A Brief History of Quantum Mechanics

Transcription

1 A Brief History of Quantum Mechanics R. J. Renka Department of Computer Science & Engineering University of North Texas 01/31/2018

2 Wave and particle theories of light In 1630 René Descartes described light as waves in a universal medium (Aristotle s aether). In 1660 Francesco Grimaldi discovered and coined the term diffraction whereby a beam of light shone through a small slit spreads out creating an interference pattern in a manner similar to the behavior of water waves and sound waves. This theory was further developed in the 1670 s by the Dutch physicist Christiaan Huygens and the English physicist Robert Hooke. Between 1666 and 1704 Isaac Newton developed what he termed a corpuscular theory of optics. His hypothesis was based on the idea that only particles could be reflected in straight lines. He explained diffraction and refraction in terms of particles being accelerated laterally.

3 Reflection and refraction For index of refraction n = c/v and phase velocity v = E/p, conservation of energy E and momentum p gives Snell s law: sin(θ i ) sin(θ refr ) = v i v refr = n refr n i = p refr p i.

4 Newton s corpuscular theory The wave theory advocates believed that the waves were made of white light, and the color spectrum seen through a prism is due to corruption within the glass. Newton disproved this with an experiment involving two prisms configured so that the spectrum from the first was recomposed into white light by the second. He concluded that light is composed of colored particles that combine to appear white. He introduced the term color spectrum and chose to divide it into the seven rainbow colors red, orange, yellow, green, blue, indigo, and violet, each with a unique angle of refraction. (He chose 7 because that s a mystical number). Newton also experimentally determined that color is a property of reflected light rather than a property of the objects that reflect the light.

5 Newton s crucial experiment

6 Young s double-slit experiment Newton s theory was dominant until 1804 when Thomas Young described a double-slit experiment in which constructive and destructive interference of a pair of circular waves produces a checkerboard pattern of high and low intensity. Using Newton s data, Young accurately calculated the wavelength range of visible light. By 1821 Young and French engineer Augustin-Jean Fresnel had shown that light waves are entirely transverse. This provided an explanation of polarization. The wave theory gained wider acceptance with experiments demonstrating diffraction. x θ P

7 Electromagnetic radiation In 1831 Michael Faraday discovered that a magnet moving across a copper wire induces a current. In 1864 James Clerk Maxwell developed an electromagnetic wave equation by combining Faraday s law of induction with Carl Friedrich Gauss s laws concerning electric and magnetic fields and Andre-Marie Ampere s law relating magnetic fields to electric current. Maxwell calculated the wave velocity to be the speed of light and concluded that light is a form of electromagnetic radiation a transverse wave (in the aether) consisting of oscillating electric and magnetic fields. Infrared and ultraviolet radiation had been discovered in 1800 and 1801, respectively, and radio waves, X-rays, and gamma rays were discovered in 1888, 1895, and 1900, respectively. In 1887 Americans Albert Michelson and Edward Morley conducted an experiment showing that the speed of light is the same in all directions, independent of the earth s movement through the aether which therefore does not exist.

8 Electric and magnetic fields Lorenz Force Law: The force on a particle with charge q is F = q(e + v B) for electric field E (Newtons/Coulomb) and magnetic field B (teslas = newtons/m-amp). The magnetic field is created by a moving charge (current) or a magnetic dipole. The electric field experienced by a positive point charge at x 0 due to a point charge q 1 is the Coulomb force per unit charge E(x 0 ) = F/q 0. This follows from Coulomb s Law: F = 1 4πɛ 0 q 0 q 1 x 1 x 0 2 x 0 x 1 x 0 x 1.

9 Maxwell s equations Gauss: E = ρ ɛ 0, B = 0 Faraday: E = B t E Ampere: B = µ 0 (J + ɛ 0 t ) ρ = charge density (Coulombs/volume) J = current density (amps/area) ɛ 0 = permitivity of free space (C 2 /newton-m 2 ) µ 0 = permeability of free space (newton-s 2 /C 2 ) c = 1/ µ 0 ɛ 0 = m/s = speed of light In a vacuum where ρ = 0 and J = 0, the fields u = E and u = B satisfy the linear wave equation 1 2 u c 2 2 t 2 u = 0. This follows from ( u) = ( u) 2 u.

