DOCSLIB.ORG

Bound-State Solutions of Dirac Equation Plan of the Lecture

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bound-State Solutions of Dirac Equation Plan of the Lecture Lecture 3 Bound-state solutions of Dirac equation Plan of the lecture • Few comments about Dirac equation • Free- and bound-state solutions • Dirac’s spectroscopic notations – Integrals of motion – Parity of states • Energy levels of the bound-state Dirac’s particle • Structure of Dirac’s wavefunction • Radial components of the Dirac’s wavefunction Four-vectors In the relativistic world it is more convenient to work with four-vectors: Contravariant vectors Covariant vectors 휇 푥 = 푡, 푥, 푦, 푧 푥휇 = 푡, −푥, −푦, −푧 휕 휕 휕 휕 휕 휕 휕 휕 휕휇 = , − , − , − 휕 = , , , 휕푡 휕푥 휕푦 휕푧 휇 휕푡 휕푥 휕푦 휕푧 휇 푝 = 퐸, 푝푥, 푝푦, 푝푧 푝휇 = 퐸, −푝푥, −푝푦, −푝푧 Lorentz transformation 푥′휇 = ෍ 푎휈 푥휇 ′ 휈 휇 푥휇 = ෍ 푎휇 푥휇 휈 휈 훾 −훾훽 0 0 훾 훾훽 0 0 −훾훽 훾 훾훽 훾 0 0 푎휈 = 0 0 푎휈 = 휇 0 0 1 0 휇 0 0 1 0 0 0 0 1 0 0 0 1 Klein-Gordon equation Based on the relativistic energy-mass equation: 퐸2 = 푝ҧ2 + 푚2 One can derive Klein-Gordon equation for scalar (zero-spin) relativistic particles: Oscar Klein 휇 2 휕 휕휇 + 푚 휑 푥 = 0 By introducing d'Alembert operator: 휕2 휕휇휕 =⊡= − 휵2 휇 휕푡2 We can re-write Klein-Gordon equation as: ⊡ + 푚2 휑 푥 = 0 Klein-Gordon equation We can derive Klein-Gordon equation for scalar (zero-spin) relativistic particles: 휇 2 휕 휕휇 + 푚 휑 푥 = 0 Free-particle solutions of this equation: Oscar Klein 휑 푥 = 푁 푒−푖 푝푥 = 푁 푒−푖퐸푡+푖풑풓 Allow particles with both positive and negative energy: 퐸 = ± 풑2 + 푚2 And with positive and negative probability density: 푗0 = 2 푁 2 퐸 How do we understand negative-energy solutions? And what is much worse, the negative probability density? Dirac equation We can re-write Klein-Gordon equation yet in the other form: 휇 2 휇 2 휕 휕휇 + 푚 휑 푥 = 0 푝 푝휇 − 푚 = 0 The idea of Dirac was to linearize the Klein-Gordon equation: 휇 2 휇 휈 푝 푝휇 − 푚 = 훾 푝휇 − 푚 훽 푝휈 + 푚 = 0 By comparing both sides of this equation (and only!!!) one can find that: 1 0 0 0 0 −1 0 0 휇 휇 휇 휈 푔 = 훾 = 훽 and 훾 , 훾 = 2푔휇휈 with 휇휈 0 0 −1 0 0 0 0 −1 Plan of the lecture • Few comments about Dirac equation • Free- and bound-state solutions • Dirac’s spectroscopic notations – Integrals of motion – Parity of states • Energy levels of the bound-state Dirac’s particle • Structure of Dirac’s wavefunction • Radial components of the Dirac’s wavefunction Dirac equation for bound electron • Stationary Dirac equation reads (let us add potential): 2 icα V(r) mec 0 (r) E (r) • Its solutions depend, of course, on the particular form of V (r) Ze2 Free particle V(r) 0 Particle in Coulomb V (r) (discussed before) potential r E ( p) m c2 e Positive continuum + mc2 + mc2 Bound (discrete) states! How0 to describe (characterize) discrete bound0 state of Dirac spectrum? - mcBy2 the way: how did we characterize Schr- mc2ödinger spectrum? 