Derivation of Quantum Mechanics

Transcription

1 Derivation of Quantum Mechanics Andrew Forrester January 28, 29 Contents 1 Explanation 1 2 Derivation of Quantum Mechanics... and Quantum Field Theory Questions Development Motivation for the Schrödinger Equation 2 4 Symmetry Transformations on Operators Vector Operators Under Rotations Quick Quantum Derivations 5 6 Perturbation Theory 7 1 Explanation The Notes and Outlines provide definitions and statements regarding the topics. The Derivations provide proofs for the statements. 2 Derivation of Quantum Mechanics... and Quantum Field Theory Ideally, this derivation should show incorrect proposals as well as the correct ones. It should be a creative trial and error process, not a presentation of fact after obvious fact. (Also, mentioning how out-of-order the historical developments were will allow the reader to see that the ideas involved really are tricky and that to make a discovery one may have to play with the ideas for a while before coming upon the most logical progression. Sometimes even the ordering of experimental or physical discoveries may seem out of order.) 2.1 Questions Motivation and Context: What are the problems that lead to the development of QM? Which problems does QM solve? What are the various uses of QM? What new questions does QM create? What are the limits of QM? 2.2 Development Quantized EM Radiation: Quantum nature of energy from blackbody radiation proposed/discovered. Continuation of the essential use of harmonic oscillators in physics with Planck s model. (!!) OR is it more correct to say that Planck s conclusion was just that the energy was proportional to frequency E = hν? (and matter is quantized in the form of atoms which correspond to harmonic oscillators of particular quantized frequencies?) 1

2 Quantum of EM Radiation: Particle nature of light discovered and proposed as photon concept. ((!!) LOOK into dis/proof for this theory.) Einstein s energy-frequency (angular tempofrequency) equation: E = ω = hν. Photoelectric Effect Particle-Wave Duality: Wave nature of particles proposed/discovered. de Broglie s momentum-wavenumber (angular spatiofrequency) equation: p = k = h/λ. Davisson-Germer Experiment (Bragg law for electrons diffracting off nickel crystal) Developing Schrödinger Equation using waves ( schr2.html#c1) (quantum mechanics: matrix versus wave mechanics, into transformation/transition? theory and relativistic quantum theory) 3 Motivation for the Schrödinger Equation Particle theory of matter (atoms and particles obeying Newton s Laws in absolute space and time) Vs. Field theory of electrodynamics (continuous vector fields obeying Maxwell Equations in Aether) These two theories would have to be connected smoothly somehow to form a theory of interaction and exchange between matter and electromagnetic radiation accounting for black-body radiation, the photoelectric effect and other photochemical reactions, light scattering (Thomson, Compton and inverse Compton), atomic spectra Before Planck, while seeking to link these two physics together, imprecise and even inadmissible conclusions were in fact arrived at in respect of the energy equilibrium between matter and radiation in a thermally insulated medium: matter, it came to be said, must yield all its energy to the radiation and so tend of its own accord to absolute zero temperature! This absurd conclusion had at all costs to be avoided. - [1] de Broglie... Planck realized the way of avoiding it: instead of assuming, in common with the classical wave theory, that a light source emits its radiation continuously, it had to be assumed on the contrary that it emits equal and finite quantities, quanta. The energy of each quantum has, moreover, a value proportional to the frequency v of the radiation. It is equal to hv, h being a universal constant since referred to as Planck s constant. (a.k.a. the quantum of action) - [1] de Broglie The existence of radiation quanta thus implies the corpuscular concept of light. On the other hand, as shown by Jeans and H. Poincaré, it is demonstrable that if the motion of the material particles in light sources obeyed the laws of classical mechanics it would be impossible to derive the exact law of black-body radiation, Planck s law. It must therefore be assumed that traditional dynamics, even as modified by Einstein s theory of relativity, is incapable of accounting for motion on a very small scale. The existence of a granular structure of light and of other radiations was confirmed by the discovery of the photoelectric effect. If a beam of light or of X-rays falls on a piece of matter, the latter will emit rapidly moving electrons. The kinetic energy of these electrons increases linearly with the frequency of the incident radiation and is independent of its intensity. This phenomenon can be explained simply by assuming that the radiation is composed of quanta hv capable of yielding all their energy to an electron of the irradiated body: one is thus led to the theory of light quanta proposed by Einstein in 195 and which is, after all, a reversion to Newton s corpuscular theory, completed by the relation for the proportionality between the energy of the corpuscles and the frequency. A number of arguments were put forward by Einstein in support of his viewpoint and in 1922 the discovery by A. H. Compton of the X-ray scattering phenomenon which bears 2

