Modern Physics But Mostly Quantum Mechanics

List of Figures1.1Black-Body Radiation (Source: Wikimedia Commons; Author: DarthKule) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1610.1 Spherical Coordinate System (Source: Wikimedia Commons; Author:Andeggs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19910.2 Spherical Coordinate System Used in Mathematics (Source: Wikimedia Commons; Author: Dmcq) . . . . . . . . . . . . . . . . . . . . . . 20010.3 Volume Element in Spherical Coordinates (Source: Victor J. Montemayor, Middle Tennessee State University) . . . . . . . . . . . . . . 20311.1 Stern-Gerlach Experiment (Diagram drawn by en wikipedia TheresaKnott.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23611.2 Results of Stern-Gerlach Experiment with and without a non-uniformexternal magnetic field (Source: Stern and Gerlach’s original paper[Gerlach and Stern, 1922, p.350]) . . . . . . . . . . . . . . . . . . . . 23711

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Chapter 1Classical Physics vs. ModernPhysicsVery roughly speaking, Classical Physics consists of Newtonian Mechanics and thetheory of Electricity and Magnetism by Maxwell, while Modern Physics typicallymeans Relativity and Quantum Mechanics.Classical Physics Newtonian Mechanics Maxwell’s Electricity and MagnetismModern Physics Relativity Quantum Mechanics1.1 Extreme Conditions, Precise Measurements,and New PhysicsQuantum Mechanics and the Theory of Relativity emerged early in the 20th century when technological advances were beginning to make laboratory experimentsunder extreme conditions possible. The same advances also made high-precisionmeasurements readily available. Under these extreme conditions, physicists foundthat Newtonian Mechanics no longer held. One such condition is a microscopicphenomenon such as the behavior of an electron, and another phenomenon involves13

14CHAPTER 1. CLASSICAL PHYSICS VS. MODERN PHYSICSobjects moving at speeds comparable to that of light. The former led to the discoveryof Quantum Mechanics, and the latter to Special Relativity1 .Two Types of Extreme Circumstances1. Microscopic Phenomena (atomic and subatomic)Tunneling: A moving electron can go through a potential barrier which requiresmore energy than the kinetic energy of the electron.Wave-particle duality: Microscopic particles such as an electron sometimesbehaves like a wave showing interference and superposition.2. At High Speeds (near the speed of light denoted by c)Fast moving particles live longer.Gallilean transformation does not work.In this book, we will only deal with Quantum Mechanics, and the following listshows some of the key events, discoveries, and theories that contributed to the birthof quantum mechanics.Historical Events, Discoveries, and Theories1. Black-body Radiation: (Quantization, Planck’s Postulate)2. Particle-Like Properties of Radiation: (Wave-Particle Duality)3. Einstein’s Quantum Theory of Light: A bundle of energy is localized in a smallvolume of space.4. Atomic spectra indicating discrete energy levels E1 , E2 , E3 , . . . as opposed tocontinuous distribution of energy levels E [0, )21There are two distinct theories of relativity; the Special Theory of Relativity and the GeneralTheory of Relativity. It is the Special Theory of Relativity that deals with high-speed phenomena.The General Theory of Relativity is an attempt to explain gravity geometrically. We only concernourselves with the Special Theory, which is a theory of space and time that successfully accountsfor the deviations from Newtonian Mechanics for fast-moving objects.2We will later see in Section 7.6.1 that E 0 is impossible in some quantum mechanical systemssuch as a simple harmonic oscillator.

