LASER PHYSICS PHY 395M
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This 5 page Class Notes was uploaded by Malvina Orn on Monday September 7, 2015. The Class Notes belongs to PHY 395M at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 68 views. For similar materials see /class/181827/phy-395m-university-of-texas-at-austin in Physics 2 at University of Texas at Austin.
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Date Created: 09/07/15
12304 Units a cgs Gaussian Units and Fundamental constants Atomic physicists have traditionally used cgsGaussian units as for example in the above formulas In this respect they are bad citizens because the various government standards agencies are working hard to standardize on the SI system of units and others systems are strongly discouraged But Gaussian units are still used because they are familiar and because many of the formulas are simpler and heater to write down without meaningless factors of 411780 oating around In this class we will stay with tradition and work mostly with cgsGaussian units These units and a few of the important constants and equations are given by length cm mass g time s energy erg 1 erg 1 g cmzs2 10 7 J charge esu potential statvolt 1 statvolt lt gt 299792458 V magnetic field Gauss 1 Gauss lt gt 10 4 T Coulomb s law F qlqzr Lorentz force law F qE Vc X B Note that in Gaussian units E and B actually have the same dimensions Boltzmann s constant kB 1380650524 X 10 16 ergK Planck s constant211 h 10545716818 X 10 27 erg s magnitude of electron charge e 48032044042 X 10 10 esu electron mass m 9109382616 X 10 28 g Bohr magneton 11B ehzrne 92740094980 X 10 21 ergG proton mass M 16726217129 X 10 24 g unified atomic mass unit 12C mass12 u 16605388628 X 10 24 g velocity of light c 299792458 X 1010 cms exactly Rydberg constant R 2 R00 nre44nh3e 1097373156852573 X 105 enr 1 ne structure constant 00 ezhc 729735256824 X 10 3 inverse ne structure constant 1x 137 0359991124 The values of the constants and their uncertainties result from an analysis of all available data by an international committee called CODATA Committee on Data for Science and Technology These values can be found at the NIST web site at physicsnist govcuu constants The values are given in SI units I have done the conversion into cgsGaussian units to obtain the values given here By convention the numbers given in parenthesis indicate the uncertainty in the last digits For example the Ryberg constant is R 1097373156852503 X 105 cm 1 10973731568525 4 00000000000073 cm 1 Most of the fundamental constants are known to about 1 part in 107 The Rydberg constant is one of the most accurately known constants with an uncertainty of about 1 part in 10 We will come back to the topic of the measurement of the fundamental constants later in this course Note that in the modern system of units c is an exactly de ned constant Therefore one no longer measures the speed of light in terms of defined time and length standards Instead time is defined in terms of periods of vibration on the Cs hyperfine transition and length is defined as the distance light travels in a certain time b Spectroscopic units Atomic and molecular physicists often use units of wavenumbers or cmil for giving transition energies This simply consists of giving the inverse of the wavelength of a transition ie giving 1 l V wavelength A with the wavelength measured in centimeters This is convenient because taking the inverse for a transition energy measured in cm 1 immediately gives the wavelength Also to add energies the wavenumbers simply add For example in hydrogen for the Lyman alpha transition 17192 82200 cm 1 and for the Balmer alpha transition we have 17293 15200 cmil The Lyman beta transition n l gt 3 has a wavenumber which is the sum ofthese 17193 82200 15200 cm 1 97500 cmil Also here are some useful unit conversions 1 cm 1lt gt 29979 GHz 8066 cm 1 lt gt 1 eV c Atomic units Atomic units are also popular in atomic and molecular physics because they yield simple uncluttered equations They are given by length Bohr a0 hzmez 0529177210818 X 10 8 cm energy Hartree ezao 43597441775 X 10 11 erg a 27211 ev speed xc 2187691263373 X 108 cms time aoxc 241888432650516 X 10 17 s frequency xcao 41tcR 4134137320627 X 1016 rads charge e 4 8032044042 X 10 10 esu mass m 9109382616 X 10 28 g electric eld eZa0 51422064244 X 109 Vcm The length and speed units are the radius of the lowest Bohr orbit of hydrogen and the speed of the electron in that orbit The electric eld unit is the eld of a proton at a distance of 1 Bohr The unit of mass is the electron mass not the uni ed atomic mass unit u Note that if you want to accurately describe the energies of the hydrogenic atom in this system of units an additional reduced mass correction is required e g the ground state energy of the hydrogen atom in this system of units is 7 12 MMm The ne structure constant 01 appears repeatedly in atomic physics as a natural scaling factor 012 gives the atomic unit of energy relative to the electron rest mass energy Since 012 E 5 X 1075 the hydrogen atom is nonrelativistic To convert from cgs to atomic units note that h gt 1 e gt 1 and m gt 1 but that c gt 101 513704 Also note that the atomic unit of energy ezao is twice the ground state energy of hydrogen The energy of hydrogen can be expressed in terms of inverse wavelength frequency or energy 2 R le i 1097373 gtlt105 cm391 0 2 a c 2 olel 6 i3289833x1015Hz 2 610 h 2 holel 6 13605eV 2 a0 d Schrodinger equation In 1926 Schrodinger wrote down his equation and solved it for hydrogen Written in atomic units for a hydrogenic atom of nuclear charge Ze and neglecting the reduced mass corrections this equation is v2 2 E Dyx0 1 r I am assuming everyone is very familiar with the solution of this equation so I will only review the barest essentials here The equation is separable in spherical coordinates with the substitution 410941 RrYzmeaP 2 The electron has a magnitude of angular momentum squared lhz with Z a non negative integer and its zcomponent of angular momentum is mh where m is an integer in the range Z The radial wavefunction Rr is found by solving the radial equation R 3R392E Ro 3 r r r where the primes denote derivatives with respect to r The limiting cases of this equation are instructive For r gt 00 Rquot 2ER E 0 so that R E A eXp sr with A a constant and s 2E12 For r gt 011quot 2rR39 1rzR 0 so that R s 0 rl These limiting solutions are illustrated below The combination of the Coulomb potential Vc and the centrifugal potential Vcem forms an effective radial potential ve veg vcema Zr lam28 4 with an attractive well The bound state energies are determined by the requirement that the wavefunctions have the short and long range forms given above ie that the coefficients of the diverging solutions eXp8r large r and 1 small r are zero so that the wavefunctions are normalizable This leads to a ground state solution for each X with no radial nodes and an infinite number of excited state solutions for each X which have some number of oscillations in the central part of the effective potential The exact solution for the radial wavefunction is Rp R p Ce PZplLi f p 5 where p 28r C is a normalization constant and Lil1 is the associated Laguerre function which is a polynomial of order n 1 The energies of the states are given by E Z22n2 6 where n 1 2 3 and for each n X takes on the values 0 1 n l R N eXp 8r Zr e Properties of the solution for hydrogen a degeneracy ofterm 11 Z 0 1 n l m n71 Total degeneracy not including spin 22 1 n2 7 1 b number of radial nodes n 1 spheres 8 number of angular nodes lml cones 9 c mean values ofrquot rn ld3r rn lulr9ql2 lr 1112 3 Vr 1r 1n2 10 note from Virial theorem 2T rVV where T kinetic energy and we find for the Coulomb potential T V 2 12n2 Hence the energy for the hydrogenic atom is
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