Chapter 2 Quantum Theory
Chapter 2 - Quantum Theory At the end of this chapter – the class will: Have basic concepts of quantum physical phenomena and arudimentary working knowledge of quantum physics Have some familiarity with quantum mechanics and itsapplication to atomic theory Quantization of energy; energy levelsQuantum states, quantum numberImplication on band theory
Chapter 2 Outline Basic concept of quantization Origin of quantum theory and key quantum phenomena Quantum mechanics Example and application to atomic theory
Concept introductionThe quantum carImagine you drive a car. You turn on engine and itimmediately moves at 10 m/hr. You step on the gaspedal and it does nothing. You step on it harder andsuddenly, the car moves at 40 m/hr. You step on thebrake. It does nothing until you flatten the brake withall your might, and it suddenly drops back to 10 m/hr.What’s going on?
Continuous vs. QuantizationConsider a billiard ball. It requires accuracy and precision. Youhave a cue stick. Assume for simplicity that there is no frictionloss. How fast can you make the ball move using the cue stick?How much kinetic energy can you give to the ball?The Newtonian mechanics answer is: any value, as much as energy as you can deliver. The ballcan be made moving with 1.000 joule, or 3.1415926535 or0.551 joule. Supposed it is moving with 1-joule energy,you can give it an extra 0.24563166 joule by hitting it with thecue stick by that amount of energy. The ball will have1.24563166 joule. The energy value is continuous.Let's shrink the table down to a very small size, and let the billiard ball become as small and lightmass as an electron. Can you still do the same thing to its energy?Answer: NO. If you don't hit the ball (electron) with the right amount of energy, IT WON'TCHANGE. Supposed it is moving with 1x10-19 Joule, and you want it to be moving with 1.014x1019 Joule which happens not to be a value that it's happy with. You can hit it as many times as youwant with 0.014x10-19 Joule, and nothing will happen. The ball just ignores your cue stick.The reason is that the energy of the quantum ball now is not continuous, but quantized intodiscrete values
Quantization of energyContinuous energy: it canbe any value on the axisEnergyDiscrete (or quantizedenergy): it can ONLY be atcertain values on the axisE1E2 E3E4E5EnergyKey concept of the quantum world: The energy can have discrete values: referred to as quantization But it doesn’t have to be always discrete! It can be continuous also, andcan be a mixed of discrete and continuum How many discrete levels can a system have?
How does a quantum billiard ball behave?Supposed that the quantum ball has energy E1If we hit it with a cue stick, what will happen?Answer:It depends, if the cue stick hits it with an energy DEsuch that E1 DE does not coincide with any of thequantized levels or E5, it won't change! Thequantum ball just ignores the cue stick. This is themajor difference between quantum physics andclassical physics.DEE1E2 E3E4E5Energy
How does a quantum billiard ball behave?DEE1Absorptionor energy transferredE2 3.4?3.5?3.6DE
General concept of quantization –quantumnumber Quantization applies not just to energy.Many fundamental physical quantities arealso quantized What is quantized electric charge? (e) Are the masses of objects quantized? mass of electron? proton, neutron, neutrino, photon A quantity that is absolutely quantized: angularmomentum – what is the unit of a.m.? Other: linear momentum Sometimes, we associate a “number” to a quantizedquantity, we call it “quantum number”
A Question A free electron in infinite space. Is it energy quantized? No, it’s not. Quantum mechanics (QM) does not say that everything has tobe quantized.A free electron has an energy value:p2where p is linear momentum, m is the mass,2justm like classical mechanics. pEnergycan be continuous, and so is the energy.8642momentum0.511.522.53
A Question A free electron in infinite space. Is it energy quantized? No But a bound electron, one which is under the influence of a force thatholds electron to a “confined” region of space, has its energy quantized.- If the electron experiences a force, but the force is not enough to hold itto a finite region of space, is its energy quantized? No. So how do weknow when the energy is quantized or not? We have to SOLVE thespecific problem to find out. All QM says is that there is the possibility ofquantization, depending on specific situations, this is different fromNewtonian mechanics in which, there is no quantization at all.
