How Electrons Spin
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Path Integrals in Quantum Mechanics
Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases. -
Philosophical Rhetoric in Early Quantum Mechanics, 1925-1927
b1043_Chapter-2.4.qxd 1/27/2011 7:30 PM Page 319 b1043 Quantum Mechanics and Weimar Culture FA 319 Philosophical Rhetoric in Early Quantum Mechanics 1925–27: High Principles, Cultural Values and Professional Anxieties Alexei Kojevnikov* ‘I look on most general reasoning in science as [an] opportunistic (success- or unsuccessful) relationship between conceptions more or less defined by other conception[s] and helping us to overlook [danicism for “survey”] things.’ Niels Bohr (1919)1 This paper considers the role played by philosophical conceptions in the process of the development of quantum mechanics, 1925–1927, and analyses stances taken by key participants on four main issues of the controversy (Anschaulichkeit, quantum discontinuity, the wave-particle dilemma and causality). Social and cultural values and anxieties at the time of general crisis, as identified by Paul Forman, strongly affected the language of the debate. At the same time, individual philosophical positions presented as strongly-held principles were in fact flexible and sometimes reversible to almost their opposites. One can understand the dynamics of rhetorical shifts and changing strategies, if one considers interpretational debates as a way * Department of History, University of British Columbia, 1873 East Mall, Vancouver, British Columbia, Canada V6T 1Z1; [email protected]. The following abbreviations are used: AHQP, Archive for History of Quantum Physics, NBA, Copenhagen; AP, Annalen der Physik; HSPS, Historical Studies in the Physical Sciences; NBA, Niels Bohr Archive, Niels Bohr Institute, Copenhagen; NW, Die Naturwissenschaften; PWB, Wolfgang Pauli, Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg a.o., Band I: 1919–1929, ed. A. Hermann, K.V. -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
Magnetism, Angular Momentum, and Spin
Chapter 19 Magnetism, Angular Momentum, and Spin P. J. Grandinetti Chem. 4300 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin In 1820 Hans Christian Ørsted discovered that electric current produces a magnetic field that deflects compass needle from magnetic north, establishing first direct connection between fields of electricity and magnetism. P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Biot-Savart Law Jean-Baptiste Biot and Félix Savart worked out that magnetic field, B⃗, produced at distance r away from section of wire of length dl carrying steady current I is 휇 I d⃗l × ⃗r dB⃗ = 0 Biot-Savart law 4휋 r3 Direction of magnetic field vector is given by “right-hand” rule: if you point thumb of your right hand along direction of current then your fingers will curl in direction of magnetic field. current P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Microscopic Origins of Magnetism Shortly after Biot and Savart, Ampére suggested that magnetism in matter arises from a multitude of ring currents circulating at atomic and molecular scale. André-Marie Ampére 1775 - 1836 P. J. Grandinetti Chapter 19: Magnetism, Angular Momentum, and Spin Magnetic dipole moment from current loop Current flowing in flat loop of wire with area A will generate magnetic field magnetic which, at distance much larger than radius, r, appears identical to field dipole produced by point magnetic dipole with strength of radius 휇 = | ⃗휇| = I ⋅ A current Example What is magnetic dipole moment induced by e* in circular orbit of radius r with linear velocity v? * 휋 Solution: For e with linear velocity of v the time for one orbit is torbit = 2 r_v. -
Samuel Goudsmit
NATIONAL ACADEMY OF SCIENCES SAMUEL ABRAHAM GOUDSMIT 1 9 0 2 — 1 9 7 8 A Biographical Memoir by BENJAMIN BEDERSON Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 2008 NATIONAL ACADEMY OF SCIENCES WASHINGTON, D.C. Photograph courtesy Brookhaven National Laboratory. SAMUEL ABRAHAM GOUDSMIT July 11, 1902–December 4, 1978 BY BENJAMIN BEDERSON AM GOUDSMIT LED A CAREER that touched many aspects of S20th-century physics and its impact on society. He started his professional life in Holland during the earliest days of quantum mechanics as a student of Paul Ehrenfest. In 1925 together with his fellow graduate student George Uhlenbeck he postulated that in addition to mass and charge the electron possessed a further intrinsic property, internal angular mo- mentum, that is, spin. This inspiration furnished the missing link that explained the existence of multiple spectroscopic lines in atomic spectra, resulting in the final triumph of the then struggling birth of quantum mechanics. In 1927 he and Uhlenbeck together moved to the United States where they continued their physics careers until death. In a rough way Goudsmit’s career can be divided into several separate parts: first in Holland, strictly as a theorist, where he achieved very early success, and then at the University of Michigan, where he worked in the thriving field of preci- sion spectroscopy, concerning himself with the influence of nuclear magnetism on atomic spectra. In 1944 he became the scientific leader of the Alsos Mission, whose aim was to determine the progress Germans had made in the development of nuclear weapons during World War II. -