3 his name confirmed it. Nevertheless, it was still necessary to adopt the wave theory to account for interference and diffraction phenomena and no way whatsoever of reconciling the wave theory with the existence of light corpuscles could be visualized.... Planck was led to assume that only certain preferred motions, quantized motions, are possible or at least stable, since energy can only assume values forming a discontinuous sequence. This concept seemed rather strange at first but its value had to be recognized because it was this concept which brought Planck to the correct law of black-body radiation and because it then proved its fruitfulness in many other fields. Lastly, it was on the concept of atomic motion quantization that Bohr based his famous theory of the atom. For de Broglie s research: The necessity of assuming for light two contradictory theories - that of waves and that of corpuscles - and the inability to understand why, among the infinity of motions which an electron ought to be able to have in the atom according to classical concepts, only certain ones were possible: such were the enigmas confronting physicists at the time I resumed my studies of theoretical physics. When I started to ponder these difficulties two things struck me in the main. Firstly the lightquantum theory cannot be regarded as satisfactory since it defines the energy of a light corpuscle by the relation W = hv which contains a frequency v. Now a purely corpuscular theory does not contain any element permitting the definition of a frequency. This reason alone renders it necessary in the case of light to introduce simultaneously the corpuscle concept and the concept of periodicity. On the other hand the determination of the stable motions of the electrons in the atom involves whole numbers, and so far the only phenomena in which whole numbers were involved in physics were those of interference and of eigenvibrations. That suggested the idea to me that electrons themselves could not be represented as simple corpuscles either, but that a periodicity had also to be assigned to them too. I thus arrived at the following overall concept which guided my studies: for both matter and radiations, light in particular, it is necessary to introduce the corpuscle concept and the wave concept at the same time. In other words the existence of corpuscles accompanied by waves has to be assumed in all cases. However, since corpuscles and waves cannot be independent because, according to Bohr s expression, they constitute two complementary forces of reality, it must be possible to establish a certain parallelism between the motion of a corpuscle and the propagation of the associated wave. The first objective to achieve had, therefore, to be to establish this correspondence.... Here again it is demonstrated that the group velocity of the waves is equal to the velocity of the corpuscle. The parallelism thus established between the corpuscle and its wave enables us to identify Fermat s principle for the waves and the principle of least action for the corpuscles (constant fields). Fermat s principle states that the ray in the optical sense which passes through two points A and B in a medium having an index n(xyz) varying from one point to another but constant in time is such that the integral B A n dl taken along this ray is extreme. On the other hand Maupertuis principle of least action teaches us the following: the trajectory of a corpuscle passing through two points A and B in space is such that the integral B A p dl taken along the trajectory is extreme, provided, of course, that only the motions corresponding to a given energy value are considered.... It follows that Fermat s and Maupertuis principles are each a translation of the other and the possible trajectories of the corpuscle are identical to the possible rays of its wave. So de Broglie s idea of particle-wave duality was that the possible paths of a particle associated with a wave are identical to the possible rays of the wave. This seems to be different from the probability interpretation of waves (maybe it s not contradictory), and keeps a mechanistic perspective

4 to the fine work done by Schrödinger. It is based on wave propagation equations and strictly defines the evolution in time of the wave associated with a corpuscle.... if in fact it is assumed that the light energy is carried by light corpuscles, photons, then the photon density in the wave must be proportional to the intensity. 4

5 Lennard, Wiechert, Larmor, Lorentz, Lorenz, Zeeman Notes from Quantum Mechanics: Concepts and Applications Complementarity principle: Waves are particles Black-body radiation, Photoelectric effect, Compton effect, Pair production Planck s postulate, Einstein s photon Particles are waves Davisson-Germer experiment, Thomson experiment, double-slit experiments de Broglie s hypothesis 4 Symmetry Transformations on Operators Given an observable Q that has eigenstates q i with eigenvalues q i, [Q, H] = implies... q 1 H q 2 = implies that transitions between q 1 and q 2 are forbidden. WHY? For a one-parameter subgroup of continuous symmetries g(ɛ) = exp( iɛg), the change of A to first order in ɛ is δa = A A = (I iɛg)a(i + iɛg) A = iɛ[a, G] 4.1 Vector Operators Under Rotations... [J i, V j ] = i k ɛ ijkv k 5 Quick Quantum Derivations Bohr Atom Energies, and Bohr Radius Using the Bohr assumptions, the angular momentum is quantized L = r p = rmv = n and the centripetal force for the electron s uniform circular motion is provided by the Coulomb force F centripetal = F electric, so m v2 r = k e2 r 2 r = ke2 mv 2 = r2 r = mv2 r 2 ke 2 = 2 n 2 mke 2 an2 ( ) E = 1 2 mv2 k e2 r = 1 2 k e2 r k e2 r = k e2 2r = k2 me4 2 2 n 2 We defined the Bohr radius a to be a 2 kme 2 5