10 Blackbody radiation A blackbody is an idealized object that perfectly absorbs light. However, thermal vibrations of the atoms in the walls cause a blackbody to emit electromagnetic radiation (mostly infrared at normal temperatures). Thus, in a cavity, radiation is absorbed and re-emitted until equilibrium is reached. According to statistical mechanics, the total amount of energy in the cavity, computed by summing over an infinite number of modes, is infinite. In 1900, in order to explain the finite energy associated with blackbody radiation, Max Planck postulated that the energy in the electromagnetic field at frequency ω should be quantized as an integer multiple of ω (for = J-s in SI or MKS units). Planck thought of the energy quantization as arising from properties of the walls of a black body. Einstein argued that the quantization was inherent in the radiation. Thus began a rebirth of the particle theory of light.

11 Photoelectric effect In 1887 Heinrich Hertz noticed that light was capable of ejecting electrons from metal surfaces. In 1901 it was discovered that electrons are only ejected if the light frequency is high enough, and the result is independent of the light intensity. Contrary to the wave theory, increasing the light intensity, and hence the strength of the electric and magnetic fields, increases the number of electrons emitted but not the energy of each electron. In 1905 Einstein published a paper describing the photoelectric effect in terms of light quanta. He predicted that the maximal kinetic energy of the ejected electrons is hν W > 0, where W is the energy barrier that confines electrons to the metal. This was confirmed experimentally. His theory also explained why photoelectrons appeared instantaneously even with low intensity. He won the Nobel prize for this work in The existence of photons was decisively proved by beam splitting experiments in 1986.

12 Properties of the photon Photons have the following properties. Energy E = hν = ω Momentum p = E/c = hν/c = k Rest mass 0 (a nonzero rest mass would imply infinite energy) Charge 0 Size 0 Spin 1 (in units of ) Spin leads to left or right circular polarization (rotation of the electric and magnetic fields) measured in a plane normal to the direction of propagation. It can be generated and detected with filters.

13 Electrons In 1911 Ernest Rutherford put forward his model of an atom in which a small nucleus contains most of the mass, has positive charge nq for atomic number n, and is surrounded by a cloud of n electrons, each having charge q. Some of the electrons of one atom may be shared with another atom to form a molecular bond. Basic to the theory is that electrons are particles. In 1913 Niels Bohr postulated that electrons orbit the nucleus with quantized angular momentum n for integer n, and could move from one orbit to another by absorbing or emitting a photon with energy ω. He received the 1922 Nobel prize for his work. It was later discovered that electrons have an intrinsic angular momentum (spin = /2), analogous to rotation about an axis, that adds to the total angular momentum.

14 Electrons continued The Planck-Einstein relation for the energy of a photon is E = ω. From special relativity, E 2 = p 2 c 2 + m 2 c 4 so that E = pc for the massless photon. The linear momentum is thus p = k for wavenumber (spatial frequency) k = 2π/λ from c = λω/(2π). In 1923 Louis de Broglie postulated that an electron (and any other particle) can be described by a matter wave with spatial frequency k related to momentum by p = k. Imagine a wave superimposed on the classical (circular) trajectory of the electron. The orbit consists of an integer number of periods: 2πr = n(2π/k) for radius r. The de Broglie hypothesis gave an alternative to Bohr s quantization of angular momentum (rp = n ) as an explanation of the allowed energies of hydrogen. Double-slit experiments involving one electron at a time display the interference pattern of a wave.

15 Stern-Gerlach experiment 1922 experiment by Otto Stern and Walther Gerlach in Frankfort: silver atoms travel through an inhomogeneous magnetic field and are deflected up or down depending on their spin. 1: furnace. 2: beam of silver atoms. 3: inhomogeneous magnetic field. 4: expected result. 5: what was actually observed.

16 Cascaded Stern-Gerlach mesurements The Stern-Gerlach experiment led to the discovery that an electron has intrinsic angular momentum (spin) with values /2 or /2 measured along any axis and which contributes to the magnetic dipole moment of an atom about which it is rotating. Spin is modeled by a qubit as demonstrated by the experiment below in which the devices are oriented along the z axis to measure spin in the computational basis ( + Z = 0, Z = 1 ) or x axis to measure in the + / basis ( + X = +, X = ).

17 Schrödinger and Heisenberg In 1925, Werner Heisenberg, Max Born, and Pascual Jordan proposed a model of quantum mechanics based on treating the position and momentum of a particle as matrices of size. Heisenberg received the 1932 Nobel prize. In 1926, Erwin Schrödinger published four papers in which he proposed a wave theory of quantum mechanics, along the lines of the de Broglie hypothesis. He showed how the waves evolve in time and showed that the energy levels of an atom can be understood as eigenvalues of a Hamiltonian operator. He also showed that the wave model was equivalent to Heisenberg s matrix model. He shared the 1933 Nobel prize with Paul Dirac.