Countable and UncountableQuantization has a major impact on physical theory.One case is the issue of countable and uncountablenumber of states.How many numbers are there in the set of all integer?How many real numbers between 0 and 1?Are the two sets, all integers and all real number from 0-1, equal?Are the set of real number countable?Quantization makes the number of states (or physicalconditions, or solutions) countable.Is this some arcane, irrelevant math? No, it is a crucialdifference between quantum and classical theory, andbasically led to the discovery of the quantum theory: MaxPlanck’s black body radiation theory
The Origin of the Quantum Theory:The black-body problemAny hot object emits electromagnetic waves; we call thisthermal radiation.Find examples of this around you“red hot” objects in a furnace, the sun, the universe (3K backgroundradiation). Do you think living thing emit radiation? How about us? Doyou think we get cancer from our own thermal radiation?People have studied this in 19th century and observed thatthe hotter the object is, the brighter is the radiation, and also“redder” (shorter wavelength)How does the glow change as you heat up an object?Do you think the RGB color we see from the TV screen thermalradiation? Why so and if not, why not?
Black-body radiationHotter object - Brighter radiation: Stefan’s lawThermal Radiation Power T 4Radiating areaemissivityStefanBoltzmannconstantA perfectly absorbing object has emissivity 1. It absorbsall radiation falling on it, thus, it is a perfectly black object,called black-body.In real life, very hard to make a black body, one way tomake it close to it is a cavityAs shown, any light falling into the cavity aperture is mostlikely absorbed, thus, the radiation from the apertureshould be like black-bodytemperature
Black-body radiationHotter object - color shifts to shorter wavelengthRayleigh –Jeans’ law based on Maxwell’s theoryu Spectraldensity8 Wavelength4k BTtemperature600Boltzmannconstant500400Ultraviolet catastrophe! Where doesit go wrong?300u( )What is the problem of this? What isthis object color?20010000.511.522.5Wavelength3
Planck quantum theoryPlanck proposed a bold concept of "quantum of action”A p*xLater, Einstein coined the corpuscular theory of light:“photon” is the most fundamental light particle- The concept of indivisible lightDoes an electromagnetic wave carry energy?Or power?If so, what is the energy or power of anelectromagnetic wave?
Planck quantum theoryAs it turns out, the energy of an electromagneticwave cannot be arbitrarily small, and cannot becontinuous according to Planck’s theory – rather,the energy of an electromagnetic wave isquantized.The amount of energy an electromagnetic wavemode in a black-body cavity can ONLY havediscrete value:A new fundamentalE n h integerconstant: Planck’sconstant
Planck quantum theoryWhat happens? The number of modessuddenly go from uncountably infinite tocountably infiniteWith this, Planck shows that the resultshould be:Planck’s constantu wavelength8 e5chh / k BTfrequency 1
Planck’s theory of black-body radiationClassical theory600500u 4003002008 5 e h / kBT 110000.511.522.530.251000 KQuantum theory0.29000.150.17000.05500246Wavelength (um)8ch10
What is Planck’s constant?h 6.63 x 10-34 Joule sec.Energy of a photon of frequency : hE h h where 2 2 Link photon energy to the color of light
The electromagnetic spectrumFrom LBL
Second crucial evidence of theQuantum theory: photoelectric effectFrank-Hertz experiment The existence of threshold: photonfrequency must exceed a certain value (notintensity) Energy of photoelectrons is proportional tofrequency - NOT intensity. The proportionalconstant is Planck constant – (Milliken exp)
Photoelectron energyThe photoelectric effect (cont.)E h 0 Frequency
Discrete Lines in Atomic Vapor SpectraKey concepts Optical spectra Continuous spectra vs. quantizedspectrahttp://library.thinkquest.org
Where do the light come from? What do thespectra tell you?http://library.thinkquest.org
Atomic spectra Most common and natural lightsources have continuous spectra. Atomic vapor spectra, howeverhave discrete lines Where do the light come from? Whatdo the spectra tell you? Some exhibit simple mathematicalrelations: Balmer series, Lymanseries, Paschen series Rydberg showed that all theseseries can be based on a“quantized” expression of energy:What observed is the difference energy (ortransition energy):11 En Ek R 2 2 h n k kn http://library.thinkquest.orgREn 2nRydberg constantn 1,2,3 quantum numberk 2: Balmer seriesk 1: Lyman seriesk 3: Paschen
