LECTURE 6 QUANTUM PHYSICS II Instructor: Shih-Chieh Hsu Development of Quantum Mechanics 2 ¨ In 1862, Kirchhoff coined black body radiation or known as cavity radiation ¤ The experiments raised the question of the failure of classical EM theories ¨ In 1887, Heinrich Hertz discovers photoelectric effect ¤ Another experiment raised the concern of the wave and particle nature of the light. Theory of quantization of light 3 ¨ In 1900, Max Planck resolves the blackbody radiation issues by introducing “quantum” concept of the discrete energy element ¤ The energy element is discrete and the energy is proportional to the frequency ¤ The invention of Planck constant h ¨ In 1905, Einstein explained photoelectric effect by using Max Planck’s light quantization concept ¤ Photon is introduced by Gilbert N. Lewis in 1926 More Quantization System 4 ¨ In 1909, Robert Millikan conducted oildrop experiment and showed that electric charge is quantized. ¨ In 1911, Ernest Rutherford’s Gold Foil Experiment disproved the plum pudding model of the atom. Bohr’s classical model of H 5 ¨ Niels Bohr proposed a model of the hydrogen atom that successfully predicted the observed spectra. ¤ The electron of the hydrogen atom moves in a circular orbit around the positive nucleus according to Coulomb’s law and classical mechanics like the planets orbiting around Sun. Flaw in the classical model 6 ¨ Classical EM theory says that an electron in a circular orbit is accelerating, so it would radiate an EM wave and loses its energy. ¨ This atom would quickly collapse as the electron spirals into the nucleus and radiates away the energy. Spectral Lines 7 http://chemistry.tutorvista.com/inorganic-chemistry/spectral-lines.html Hydrogen Spectrum 8 Bohr’s semi-classical model (1913) 9 Energy Quantization In Aton En energy level |ΔEn| = hf=hc/λ 10 Old Quantum Theory Old Quantum Theory 11 ¨ In 1913, Niels Bohr explains the spectra line of the hydrogen atom – using quantization ideas. ¨ 1918-1923 expansion of quantum mechanics research works! e.g. Stern-Gerlach demonstrated spin property of electrons in 1920 Stern Gerlach Three Failures of Classical Physics 12 ¨ Black Body Radiation. ¨ The Hydrogen Atom ¨ Photoelectric Effect. 13 New Quantum Theory New Quantum Theory 14 ¨ In 1924, Louise de Broglie proposes matter wave theory. ¨ In 1925, Matrix Mechanics is invented ¤ Heisenberg Uncertainty was proposed in 1927 Werner Heisenberg Max Born Pascual Jordan Completion of Quantum Mechanics 15 ¨ In 1925, Erwin Schrödinger invented wave mechanics and non-relativistic Schrödinger equation as generalization of de Broglie’s theory ¨ 1927, Paul Dirac began the process of unifying quantum mechanics with special relativity by proposing the Dirac equation for the electron. de Broglie hypothesis 16 ¨ In 1924 Louis de Broglie hypothesized: ¤ Since light exhibits particle-like properties and act as a photon, particles could exhibit wave-like properties and have a definite wavelength. ¨ The wavelength and frequency of matter: h E λ = f = p h ¤ For macroscopic objects, de Broglie wavelength is too small to be observed. Example 1 17 ¨ One of the smallest composite microscopic particles we could imagine using in an experiment would be a particle of smoke or soot. These are about 1 µm in diameter, barely at the resolution limit of most microscopes. A particle of this size with the density of carbon has a mass of about 10-18 kg. What is the de Broglie wavelength for such a particle, if it is moving slowly at 1 mm/s? −34 h h h 6.626×10 Js 13 λ = = = = = 6.626×10− m p mv 10−18 kg ×10−3 m / s 10−18 kg ×10−3 m / s h = 6.626 ´ 10-34 J·s = 4.136 ´ 10-15 eV·s Diffraction of matter 18 ¨ In 1927, C. J. Davisson and L. H. Germer first observed the diffraction of electron waves using electrons scattered from a particular nickel crystal. ¨ G. P. Thomson (son of J. J. Thomson) showed electron diffraction when the electrons pass through a thin metal foils. ¨ Diffraction has been seen for neutrons, hydrogen atoms, and alpha particles. ¨ In all cases, the measured l matched de Broglie’s prediction. X-ray diffraction electron diffraction neutron diffraction Interference and diffraction of matter 19 ¨ If the wavelengths are made long enough (by using very slow moving particles), interference patters of particles can be observed. ¨ These figures show the build up of the electron two-slit interference pattern as the electrons arrive at the detector. Electron microscope 20 ¨ When viewing details of objects with visible light, the details can be resolved only if they are larger than the wavelength of the light. ¨ In electron microscopes, beams of electrons, with small wavelength is used to “see” small objects. Pollen grains Clicker Question 18-1 ¨ The electron microscope is a welcome addition to the field of microscopy because