2 Negative continuum E ( p) mec Schrödinger equation: Quantum numbers • One needs three quantum numbers (r) (r,,) R (r)Y (,) to define the state of hydrogen nl lml (hudrogen-like) atom: • n = 1, 2, 3… (principal) • l = 0, … n-1 (orbital) Electron is free 0 (au) Energy • ml = -l, …. +l (magnetic) Electron is bound to ion ……….. ……….. 3s (n=3, l=0) 3p (n=3, l=1) 3d (n=3, l=2) -1/18 • The energy depends only on the 2s (n=2, l=0) 2p (n=2, l=1) principal quantum number: -1/8 Z 2 E 0 n 2n2 • i.e. in nonrelativistic theory the states are degenerate (l, m)! -1/2 1s (n=1, l=0) Can we use the same set of quantum numbers (n,l,m) for Dirac spectrum? Constants on motion (1) • For the description of the (stable) atom we need to have a set of quantum numbers which do not change as time evolves. • Let us take some observable (operator which represents some physical quantity) Q and its expectation value in some quantum state: Q Qˆ • To find the general requirement for Q being not dependent on time, let us first derive the (matrix form of) Heisenberg equation of motion: d d i Qˆ Q Qˆ Hˆ ,Qˆ dt dt t Constants on motion (2) • To find the general requirement for being not dependent on time, let us first derive the (matrix form of) Heisenberg equation of motion: • Therefore, if H ˆ , Q ˆ 0 and Q ˆ do not depend (directly) on time, we find: Q d Q 0 dt d d i Qˆ Q Qˆ Hˆ ,Qˆ dt dt t • Expectation value q Q does not change with time and provides us a “good quantum number” for the description of quantum system! Non-relativistic hydrogen • Schrödinger Hamiltonian (r )in spherical (r,, )coordinates: R (r)Y (,) nl lml 2 Ze2 2 1 Lˆ2 Ze2 Hˆ 2 r 2 2 2 2m r 2m r r r 2mr r • Its eigenfunctions: ˆ ˆ ˆ2 ˆ2 ˆ ˆ ˆ ˆ2 ˆ • Operators H , L Z , L commute with each other: [L , H] 0, [Lz , H] 0,[L , Lz ] 0 ˆ2 2 ˆ ˆ • And: L (r) l(l 1) (r), Lz (r) ml (r), H (r) E (r) (n, l, m) are good quantum numbers. … but only in the nonrelativistic case! Relativistic hydrogen 2 Ze2 2 ˆ 2 ˆ Ze 2 H S H icα m c 2m r D r e 0 (r) R (r)Y (,) () (r) R (r)Y (,) nlml sms nl lml sms nlml nl lml • In Dirac’s theory, however, neither the components of L nor L2 commute with Hamiltonian. Instead, one can show that: [Lˆ, Hˆ ] icα p ˆ ˆ σ 0 • The same is true for the spin operator: S 2 0 σˆ [Sˆ, Hˆ ] icα p Spin-orbit interaction (1) ✓ Please, remind yourself discussion from the last ✓ Let us move to the rest frame of electron lecture concerning magnetic dipole moment (we are riding with electron) In classical electrodynamics: | | I A current vector area of the current In the rest frame of electron there is a loop magnetic filed caused by the relative motion of the nucleus (magnetic field of current loop)! q 2 qv 2 q | | I A r r L I T 2r 2m B 0 2r In quantum mechanics, where for electron: q=-e B Ze Zev I e T 2r μˆ Lˆ / , l 0 0 2m Ze e mvr Bohr magneton 2r 2 m Spin-orbit interaction (2) In the rest frame of electron there is a magnetic filed caused by the relative motion of the nucleus (magnetic field of current loop)! B ZeL B 0 (r) L 4r 3m • Electron has spin (intrinsic