Hydrogen atom emission spectra11 En Ek R 2 2 kn Which series have longer wavelengths,shorter wavelengths?What wavelength range is the Lymanseries? What do we typically call thatwavelength range?How about Balmer series?And Paschen series?physics.colorado
Birth of the Quantum Theory Evidences were mounting that energy of light, electronsin atoms are quantized Discovery of fundamental particles smaller than atom Quantized charges Classical physics: Newtonian mechanics, Maxwellelectromagnetic theories could not account for quantumphenomena
Evolution of Quantum Mechanics Old quantum theory: Bohr-Sommerfeld’s atomic model. Newtonianmechanics for electron motion relative to the proton. Postulate: angularmomentum is quantized, which leads to energy quantization and orbitalquantum number Unfortunately, it did not takelong to see that the modelcould not explain many otherphenomena. Magnetic quantum number andAnomalous Zeeman effect:atomic spectral lines split under a magnetic field. The split does notconform to Bohr-Sommerfeld model. Discovery of electron spin. Zeeman split into even, rather than oddnumber of lines: anomalous Zeeman effect Old quantum theory appeared artificial with many postulates. Birth ofthe new quantum mechanical theory: DeBroglie: wave-particle duality,Heisenberg: uncertainty principle, Schroedinger: wave equation
Anomalous Zeeman Effect Electron has spin ½: Uhlenbeck &Goudsmit Spin-orbit coupling: L ½increment: 3/2, 5/2, J in ½If angular orbital quantumnumber is l, then the number ofmagnetic quantum level is:mL l , l 1, , l 1, lExample: l 2, mL -2,-1,0,1,2The number of magnetic quantum states is2l 1What is l if the number of magnetic quantum states is even?1For example: 2: 2s 1 2 s 2Electron has a spin (no classical analog), and it is 1/2Classical theory, old quantum theory are untenable!A new, systematic approach (not piecemeal) is needed
Relevance of Quantum Theory to Solid StateElectronics Essential to semiconductor physics is the concept of energy bands (band theory) and the energy band-gap, density-of-state: explained only withquantum theoryBehavior of electrons (individual): band theory explains the electronconduction behavior in semiconductors (classical conduction theory – Druidmodel – is insufficient) – conduction in nearly-filled band: hole modelBehavior of electrons (ensemble): Pauli exclusion principle and Fermi-Diracstatistics describes electron ensemble statistical behaviorQuantum mechanical transition theory: semi-classical quantumelectrodynamics (as opposed to quantum electrodynamics) explains opticaltransitions in semiconductorQuantization of elementary excitations in condensed matters: phonons,plasmons, excitons, and polaritons quantum mechanical excitation andrelaxation theory
What is it?It is a set of rules and equations (like Newton’slaws) that allow us to calculate and predict thebehavior of a quantum mechanical system
Overview of Topics Concept of wave-particle duality; Heisenberguncertainty principle Wave mechanics: DeBroglie wavelength, electrondiffraction, Schroedinger equation Basic concepts of wave mechanics Examples: Quantum wells The hydrogen atom Particle spin and statistics; Fermi-Dirac statistics
Heisenberg’s uncertainty principle- It is linked to Planck’s hypothesis of quantum of action:Dx Dp hDE Dt h- This principle completely decouples the modern quantum mechanics from the oldconcept, including Bohr’s model.- It says that it is fundamentally impossible and there is no need for a theory to beable to specify exactly the position (x) AND momentum (p) of a particle moreaccurately than the uncertainty given above.- For that matter, any other pair of quantities whose product has the unit of action(energy x time or position x momentum) obey the same principle. e. g. we know thatangular momentum is quantized- Heisenberg theory: matrix approach to quantum mechanics
Wave-particle duality De Broglie observed that an electromagnetic wave has particle-likebehavior, which is the photon So, can a particle (such as an electron) behave like a wave?De Broglie came up with a hypothesis (1924): the wavelengthassociated with a particle of momentum p is: hpWhere does this come from? He observed that for light:E mc2 h hmc h orc pphoton h If this is the relation of photon momentum to its wavelength, thenwhy not the same for a particle to its wavelength?