Otto Stern (1888–1969): the Founding Father of Experimental Atomic Physics
Ann. Phys. (Berlin) 523, No. 12, 1045 – 1070 (2011) / DOI 10.1002/andp.201100228 Historical Article Otto Stern (1888–1969): The founding father of experimental atomic physics J. Peter Toennies1, Horst Schmidt-Bocking¨ 2, Bretislav Friedrich3,∗, and Julian C. A. Lower2 1 Max-Planck-Institut f¨ur Dynamik und Selbstorganisation, Bunsenstrasse 10, 37073 G¨ottingen, Germany 2 Institut f¨ur Kernphysik, Goethe Universit¨at Frankfurt, Max-von-Laue-Strasse 1, 60438 Frankfurt, Germany 3 Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, 14195 Berlin, Germany Received 22 September 2011, revised 1 November 2011, accepted 2 November 2011 by G. Fuchs Published online 15 November 2011 Key words History of science, atomic physics, Stern-Gerlach experiment, molecular beams, magnetic dipole moments of nucleons, diffraction of matter waves. We review the work and life of Otto Stern who developed the molecular beam technique and with its aid laid the foundations of experimental atomic physics. Among the key results of his research are: the experimental test of the Maxwell-Boltzmann distribution of molecular velocities (1920), experimental demonstration of space quantization of angular momentum (1922), diffraction of matter waves comprised of atoms and molecules by crystals (1931) and the determination of the magnetic dipole moments of the proton and deuteron (1933). c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Introduction Short lists of the pioneers of quantum mechanics featured in textbooks and historical accounts alike typi- cally include the names of Max Planck, Albert Einstein, Arnold Sommerfeld, Niels Bohr, Max von Laue, Werner Heisenberg, Erwin Schr¨odinger, Paul Dirac, Max Born, and Wolfgang Pauli on the theory side, and of Wilhelm Conrad R¨ontgen, Ernest Rutherford, Arthur Compton, and James Franck on the experimental side. -
Magnetic Moment of a Spin, Its Equation of Motion, and Precession B1.1.6
Magnetic Moment of a Spin, Its Equation of UNIT B1.1 Motion, and Precession OVERVIEW The ability to “see” protons using magnetic resonance imaging is predicated on the proton having a mass, a charge, and a nonzero spin. The spin of a particle is analogous to its intrinsic angular momentum. A simple way to explain angular momentum is that when an object rotates (e.g., an ice skater), that action generates an intrinsic angular momentum. If there were no friction in air or of the skates on the ice, the skater would spin forever. This intrinsic angular momentum is, in fact, a vector, not a scalar, and thus spin is also a vector. This intrinsic spin is always present. The direction of a spin vector is usually chosen by the right-hand rule. For example, if the ice skater is spinning from her right to left, then the spin vector is pointing up; the skater is rotating counterclockwise when viewed from the top. A key property determining the motion of a spin in a magnetic field is its magnetic moment. Once this is known, the motion of the magnetic moment and energy of the moment can be calculated. Actually, the spin of a particle with a charge and a mass leads to a magnetic moment. An intuitive way to understand the magnetic moment is to imagine a current loop lying in a plane (see Figure B1.1.1). If the loop has current I and an enclosed area A, then the magnetic moment is simply the product of the current and area (see Equation B1.1.8 in the Technical Discussion), with the direction n^ parallel to the normal direction of the plane. -
4 Nuclear Magnetic Resonance
Chapter 4, page 1 4 Nuclear Magnetic Resonance Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet. Since then, the splitting of energy levels proportional to an external magnetic field has been called the "Zeeman effect". The "Zeeman resonance effect" causes magnetic resonances which are classified under radio frequency spectroscopy (rf spectroscopy). In these resonances, the transitions between two branches of a single energy level split in an external magnetic field are measured in the megahertz and gigahertz range. In 1944, Jevgeni Konstantinovitch Savoiski discovered electron paramagnetic resonance. Shortly thereafter in 1945, nuclear magnetic resonance was demonstrated almost simultaneously in Boston by Edward Mills Purcell and in Stanford by Felix Bloch. Nuclear magnetic resonance was sometimes called nuclear induction or paramagnetic nuclear resonance. It is generally abbreviated to NMR. So as not to scare prospective patients in medicine, reference to the "nuclear" character of NMR is dropped and the magnetic resonance based imaging systems (scanner) found in hospitals are simply referred to as "magnetic resonance imaging" (MRI). 4.1 The Nuclear Resonance Effect Many atomic nuclei have spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is L =+h II(1). (4.01) The component in the direction of an applied field is Lz = Iz h ≡ m h. (4.02) The external field is usually defined along the z-direction. The magnetic quantum number is symbolized by Iz or m and can have 2I +1 values: Iz ≡ m = −I, −I+1, ..., I−1, I. -