moment) and, hence, spin magnetic moment: ˆ μˆ s gs 0 S / • Which interacts with external field as: ˆ ˆ ˆ H μˆ s B (r) L S Spin-orbit term! (A more rigorous derivation requires detailed analysis of Dirac equation.) Spin-orbit interaction (3) • Coming back to Dirac equation for the hydrogen-like ions: Ze2 2 icα mec 0 (r) E (r) r ˆ ' ˆ ˆ • Which should include the spin-orbit term: H μˆ s B (r) L S Let us recall commutation relations for the operators L and S: [Lˆ, Hˆ ] icα p This relations indicate that we the form of “good” operator (the one which commutes) with the Hamiltonian: J L S [Sˆ, Hˆ ] icα p Spin-orbit interaction (3) • Coming back to Dirac equation for the hydrogen-like ions: Ze2 2 icα mec 0 (r) E (r) r ' • Which should include the spin-orbit term: Hˆ μˆ B (r) Lˆ Sˆ (r) sR (r)Y (,) () nlml sms nl lml sms Now it becomes clear why the wavefunction is not adequate for Dirac’s case and, hence, l, ml, s, ms are “bad” quantum numbers. The reason is: L ˆ S ˆ does not commute with Lz or Sz. What to do? Obviously: we have to build from L and S operator which commutes with LS. Total angular momentum • We shall introduce the total angular momentum : S Total angular momentum J Orbital angular momentum Spin L 2 • Operators J and Jz commutes with LS and with Dirac Hamiltonian! Jˆ is a “right” observable for the Dirac equation! • Since like any other angular momentum it satisfies: Jˆ 2 j( j 1)2 Jˆ m jmj jmj z jmj j jmj Now we can describe theJ state Lof relativisticS hydrogen atom (ion) by set of quantum numbers: n, j, mj … and by parity. Parity operator • Solutions of the Dirac (as well as Schrödinger) equation may be separated on the basis of their response to spatial coordinate inversion. • Parity operator: 2 2 x x r r ˆ 2 Ze H S ˆ ˆ Pˆr Pˆ 2m r Pr P y y z z Cartesian coordinates Spherical coordinates • For the Schrödinger case the parity operator commutes with Hamiltonian: ෠ ෡ where Parity is a good 푃, 퐻푠 = 0 quantum number! • Hence, solutions of Schrödinger equation are - at the same time – eigenfunctions of permutation operator: Pˆ (r) (r) (r) nlml nlml nlml Parity operator • For the Schrödinger case the parity operator commutes with Hamiltonian: ෠ ෡ Parity is a good 푃, 퐻푠 = 0 where quantum number! 2 Ze2 H•ˆ Hence, solutions2 of Schrödinger equation are - at the same time – S eigenfunctions2m rof permutation operator: • How to find eigenvalue ? Pˆ 2 (r) Pˆ (r) 2 (r) nlml nlml nlml Solutions of Schrödinger equation are either having 1 even or odd parity! Why we usually don’t use as an additional quantum number? • By employing properties of spherical harmonics we may find: Pˆ (r) PˆR (r)Y (,) R (r)Y ( , ) (1)l R (r)Y (,) (1)l (r) nlml nl lml nl lml nl lml nlml Pˆ (r) (r) (r) Orbital momentum l definesnlml also parity!nlml How it is fornlm Diracl case? Parity of Dirac states • Solutions of the Dirac (as well as Schrödinger) equation may be separated on the basis of their response to spatial coordinate inversion.