6 1687 Sir Isaac Newton s Philosophiae Naturalis Principia Mathematica published (Newton s laws of motion and his law of universal gravitation appear) 188 John Dalton s A New System of Chemical Philosophy published (modern atomic theory of elements proposed to explain why elements always react in simple proportions, and why certain gases dissolved better in water than others) 1865 James Clerk Maxwell s A Dynamical Theory of the Electromagnetic Field published (Maxwell Eqns appear) 1869 Dmitri Mendeleev s The Dependence Between the Properties of the Atomic Weights of the Elements published (modern periodic table appears and is used by Mendeleev to predict properties of undiscovered elements) 1874 electron as unit of charge posited by G. Johnstone Stoney, then he developed a theory of the electron and estimates its mass (1894? 1 ). 188s? Heinrich Hertz provides experimental basis for Maxwell s electrodynamic wave theory 1886 cathode rays and canal rays discovered by Eugene Goldstein; explained as negative (electron?) particles and positive (proton?) particles 1887 photoelectric effect discovered by Heinrich Hertz (1899: Lenard demonstrated the cause to be emission of electrons at a certain velocity from the negatively charged body; velocity increases with frequency, number of electrons increases with intensity of light; stressed that this phenomenon was not in good agreement with the then prevailing concepts.) 1897 electron and its subatomic nature discovered by Joseph J. Thomson, and he proposes plum-pudding model of atom; Pieter Zeeman shows that light is radiated by the motion of charged particles in an atom (Zeeman effect, using Lorentz electron E&M theory) 191 Max Planck s On the Law of Distribution of Energy in the Normal Spectrum published (Planck s Law of black-body radiation appears: he discovered his equation was a good interpolation between Rayleigh-Jean s Law [correct for low frequencies] and Wien s Law [correct for high frequencies], and was able to find assumptions that would generate his equation:...) 199 nuclear theory of the atom suggested by the gold foil experiment of Ernest Rutherford - the positive charge of an atom and most of its mass is concentrated in a nucleus at the centre of the atom 1913 quantized circular solar system model of atom proposed by Niels Bohr 1918 proton discovered by Ernest Rutherford 2 (alpha particles shot into nitrogen gas, hydrogen nuclei detected; suggested hydrogen nucleus is an elementary particle) 1922 Stern-Gerlach experiment 1923 Compton effect Wave nature of matter proposed by Louis de Broglie 1925 quantum mechanics (matrix mechanics form) developed by Werner Heisenberg 1925 elementary particle spin postulated by Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit (both angular momentum and magnetic moment, with g = 2 correction made by Llewellyn Thomas using Thomas precession) 1926 quantum mechanics (wave mechanics form) developed by Erwin Schrödinger 1927 uncertainty principle stated by Werner Heisenberg 193 Paul Dirac Table 1: Relevant (Approximate and Simplified) Historical Development 6

7 Bohr Magneton We derive the electron orbit magnetic dipole moment and thereby define the Bohr magneton. Using the Bohr assumptions, the electron circulates in a flat orbit of radius r, with quantized angular momentum L = r p = mrv = n (it isn t l(l + 1)???), and µ = IA = ( e v 2πr ) (πr 2 ) = evr 2 = e 2m L = e l(l + 1) µb l(l + 1) 2m (I got this from WAIT A MINUTE! ISN T L = n IN THE BOHR MODEL, NOT L = l(l + 1)??? We defined the Bohr magneton µ B to be Note: When we include spin, we get Compton Effect µ b e 2m µ = e 2m L g e 2m S = µ B L g µ B S λ = λ λ = h (1 cos θ) m e c Uses conservation of energy-momentum, p γ = E/c, E e = m e 2 c 4 + p e 2 c 2, and the law of cosines. 6 Perturbation Theory See Quantum HW Phys 221B Prob Set 11 (2-1-6) Prob 7.1 ψ n = = ψn + ψ 1 n ψn m n + ψ 2 n H 1 mn E m E n ψm Hmn 1 ψ m H 1 ψ n + Hln 1 Hml 1 (E m n l n l E n) (Em En) H1 nnhmn 1 ψ (Em En) 2 m E 1 n = H 1 nn H 1 mn En 2 = E m n m En References [1] Louis de Broglie: The Wave Nature of the Electron: Nobel Lecture, December 12, 1929, (http: //nobelprize.org/nobel_prizes/physics/laureates/1929/broglie-lecture.pdf, Accessed January 27) 7