151.2. BLACK-BODY RADIATION1.2 Black-Body RadiationA black body is an idealization of “real-life” black objects in that it does not reflectany light; i.e. perfectly black. When an object is perfectly black, all the radiationcoming from the object is due to emission, and the pattern of emission depends onlyon its temperature. The thermal emission patterns are shown for 3000 K(red), 4000K (green), and 5000 K (blue) in Figure 1.1. The figure also shows the pattern predicted by the classical theory (dark brown). As you can see, the classical theorypredicts that the emitted radiation increases rapidly without bound as the frequencyincreases. This phenomenon is called ultraviolet catastrophe, which does not agreewith the actual observation.In order to circumvent this problem, in 1901 Max Planck found a mathematicalexpression that fit the data, in which he assumed that energy levels are not continuously distributed but are discrete, with each level being an integer multiple of aquantity called Planck constant denoted by h. The Planck constant is a proportionality constant between the energy E and the frequency ν such thatE hν.(1.1)For curiosity, Planck’s formula for the spectral radiance I(ν, t), the energy per unittime (or the power) radiated per unit area of emitting surface in the normal directionper unit solid angle per unit frequency by a black body at temperature T , is givenby2hν 31I(ν, T ) 2 hν;c e kt 1(1.2)where ν is the frequency of the radiation, h 6.62606957 10 34 (Js: joule-seconds)is the Planck constant, c 299792458 (ms 1 : meters per second) is the speed oflight in vacuum, and k 1.3806488 10 23 (m2 kg s 2 K 1 : meter2 kilograms persecond per second per Kelvin) is the Boltzmann constant.1.3 Quantum Theory of LightWhile it was undeniable that light behaved as a wave, a physical phenomenon calledphotoelectric effect was not consistent with this view.

16CHAPTER 1. CLASSICAL PHYSICS VS. MODERN PHYSICSUVVISIBLEINFRARED14Spectral radiance (kW · sr-¹ · m-² · nm-¹)5000 K12Classical theory (5000 K)10864000 K423000 K000.511.522.53Wavelength ( m)Figure 1.1: Black-Body Radiation (Source: Wikimedia Commons; Author: DarthKule)

171.3. QUANTUM THEORY OF LIGHT1.3.1Photoelectric EffectPhotoelectric effect is a phenomenon where electrons, called photoelectrons, areejected from a metal surface when light shines on it. The following two observations were made. The intensity of the incident light had no effect on the maximum kinetic energyof the photoelectrons. The brightness or dimness of the light had no effect onthe maximum kinetic energy if the frequencies are the same. No electron was ejected by light with frequencies below a certain cutoff value,called the threshold frequency.These were not consistent with the wave nature of light. First, the classical theory predicted that light with higher intensity ejects electrons with greater energy.Secondly, according to the wave picture, light with lower frequency can also ejectelectrons except that it takes longer.1.3.2Einstein’s Idea of QuantizationThese discrepancies were resolved by Einstein’s idea of light quantum, which claimsthat radiation energy is not continuously distributed over the wavefront, but is localized forming bundles of energy or particles. They were later called photons. Einsteinpostulated that a single photon interacts with a single electron transferring its energyto the electron instantaneously. The amount of energy E carried by the photon isgiven by the Planck’s formula(1.3)E hν.Another important relation, which is a direct consequence of (1.3) isp h;λwhere p is the momentum and λ is the wavelength.3(1.4)3

18CHAPTER 1. CLASSICAL PHYSICS VS. MODERN PHYSICS1.4 de Broglie-Einstein PostulatesEinstein showed that light, which had classically been regarded as an electromagneticwave, can also be viewed as a collection of particles, later called photons. This isknown as wave-particle duality, i.e. light possesses both a wave nature and a particlenature.A French physicist named de Broglie tried to generalize this idea. As light, classicallya wave, is also particle-like, he conjectured that things such as an electron, so farconsidered to be a particle, could also possess a wave-nature. The wave associatedwith a particle is called de Broglie wave or de Broglie’s matter wave.In order to compute the frequency and the wavelength of the matter wave, de Broglieassumed that Einstein’s relations (1.3) and (1.4) also apply here.E hνp hλ(1.7)The relations (1.7) are known as de Broglie-Einstein relations.The correctness of these formulas were soon confirmed by two American physicistsDavisson and Germer, who confirmed that the interference pattern for an electronbeam reflected from a single crystal of nickel fitted perfectly to the de Broglie-Einsteinrelations. Note that an interference phenomenon is possible only if electrons exhibita wave nature.Development of Quantum Mechanics was an attempt to merge this wave-particleduality and classical Newtonian Mechanics in one equation as we will see in Chapter5.The relation (1.4) is derived from (1.3) and the relativistic energy formula()2E 2 mc2 (pc)2 ;(1.5)where m is the rest mass, and c is the speed of light. Because m 0 for a photon, we haveE 2 (pc)2 E pc E pνλ hν pνλ p h.λ(1.6)