Load more

Recommended publications
  • Unit 1 Old Quantum Theory
    UNIT 1 OLD QUANTUM THEORY Structure Introduction Objectives li;,:overy of Sub-atomic Particles Earlier Atom Models Light as clectromagnetic Wave Failures of Classical Physics Black Body Radiation '1 Heat Capacity Variation Photoelectric Effect Atomic Spectra Planck's Quantum Theory, Black Body ~diation. and Heat Capacity Variation Einstein's Theory of Photoelectric Effect Bohr Atom Model Calculation of Radius of Orbits Energy of an Electron in an Orbit Atomic Spectra and Bohr's Theory Critical Analysis of Bohr's Theory Refinements in the Atomic Spectra The61-y Summary Terminal Questions Answers 1.1 INTRODUCTION The ideas of classical mechanics developed by Galileo, Kepler and Newton, when applied to atomic and molecular systems were found to be inadequate. Need was felt for a theory to describe, correlate and predict the behaviour of the sub-atomic particles. The quantum theory, proposed by Max Planck and applied by Einstein and Bohr to explain different aspects of behaviour of matter, is an important milestone in the formulation of the modern concept of atom. In this unit, we will study how black body radiation, heat capacity variation, photoelectric effect and atomic spectra of hydrogen can be explained on the basis of theories proposed by Max Planck, Einstein and Bohr. They based their theories on the postulate that all interactions between matter and radiation occur in terms of definite packets of energy, known as quanta. Their ideas, when extended further, led to the evolution of wave mechanics, which shows the dual nature of matter
    [Show full text]
  • Quantum Theory of the Hydrogen Atom
    Quantum Theory of the Hydrogen Atom Chemistry 35 Fall 2000 Balmer and the Hydrogen Spectrum n 1885: Johann Balmer, a Swiss schoolteacher, empirically deduced a formula which predicted the wavelengths of emission for Hydrogen: l (in Å) = 3645.6 x n2 for n = 3, 4, 5, 6 n2 -4 •Predicts the wavelengths of the 4 visible emission lines from Hydrogen (which are called the Balmer Series) •Implies that there is some underlying order in the atom that results in this deceptively simple equation. 2 1 The Bohr Atom n 1913: Niels Bohr uses quantum theory to explain the origin of the line spectrum of hydrogen 1. The electron in a hydrogen atom can exist only in discrete orbits 2. The orbits are circular paths about the nucleus at varying radii 3. Each orbit corresponds to a particular energy 4. Orbit energies increase with increasing radii 5. The lowest energy orbit is called the ground state 6. After absorbing energy, the e- jumps to a higher energy orbit (an excited state) 7. When the e- drops down to a lower energy orbit, the energy lost can be given off as a quantum of light 8. The energy of the photon emitted is equal to the difference in energies of the two orbits involved 3 Mohr Bohr n Mathematically, Bohr equated the two forces acting on the orbiting electron: coulombic attraction = centrifugal accelleration 2 2 2 -(Z/4peo)(e /r ) = m(v /r) n Rearranging and making the wild assumption: mvr = n(h/2p) n e- angular momentum can only have certain quantified values in whole multiples of h/2p 4 2 Hydrogen Energy Levels n Based on this model, Bohr arrived at a simple equation to calculate the electron energy levels in hydrogen: 2 En = -RH(1/n ) for n = 1, 2, 3, 4, .
    [Show full text]
  • Vibrational Quantum Number
    Fundamentals in Biophotonics Quantum nature of atoms, molecules – matter Aleksandra Radenovic [email protected] EPFL – Ecole Polytechnique Federale de Lausanne Bioengineering Institute IBI 26. 03. 2018. Quantum numbers •The four quantum numbers-are discrete sets of integers or half- integers. –n: Principal quantum number-The first describes the electron shell, or energy level, of an atom –ℓ : Orbital angular momentum quantum number-as the angular quantum number or orbital quantum number) describes the subshell, and gives the magnitude of the orbital angular momentum through the relation Ll2 ( 1) –mℓ:Magnetic (azimuthal) quantum number (refers, to the direction of the angular momentum vector. The magnetic quantum number m does not affect the electron's energy, but it does affect the probability cloud)- magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field-Zeeman effect -s spin- intrinsic angular momentum Spin "up" and "down" allows two electrons for each set of spatial quantum numbers. The restrictions for the quantum numbers: – n = 1, 2, 3, 4, . – ℓ = 0, 1, 2, 3, . , n − 1 – mℓ = − ℓ, − ℓ + 1, . , 0, 1, . , ℓ − 1, ℓ – –Equivalently: n > 0 The energy levels are: ℓ < n |m | ≤ ℓ ℓ E E 0 n n2 Stern-Gerlach experiment If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by a different amount, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount.