electrons have a __________ wavelength than light, thereby increasing the __________ of the microscope. ¤ longer; resolving power ¤ longer; breadth of field ¤ shorter; resolving power ¤ longer; intensity Classical waves vs. particles 22 ¨ A classical wave behaves like a sound wave. • It exhibits diffraction and interference. • Its energy is spread out continuously in space and time. ¨ A classical particle behaves like a piece of shot. ¤It can be localized and scattered. ¤It exchanges energy suddenly at a point in space. ¤It obeys the laws of conservation of energy and momentum in collisions. ¤It does not exhibit interference or diffraction. Wave-particle duality 23 ¨ Light, normally thought of as a wave, exhibits particle properties when it interacts with matter. • photoelectric effect ¨ Electrons, normally thought of as particles, exhibit the wave properties when they pass near the edges of obstacles. • interference and diffraction ¨ All carriers of p and E exhibit both wave and particle characteristics. ¨ In classical physics, the concepts of waves and particles are mutually exclusive. Wave-particle duality 24 ¨ The classical concepts of waves and particles do not adequately describe the complete behavior of any phenomenon. Everything propagates like a wave and exchanges energy like a particle. ¨ Often the concepts of the classical particle and the classical wave give the same results. ¤ If l is very small, diffraction and interference are not observable. ¤ If there are a lot of particles, they can be treated as a wave. Clicker Question 19-1 ¨ If the wavelength of an electron is equal to the wavelength of a proton, then. 1. the speed of the proton is greater than the speed of the electron 2. the speeds of the proton and the electron are equal 3. the speed of the proton is less than the speed of the electron 4. the energy of the proton is greater than the energy of the electron, 5. both (1) and (4) are correct. h h mv2 p2 h2 λ = = E = = = 2 p mv 2 2m 2mλ Uncertainty principle 26 ¨ If we use light with l to measure the position of an object, x, its uncertainty, Δx, cannot be less than ~l because of diffraction. ¨ If we use photons with pg = h/λ to measure the momentum of an object, p, Δp of the object cannot be less than ~h/λ since the photon changes the momentum of the object upon scattering. ¨ The Heisenberg uncertainty principle states that: It is impossible to simultaneously measure both the position and the momentum of a particle with unlimited precision. h ΔxΔpx ≥ , where ≡ 2 2π Quantum Mechanics (1923) 27 ¨ In quantum mechanics, a particle is described by a wave function y that obeys a wave equation called the Schrödinger equation. 2 2 ∂ − ∇ Ψ(r ,t) + U (r )Ψ(r ,t) = i Ψ(r ,t) 2m ∂t You absolutely do not need to memorize the formula. ¨ The solution of the equation by itself has no physical meaning. However, the probability to find a particle in a certain space- time is: Time-Independent Schrodinger Equation 28 ¨ Solution of the Schrödinger equation. 2 2 ∂ − ∇ Ψ(r ,t) + U (r )Ψ(r ,t) = i Ψ(r ,t) 2m ∂t Wave function 29 ¨ The Schrödinger equation describes a single particle. ¨ The probability density P(x), the probability per unit volume (or length in 1-D), of finding the particle as a function of position is given by P x = ψ 2 x ( ) ( ) ¨ The probability is probability times unit volume, i.e. P(x) Δx Normalization condition 30 ¨ If we have a particle, the probability of finding the particle somewhere must be 1. Therefore the wave function must satisfy the normalization condition. ∞ ∞ ∫ P(x)dx = ∫ ψ 2 (x)dx = 1 −∞ −∞ ¨ For y to satisfy the normalization condition, it must approach zero as |x| approaches infinity. Copenhagen Interpretation 31 ¨ 1927, Bohr, Heisenberg, Pauli had converged to a consensus based on Bohr's concept of complementarity, which states that a physical phenomenon may manifest itself in two different ‘complementary' ways depending on the experiment set up to investigate it. ¨ Thus light, for example, could appear sometimes as a wave and sometimes as a particle. Although mutually exclusive, both pictures were necessary to obtain a full description of the phenomenon. Schrödinger’s Cat 32 ¨ 1935, Austrian physicist Erwin Schrödinger proposed this thought experiment, often described as a paradox, to illustrate what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. Schrödinger’s Poor Cat 33 ¨ The material doesn’t ¨ The material has decayed. decay. The cat is alive The cat has been killed by the poison. Schrödinger’s Cat: Live or Dead 34 ¨ According to the Copenhagen interpretation, the cat is both alive and dead. It exists in a state of “superposition” Probability Calculation for a Classical Particle 35 ¨ A classical point particle moves back and forth with constant speed between two walls at x = 0 and x = 8.0 cm.