, an electron can be given a name with the use of quantum numbers. Four types of quantum numbers are used in this; Principle quantum number, n Angular momentum quantum number, I Magnetic quantum number, m l Spin quantum number, m s The principle quantum