    [Show full text]
  • Lecture 3: Particle in a 1D Box
    Lecture 3: Particle in a 1D Box First we will consider a free particle moving in 1D so V (x) = 0. The TDSE now reads ~2 d2ψ(x) = Eψ(x) −2m dx2 which is solved by the function ψ = Aeikx where √2mE k = ± ~ A general solution of this equation is ψ(x) = Aeikx + Be−ikx where A and B are arbitrary constants. It can also be written in terms of sines and cosines as ψ(x) = C sin(kx) + D cos(kx) The constants appearing in the solution are determined by the boundary conditions. For a free particle that can be anywhere, there is no boundary conditions, so k and thus E = ~2k2/2m can take any values. The solution of the form eikx corresponds to a wave travelling in the +x direction and similarly e−ikx corresponds to a wave travelling in the -x direction. These are eigenfunctions of the momentum operator. Since the particle is free, it is equally likely to be anywhere so ψ∗(x)ψ(x) is independent of x. Incidently, it cannot be normalized because the particle can be found anywhere with equal probability. 1 Now, let us confine the particle to a region between x = 0 and x = L. To do this, we choose our interaction potential V (x) as follows V (x) = 0 for 0 x L ≤ ≤ = otherwise ∞ It is always a good idea to plot the potential energy, when it is a function of a single variable, as shown in Fig.1. The TISE is now given by V(x) V=infinity V=0 V=infinity x 0 L ~2 d2ψ(x) + V (x)ψ(x) = Eψ(x) −2m dx2 First consider the region outside the box where V (x) = .
    [Show full text]
  • Further Quantum Physics
    Further Quantum Physics Concepts in quantum physics and the structure of hydrogen and helium atoms Prof Andrew Steane January 18, 2005 2 Contents 1 Introduction 7 1.1 Quantum physics and atoms . 7 1.1.1 The role of classical and quantum mechanics . 9 1.2 Atomic physics—some preliminaries . .... 9 1.2.1 Textbooks...................................... 10 2 The 1-dimensional projectile: an example for revision 11 2.1 Classicaltreatment................................. ..... 11 2.2 Quantum treatment . 13 2.2.1 Mainfeatures..................................... 13 2.2.2 Precise quantum analysis . 13 3 Hydrogen 17 3.1 Some semi-classical estimates . 17 3.2 2-body system: reduced mass . 18 3.2.1 Reduced mass in quantum 2-body problem . 19 3.3 Solution of Schr¨odinger equation for hydrogen . ..... 20 3.3.1 General features of the radial solution . 21 3.3.2 Precisesolution.................................. 21 3.3.3 Meanradius...................................... 25 3.3.4 How to remember hydrogen . 25 3.3.5 Mainpoints.................................... 25 3.3.6 Appendix on series solution of hydrogen equation, off syllabus . 26 3 4 CONTENTS 4 Hydrogen-like systems and spectra 27 4.1 Hydrogen-like systems . 27 4.2 Spectroscopy ........................................ 29 4.2.1 Main points for use of grating spectrograph . ...... 29 4.2.2 Resolution...................................... 30 4.2.3 Usefulness of both emission and absorption methods . 30 4.3 The spectrum for hydrogen . 31 5 Introduction to fine structure and spin 33 5.1 Experimental observation of fine structure . ..... 33 5.2 TheDiracresult ..................................... 34 5.3 Schr¨odinger method to account for fine structure . 35 5.4 Physical nature of orbital and spin angular momenta .
    [Show full text]
  • The Quantum Mechanical Model of the Atom
    The Quantum Mechanical Model of the Atom Quantum Numbers In order to describe the probable location of electrons, they are assigned four numbers called quantum numbers. The quantum numbers of an electron are kind of like the electron’s “address”. No two electrons can be described by the exact same four quantum numbers. This is called The Pauli Exclusion Principle. • Principle quantum number: The principle quantum number describes which orbit the electron is in and therefore how much energy the electron has. - it is symbolized by the letter n. - positive whole numbers are assigned (not including 0): n=1, n=2, n=3 , etc - the higher the number, the further the orbit from the nucleus - the higher the number, the more energy the electron has (this is sort of like Bohr’s energy levels) - the orbits (energy levels) are also called shells • Angular momentum (azimuthal) quantum number: The azimuthal quantum number describes the sublevels (subshells) that occur in each of the levels (shells) described above. - it is symbolized by the letter l - positive whole number values including 0 are assigned: l = 0, l = 1, l = 2, etc. - each number represents the shape of a subshell: l = 0, represents an s subshell l = 1, represents a p subshell l = 2, represents a d subshell l = 3, represents an f subshell - the higher the number, the more complex the shape of the subshell. The picture below shows the shape of the s and p subshells: (notice the electron clouds) • Magnetic quantum number: All of the subshells described above (except s) have more than one orientation.