Chapter 2Mathematical PreliminariesThis chapter is a summary of the mathematical tools used in quantum mechanics.It covers various important aspects of basic linear algebra, defines linear operators,and sets convenient notations for the rest of the book.2.1 Linear Vector SpacesConsider the three dimensional Cartesian space R3 equipped with the x-, y-, and zaxes. We have three unit vectors, perpendicular/orthogonal to one another, denotedby (i, j, k), (ı̂, ȷ̂, k̂), or (x̂, ŷ, ẑ). In the component notation, we have i (1, 0, 0),j (0, 1, 0) and k (0, 0, 1). Any point p in this space identified by three coordinates x, y, and z such that p (x, y, z) can also be regarded as a vector xi yj zkor (x, y, z) in the component notation. Note that we use (x, y, z) both for the coordinates and components by abuse of notation. You must be familiar with sucharithmetic operations as addition, subtraction, and scalar multiplication for vectors;which are basically component-wise operations1 . This is a canonical example of alinear vector space, and you should keep this example in mind as you study thefollowing definition of a linear vector space. If you are not so inclined, you can safelyskip this definition and still can understand the rest of this book fully.Definition 2.1 (Linear Vector Space)A linear vector space V is a set {v 1 , v 2 , v 3 , . . .} of objects called vectors for which“addition” and scalar multiplication are defined, such that1To be more precise, these operations can be done component-wise though there are otherequivalent definitions/formulations for these arithmetic operations.19

20CHAPTER 2. MATHEMATICAL PRELIMINARIES1. Both addition and scalar multiplication generate another member of V. Thisproperty is referred to as “closure” under addition and scalar multiplication.2. Addition and scalar multiplication obey the following axioms.Axioms for Addition: Consider v i , v j , and v k taken from V.(i) v i v j v j v i(commutativity)(ii) v i (v j v k ) (v i v j ) v k(associativity)(iii) There exists a unique null vector, denoted by 0, in V such that0 vi vi 0 vi.(existence of the identity element)(iv) For each v i , there exists a unique inverse ( v i ) in V such thatv i ( v i ) 0.(existence of the inverse)aAxioms for Scalar Multiplication: Consider arbitrary vectors v i , v jand arbitrary scalars α, β.(v) α(v i v j ) αv i αv j(vi) (α β)v i αv i βv i(vii) α(βv i ) (αβ)v iFact 2.1 The following facts can be proved using the axioms.1. 0v 02. α0 03. ( 1)v ( v)aThe axioms (i) through (iv) means that a linear vector space forms an abelian group underaddition.Definition 2.2 If the allowed values for the scalars {α, β, γ, . . .} come from somefield F2 , we say the linear vector space V is defined over the field F. In particular, ifF is the field of real numbers R, V is a real vector space. Likewise, if F is the field ofcomplex numbers C, V is a complex vector space.Clearly, R3 is not a complex vector space, but a real vector space. Complex vectorspaces are the canonical vector spaces in quantum mechanics.2There is an abstract mathematical definition of a field. However, it suffices for elementaryquantum mechanics to know the real numbers and the complex numbers respectively form a field.

212.1. LINEAR VECTOR SPACESDefinition 2.3 A set of vectors {v 1 , v 2 , . . . , v n } is linearly independent (LI) ifn αi v i 0 α1 α2 . . . αn 0.(2.1)i 1Definition 2.3 is equivalent to saying that no vector in {v 1 , v 2 , . . . , v n } can beexpressed as a linear combination of the other vectors in the set.Vectors i, j, and k are linearly independent becauseai bj ck 0 or (a, b, c) (0, 0, 0) a b c 0.(2.2)Generally speaking, vectors in R3 that are perpendicular to one another are linearlyindependent. However, these are not the only examples of a linearly independent setof vectors. To give a trivial example, vectors i and i j are linearly independent,though they are not perpendicular to each other. This is becauseai b(i j) 0 (a b, b) (0, 0) a b 0.(2.3)Definition 2.4 A vector space is n-dimensional if it has at most n vectors that arelinearly independent.Notation: An n-dimensional vector space over a field F is denoted by Vn (F). Theones we encounter in this course are usually members of Vn (C), including the caseswhere n .It is often clear what the field F is. In this case, we simply write Vn .Theorem 2.1 Suppose {v 1 , v 2 , . . . , v n } are linearly independent vectors in an ndimensional vector space Vn . Then, any vector v in Vn can be written as a linearcombination of {v 1 , v 2 , . . . , v n }.ProofAs Vn is n-dimensional, you can find a set of n 1 scalars {α1 , α2 , . . . , αn , αn 1 },not all zero, such that(n )αi v i αn 1 v 0.(2.4)i 1Else, {v 1 , v 2 , . . . , v n , v} are linearly independent, and Vn is at least n 1-dimensional,which is a clear contradiction. Furthermore, αn 1 , 0. If not, at least one of