Wave-particle duality (cont.)- As it turned out, the orbit of Bohr-Sommerfeld’s atom(old quantum theory) is an exact integral multiple of theDe Broglie’s electron wavelength! This must be ontosomething, because it is such a natural explanation forthe Bohr-Sommerfeld’s atom quantization modelepeph2 r n nphpr L n n 2 Bohr-Sommerfeld angular momentumquantization postulate explained!
Wave-particle duality (cont.)- This gave motivation for Schroedinger to develop adifferent math to model quantum mechanics: wavemechanics- 3 years later (1927), Davidson and Germerdemonstrated electron diffraction: experimentallyshowed that particles can indeed diffract like a wave!What is diffraction?What is the difference between particle andwave?
DiffractionDiffraction and InterferenceElectron diffraction from a crystal
An analogy to uncertainty principleSuppose we have a sine wave10.5100200300400500-0.5-1What is its .511.511.520.511.52D Dt 1hD Dt h DE Dt h2
Shroedinger’s equation and wave mechanicsHeisenberg’s matrix-based quantum mechanics is too complicated.With De Broglie’s wave theory, Schroedinger in 1926 came up with analternative description: wave equationpotential energymomentum operatorparticle mass pˆ 2 ˆ ˆ 2m V E kinetic energy operatortotal energy operator“the” wave ˆ is the potential operator (problem specific) ; Eˆ pˆ ; Vii tWhat does it mean? It’s an equation describing how a wave shouldbehave. Like Maxwell’s equations for EM wavesWhat wave is it? It’s a wave from which one can extract measurablephysical quantities (energy, momentum, ) using a set of rulesWhat are the rules?
OWave mechanics (cont.)What are the rules?A physical quantity is determined only by its “expected” value, given by:ˆ x, y , z ; t ˆ dxdydz *x,y,z;tO O O t dxdydz * x, y, z; t x, y, z; t where O-hat is the operator associated with that physical quantity.The wave itself does not necessarily have any physical meaning or ismeasurableThe wavefunction can be normalized: (if squared-integrable) dxdydz * x, y, z; t x, y, z; t 1A common interpretation of2 x, y, z; t is that x, y, z; t represents the particle “probability density”.
OWave mechanics – Common basic features Space and time can be decoupled (solution by part) For the space component, many systems (problems)have eigenfunctions, which are quantum statesassociated with definite energy levels (often quantized,but not always) Often, they form an orthogonal basis for the vectorspace of all solutions A general solution can be expressed as a linearcombination of the eigensolutions or eigenvectors Equivalence between Schroedinger wave mechanicsand Heisenberg matrix representation: eachwavefunction is a vector, and yield the matrix operatorsin Heisenberg representation
Example: free electronWhat is a free electron?An electron that experiences zero potential energyeverywherepotential energymomentum operatorparticle mass pˆ 2 ˆ ˆ 2m V E kinetic energy operatortotal energy operator“the” wave ˆ is the potential operator (problem specific) ; Eˆ pˆ ; Vii tV is zero:2pˆ 2 ˆ E 2 i 2m2m t
Example: free electron (cont.)V is zero: 2 2 i 2m t 2 Let’s solve for one dimension:Space-time separation: 2 2m ff t 2 x i t x2m i t 2 2m i2 t x x f t 2 f t 2 k 2 f t x x x 2m fi k 2 f t x t
Example: free electron (cont.) 2 f t 2 k 2 f t x x x 2 2 x k x 22m fi k 2 f t x t 2m fi k 2 f t t22m fi k 2 f t f t C1e i k t / 2m t 2 2 x x C e ikx k2 x 22t / 2m ikx i k x, t x f t Ce e 2 / 2m ti kx k x, t Ce
Example: free electron (cont.) 2 / 2m ti kx k x, t CeWhat is its wavelength? Is a plane wave! Cei kx k 2 / 2m t 2 / 2m ti k x k x , t Ceei k 1 k 2 What is its frequency? k 2 k 2 1 2 k 2 2 2m 2 mWhat does it look like?