    [Show full text]
  • A Relativistic Electron in a Coulomb Potential
    A Relativistic Electron in a Coulomb Potential Alfred Whitehead Physics 518, Fall 2009 The Problem Solve the Dirac Equation for an electron in a Coulomb potential. Identify the conserved quantum numbers. Specify the degeneracies. Compare with solutions of the Schrödinger equation including relativistic and spin corrections. Approach My approach follows that taken by Dirac in [1] closely. A few modifications taken from [2] and [3] are included, particularly in regards to the final quantum numbers chosen. The general strategy is to first find a set of transformations which turn the Hamiltonian for the system into a form that depends only on the radial variables r and pr. Once this form is found, I solve it to find the energy eigenvalues and then discuss the energy spectrum. The Radial Dirac Equation We begin with the electromagnetic Hamiltonian q H = p − cρ ~σ · ~p − A~ + ρ mc2 (1) 0 1 c 3 with 2 0 0 1 0 3 6 0 0 0 1 7 ρ1 = 6 7 (2) 4 1 0 0 0 5 0 1 0 0 2 1 0 0 0 3 6 0 1 0 0 7 ρ3 = 6 7 (3) 4 0 0 −1 0 5 0 0 0 −1 1 2 0 1 0 0 3 2 0 −i 0 0 3 2 1 0 0 0 3 6 1 0 0 0 7 6 i 0 0 0 7 6 0 −1 0 0 7 ~σ = 6 7 ; 6 7 ; 6 7 (4) 4 0 0 0 1 5 4 0 0 0 −i 5 4 0 0 1 0 5 0 0 1 0 0 0 i 0 0 0 0 −1 We note that, for the Coulomb potential, we can set (using cgs units): Ze2 p = −eΦ = − o r A~ = 0 This leads us to this form for the Hamiltonian: −Ze2 H = − − cρ ~σ · ~p + ρ mc2 (5) r 1 3 We need to get equation 5 into a form which depends not on ~p, but only on the radial variables r and pr.
    [Show full text]
  • Higher Levels of the Transmon Qubit
    Higher Levels of the Transmon Qubit MASSACHUSETTS INSTITUTE OF TECHNirLOGY by AUG 15 2014 Samuel James Bader LIBRARIES Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2014 @ Samuel James Bader, MMXIV. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Signature redacted Author........ .. ----.-....-....-....-....-.....-....-......... Department of Physics Signature redacted May 9, 201 I Certified by ... Terr P rlla(nd Professor of Electrical Engineering Signature redacted Thesis Supervisor Certified by ..... ..................... Simon Gustavsson Research Scientist Signature redacted Thesis Co-Supervisor Accepted by..... Professor Nergis Mavalvala Senior Thesis Coordinator, Department of Physics Higher Levels of the Transmon Qubit by Samuel James Bader Submitted to the Department of Physics on May 9, 2014, in partial fulfillment of the requirements for the degree of Bachelor of Science in Physics Abstract This thesis discusses recent experimental work in measuring the properties of higher levels in transmon qubit systems. The first part includes a thorough overview of transmon devices, explaining the principles of the device design, the transmon Hamiltonian, and general Cir- cuit Quantum Electrodynamics concepts and methodology. The second part discusses the experimental setup and methods employed in measuring the higher levels of these systems, and the details of the simulation used to explain and predict the properties of these levels. Thesis Supervisor: Terry P. Orlando Title: Professor of Electrical Engineering Thesis Supervisor: Simon Gustavsson Title: Research Scientist 3 4 Acknowledgments I would like to express my deepest gratitude to Dr.