5 the Dirac Equation and Spinors
5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory. -
New Measurements of the Electron Magnetic Moment and the Fine Structure Constant
S´eminaire Poincar´eXI (2007) 109 – 146 S´eminaire Poincar´e Probing a Single Isolated Electron: New Measurements of the Electron Magnetic Moment and the Fine Structure Constant Gerald Gabrielse Leverett Professor of Physics Harvard University Cambridge, MA 02138 1 Introduction For these measurements one electron is suspended for months at a time within a cylindrical Penning trap [1], a device that was invented long ago just for this purpose. The cylindrical Penning trap provides an electrostatic quadrupole potential for trapping and detecting a single electron [2]. At the same time, it provides a right, circular microwave cavity that controls the radiation field and density of states for the electron’s cyclotron motion [3]. Quantum jumps between Fock states of the one-electron cyclotron oscillator reveal the quantum limit of a cyclotron [4]. With a surrounding cavity inhibiting synchrotron radiation 140-fold, the jumps show as long as a 13 s Fock state lifetime, and a cyclotron in thermal equilibrium with 1.6 to 4.2 K blackbody photons. These disappear by 80 mK, a temperature 50 times lower than previously achieved with an isolated elementary particle. The cyclotron stays in its ground state until a resonant photon is injected. A quantum cyclotron offers a new route to measuring the electron magnetic moment and the fine structure constant. The use of electronic feedback is a key element in working with the one-electron quantum cyclotron. A one-electron oscillator is cooled from 5.2 K to 850 mK us- ing electronic feedback [5]. Novel quantum jump thermometry reveals a Boltzmann distribution of oscillator energies and directly measures the corresponding temper- ature. -
1 Introduction 2 Spins As Quantized Magnetic Moments
Spin 1 Introduction For the past few weeks you have been learning about the mathematical and algorithmic background of quantum computation. Over the course of the next couple of lectures, we'll discuss the physics of making measurements and performing qubit operations in an actual system. In particular we consider nuclear magnetic resonance (NMR). Before we get there, though, we'll discuss a very famous experiment by Stern and Gerlach. 2 Spins as quantized magnetic moments The quantum two level system is, in many ways, the simplest quantum system that displays interesting behavior (this is a very subjective statement!). Before diving into the physics of two-level systems, a little on the history of the canonical example: electron spin. In 1921, Stern proposed an experiment to distinguish between Larmor's classical theory of the atom and Sommerfeld's quantum theory. Each theory predicted that the atom should have a magnetic moment (i.e., it should act like a small bar magnet). However, Larmor predicted that this magnetic moment could be oriented along any direction in space, while Sommerfeld (with help from Bohr) predicted that the orientation could only be in one of two directions (in this case, aligned or anti-aligned with a magnetic field). Stern's idea was to use the fact that magnetic moments experience a linear force when placed in a magnetic field gradient. To see this, not that the potential energy of a magnetic dipole in a magnetic field is given by: U = −~µ · B~ Here, ~µ is the vector indicating the magnitude and direction of the magnetic moment. -
Spin the Evidence of Intrinsic Angular Momentum Or Spin and Its
Spin The evidence of intrinsic angular momentum or spin and its associated magnetic moment came through experiments by Stern and Gerlach and works of Goudsmit and Uhlenbeck. The spin is called intrinsic since, unlike orbital angular momentum which is extrinsic, it is carried by point particle in addition to its orbital angular momentum and has nothing to do with motion in space. The magnetic moment ~µ of silver atom was measured in 1922 experiment by Stern and Gerlach and its projection µz in the direction of magnetic fieldz ^ B (through which the silver beam was passed) was found to be just −|~µj and +j~µj instead of continuously varying between these two as limits. Classically, magnetic moment is proportional to the angular momentum (12) and, assuming this proportionality to survive in quantum mechanics, quan- tization of magnetic moment leads to quantization of corresponding angular momentum S, which we called spin, q q µ = g S ) µ = g ~ m (57) z 2m z z 2m s as is the case with orbital angular momentum, where the ratio q=2m is called Bohr mag- neton, µb and g is known as Lande-g factor or gyromagnetic factor. The g-factor is 1 for orbital angular momentum and hence corresponding magnetic moment is l µb (where l is integer orbital angular momentum quantum number). A surprising feature here is that spin magnetic moment is also µb instead of µb=2 as one would naively expect. So it turns out g = 2 for spin { an effect that can only be understood if linearization of Schr¨odinger equation is attempted i.e.