Example: free electron (cont.)What are the differences between these two electron 60.60.40.40.20.200 2 / 2m tis an electron plane wave!i kx k x, t Ce p̂ p xWhat is its momentum?ii x Cei kx t kCei kx t i x2 hp x k px It’s the DeBroglie’s hypothesis!
Example: free electron (cont.) x, t 2 / 2m ti kx k Ceis an electron plane wave! i kx k 2 / 2m t 2 k 2ˆ What is its energy? E i Ce t2mIt’s the quantum 2k 2E hypothesis!2mWhat is the relationship between energy and momentum? 2k 2p kE 2mp2E 2mRelevance of electron plane-wave to solid state electronics: In many crystalline semiconductors, electrons behave as if they arefree (sort of!) and therefore, simple plane wave solutions can beapplied in many problems Simple relationship between energy and momentum allows simplecalculation
A small testAn electron wave with momentum k1What is the sum of the two 03020An electron wave with momentum 0100
Example: semiconductorquantum wellp2E 2mIt looks like quantum mechanics produces the sameresults as classical mechanics for free electronsSo, why do we need quantum mechanics?Example:Semiconductor quantum wells- Application: diode lasers used in printers,CD players; 2D HEMT (high-electronmobility transistor)- Consist of layers of differentsemiconductor alloys- Electrons behave like in a simplepotential well, called “quantum well”
Example: hydrogen-like atomThe Schroedinger equation2 2mZe 0 2 2 E r We can get rid of constants by using the "natural" unit:Length: a B 2me 22eEnergy: Ry 2a BUse spherical coordinate: it’s the natural symmetry of the problem2 2 2 1 1 2 2 sin 222 rr r r sin r sin 2Separation of radial andangular variables R r Y , r 2
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
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
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
1. Quantum bits In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics.
Quantum Theory of Light 38.1 Quantum Theory of Light 38.1.1 Historical Background Quantum theory is a major intellectual achievement of the twentieth century, even though we are still discovering new knowledge in it. Several major experimental ndings led to the revelation of quantum theory or quantum mechanics of nature. In nature, we know that
Quantum Integrability Nekrasov-Shatashvili ideas Quantum K-theory . Algebraic method to diagonalize transfer matrices: Algebraic Bethe ansatz as a part of Quantum Inverse Scattering Method developed in the 1980s. Anton Zeitlin Outline Quantum Integrability Nekrasov-Shatashvili ideas Quantum K-theory Further Directions
Quantum Field Theory I Chapter 0 ETH Zurich, HS14 Prof. N. Beisert 18.12.2014 0 Overview Quantum eld theory is the quantum theory of elds just like quantum mechanics describes quantum particles. Here, a the term \ eld" refers to one of the following: A eld of a classical eld
A02 x 2 One mark for the purpose, which is not simply a tautology, and one for development. e.g. The Profit and Loss Account shows the profit or loss of FSC over a given period of time e.g. 3 months, 1 year, etc. (1) It describes how the profit or loss arose – e.g. categorising costs between cost of sales and operating costs/it shows both revenues and costs (1) (1 1) (2) 3(b) AO2 x 2 The .