    [Show full text]
  • The Principal Quantum Number the Azimuthal Quantum Number The
    To completely describe an electron in an atom, four quantum numbers are needed: energy (n), angular momentum (ℓ), magnetic moment (mℓ), and spin (ms). The Principal Quantum Number This quantum number describes the electron shell or energy level of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that atom. For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron incaesium can have an n value from 1 to 6. For particles in a time-independent potential, as per the Schrödinger equation, it also labels the nth eigen value of Hamiltonian (H). This number has a dependence only on the distance between the electron and the nucleus (i.e. the radial coordinate r). The average distance increases with n, thus quantum states with different principal quantum numbers are said to belong to different shells. The Azimuthal Quantum Number The angular or orbital quantum number, describes the sub-shell and gives the magnitude of the orbital angular momentum through the relation. ℓ = 0 is called an s orbital, ℓ = 1 a p orbital, ℓ = 2 a d orbital, and ℓ = 3 an f orbital. The value of ℓ ranges from 0 to n − 1 because the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n = 3), and so on. This quantum number specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles.
    [Show full text]
  • Relativistic Quantum Mechanics 1
    Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4).
    [Show full text]
  • Total Angular Momentum
    Total Angular momentum Dipanjan Mazumdar Sept 2019 1 Motivation So far, somewhat deliberately, in our prior examples we have worked exclusively with the spin angular momentum and ignored the orbital component. This is acceptable for L = 0 systems such as filled shell atoms, but properties of partially filled atoms will depend both on the orbital and the spin part of the angular momentum. Also, when we include relativistic corrections to atomic Hamiltonian, both spin and orbital angular momentum enter directly through terms like L:~ S~ called the spin-orbit coupling. So instead of focusing only on the spin or orbital angular momentum, we have to develop an understanding as to how they couple together. One of the ways is through the total angular momentum defined as J~ = L~ + S~ (1) We will see that this will be very useful in many cases such as the real hydrogen atom where the eigenstates after including the fine structure effects such as spin-orbit interaction are eigenstates of the total angular momentum operator. 2 Vector model of angular momentum Let us first develop an intuitive understanding of the total angular momentum through the vector model which is a semi-classical approach to \add" angular momenta using vector algebra. We shall first ask, knowing L~ and S~, what are the maxmimum and minimum values of J^. This is simple to answer us- ing the vector model since vectors can be added or subtracted. Therefore, the extreme values are L~ + S~ and L~ − S~ as seen in figure 1.1 Therefore, the total angular momentum will take up values between l+s Figure 1: Maximum and minimum values of angular and jl − sj.
    [Show full text]
  • 1. the Hamiltonian, H = Α Ipi + Βm, Must Be Her- Mitian to Give Real
    1. The hamiltonian, H = αipi + βm, must be her- yields mitian to give real eigenvalues. Thus, H = H† = † † † pi αi +β m. From non-relativistic quantum mechanics † † † αi pi + β m = aipi + βm (2) we know that pi = pi . In addition, [αi,pj ] = 0 because we impose that the α, β operators act on the spinor in- † † αi = αi, β = β. Thus α, β are hermitian operators. dices while pi act on the coordinates of the wave function itself. With these conditions we may proceed: To verify the other properties of the α, and β operators † † piαi + β m = αipi + βm (1) we compute 2 H = (αipi + βm) (αj pj + βm) 2 2 1 2 2 = αi pi + (αiαj + αj αi) pipj + (αiβ + βαi) m + β m , (3) 2 2 where for the second term i = j. Then imposing the Finally, making use of bi = 1 we can find the 26 2 2 2 relativistic energy relation, E = p + m , we obtain eigenvalues: biu = λu and bibiu = λbiu, hence u = λ u the anti-commutation relation | | and λ = 1. ± b ,b =2δ 1, (4) i j ij 2. We need to find the λ = +1/2 helicity eigenspinor { } ′ for an electron with momentum ~p = (p sin θ, 0, p cos θ). where b0 = β, bi = αi, and 1 n n idenitity matrix. ≡ × 1 Using the anti-commutation relation and the fact that The helicity operator is given by 2 σ pˆ, and the posi- 2 1 · bi = 1 we are able to find the trace of any bi tive eigenvalue 2 corresponds to u1 Dirac solution [see (1.5.98) in http://arXiv.org/abs/0906.1271].
    [Show full text]