22CHAPTER 2. MATHEMATICAL PRELIMINARIES {α1 , α2 , . . . , αn } is not 0, and yet ni 1 αi v i 0, contradicting the assumption that{v 1 , v 2 , . . . , v n } are linearly independent. We now havev n αii 1αn 1(2.5)vi.We will next show that there is only one way to express v as a linear combinationof {v 1 , v 2 , . . . , v n } in Theorem 2.1.Theorem 2.2 The coefficients of Equation 2.5 are unique.ProofConsider any linear expression of v in terms of {v 1 , v 2 , . . . , v n }v n (2.6)βi v i .i 1Subtracting Equation 2.6 from Equation 2.5,0 n i 1() αi βi v i .αn 1(2.7)However, we know {v 1 , v 2 , . . . , v n } are linearly independent. Hence,n i 1() αi αi αi βi v i 0 βi 0 or βi for all i 1, 2, . . . , n.αn 1αn 1αn 1(2.8)In order to understand Definition 2.5 below, just imagine how any vector in R3can be expressed as a unique linear combination of i, j, and k.Definition 2.5 A set of linearly independent vectors {v 1 , v 2 , . . . , v n }, which canexpress any vector v in Vn as a linear combination (Equation 2.6), is called a basisthat spans Vn . The coefficients of the linear combination βi are called the componentsof v in the basis {v 1 , v 2 , . . . , v n }. We also say that Vn is the vector space spannedby the basis {v 1 , v 2 , . . . , v n }3 .3This means that we can define Vn as the collection of all objects of the form ni 1αi v i .

232.2. INNER PRODUCT SPACESOnce we pick a basis {v 1 , v 2 , . . . , v n }, and express v as a linear combination of the basis vectors v ni 1 βi v i , we get a component notation of v as follows.v n βi v i (β1 , β2 , . . . , βn ).(2.9)i 1With this notation, vector additions and scalar multiplications can be done component by component. For vectors v (β1 , β2 , . . . , βn ), w (γ1 , γ2 , . . . , γn ) and ascalar α, we simply havev w (β1 γ1 , β2 γ2 , . . . , βn γn ) and αv (αβ1 , αβ2 , . . . , αβn ).(2.10)2.2 Inner Product SpacesThis is nothing but a generalization of the familiar scalar product for V3 (R)4 . Namely,it is an extension of the inner product defined on R3 to complex vectors. Readersshould be familiar with the component-wise definition of the usual inner product onR3 , also called dot product, such that a·b (a1 , a2 , a3 )·(b1 , b2 , b3 ) a1 b1 a2 b2 a3 b3 .We can express this as a matrix product of a row vector a and a column vector b.[a1 a2 a3] b1[] b2 a1 b1 a2 b2 a3 b3 ;b3(2.11)where the one-by-one matrix on the right-hand side is a scalar by definition. Thisdefinition of inner product is extended to include complex components by taking thecomplex conjugate of the components of a. So, the extended inner product, whichwe simply call an inner product in the rest of the book, is given by[a1 a2 a3] b1[] b1[] baaa 2 b2 a 1 b1 a 2 b2 a 3 b3 .123b3b3(2.12)Definition 2.6 below is an abstract version of the familiar inner product defined on R3 .This definition contains the inner product defined on R3 , but not all inner productsare of that type. Nevertheless, you should keep the inner

Modern Physics Relativity Quantum Mechanics 1.1 Extreme Conditions, Precise Measurements, and New Physics Quantum Mechanics and the Theory of Relativity emerged early in the 20th cen-tury when technological advances were beginning to make laboratory experiments under extreme conditions possible. The same advances also made high-precision