2.4. Quantum Mechanical description of hydrogen atom

Size: px
Start display at page:

Download "2.4. Quantum Mechanical description of hydrogen atom"

Transcription

1 2.4. Quantum Mechanical description of hydrogen atom Atomic units Quantity Atomic unit SI Conversion Ang. mom. h [J s] h = 1, Js Mass m e [kg] m e = 9, kg Charge e [C] e = 1, C Permittivity 4πε 0 [ C 2 Jm From these one can derive: Length a 0 (bohr) [m] 1 bohr = 4πε 0 h 2 m e e 2 Energy E h (hartree) [J] 1 hartree = e2 4πε 0 a 0 ] 4πε 0 = 1, C2 Jm = 0, m = 4, J 1 E h 27, 21 ev E h 627 kcal/mol The model of the hydrogen atom: an electron is situated around the nuclei which is not moving; the interaction potential is given by the Coulomb interaction: V = e2 r In quantum mechanics we have to solve the Schrödinger equation: with Notes: ĤΨ i = E i Ψ i Ĥ is the Hamiltonian including the interactions within the system (kinetic and potential energy): Ĥ = ˆT + V E i is the total energy of the system Ψ i (x, y, z) is the wave function describing the system, also called state function, here also can be called the orbital of the electron. 1. the energy is quantized; 2. i index denotes that there are several such states. The one with the lowest energy is called the ground state, the others are the excited states. About the solution: during the calculations it turns out that the states should not be labeled with a simple index i, but rather with a triplet of numbers, the so called quantum numbers: i (n, l, m) It also comes out from the calculation that quantum numbers can not have arbitrary values: this is where the name is from! For the hydrogen atom the possible values of the quantum numbers are: 18

2 n principal quantum number: 1, 2, 3,.... l angular momentum quantum number: 0, 1, 2,...(n 1) m magnetic quantum number: l, l + 1,..., 0, 1,..., l (2l+1 different values) The quantum numbers are related to physical quantities: n: determines the energy: E n = 1 2n 2 (E h ) EXACTLY LIKE IN BOHR THEORY!!! l: determines the size of the angular momentum: l = l(l + 1)( h) m: determines the z component of the angular momentum: l z = m( h) m = l, l + 1,..., 0, 1,...l What is the angular momentum? Classical definition of the angular momentum: L = r p = mr v where m is the mass, v is the speed, p is the momentum, r is the position of the particle (see figure). Why is m called the magnetic quantum number? m determines the z component of the angular momentum. Since the electron is moving around the nuclei, and has a charge, it creates magnetic moment. There is a proportional relation between angular momentum and magnetic moment: µ = µ B l µ z = m µ B 19

3 where µ B is a constant (called the Bohr-magneton). How many different values m can have? m = l + 1,..., 0,..., l, i.e. 2l + 1 values. Since the interaction with the magnetic filed will be proportional to the magnetic moment, its magnitude depends on m. in magnetic field the energy levels split up to 2l+1 different values. This is the so called Zeeman-effect. l=0 1 energy level l=1 3 energy levels l=2 5 energy levels etc. Notation of the orbitals: principal ang. mom. subshell magnetic number of orbitals quant. number (n) quant. number (l) l quant. number (m) on the subshell 1 0 1s s p -1,0, s p -1,0, d -2,-1,0,1, s p -1,0, d -2,-1,0,1, f -3,-2,-1,0,1,2,3 7 20

4 Representation of the orbitals: 1s and 2s orbitals There is a node on the 2s orbital, where the value of the wave function gets zero. 21

5 Representation of the orbitals: 2p orbitals 22

6 Representation of the orbitals: d orbitals 23

7 24

8 Representation of the orbitals: dotting the frequency of the dots represent the value: more points mean larger value of the wave function. 25

9 Radial electron density: probability of finding the electron at distance r from the nuclei (i.e. in a shell of the spere). Radial density for orbitals 1s, 2s and 2p: Radial density for orbitals 3s, 3p and 3d: 26

10 The spin of the electron We want to prove that in the ground state of hydrogen atom l=0: we put it into the magnetic field. We assume one beam: This is the so called Stern-Gerlach experiment. The beam of ground state hydrogen atom splits into two beams. This contradicts the theory, since we have expected 1, 3, 5,... beams! Conclusion: Pauli (1925): a fourth quantum number is needed; Goudsmit and Uhlenbeck suggested the concept of spin, as the internal angular momentum Classically: if the electron is not a pointwise particle, it can rotate around its axis, either to the right or to the left. In quantum mechanics: the electron as a particle has intrinsic angular momentum, which is its own property, like its charge. What do we know about it? it is like the angular momentum since there is magnetic moment associated with it; its projection can have two different values. magnitude: s(s + 1) h s quantum number z component: m s h m s quantum number, or spin quantum number m s = s, s + 1,..., s 1, s s = 1 2 since in this case m s = 1 2, FOR ELECTRONS s = 1 2 always!!!!! The fourth quantum number is: m s Thus, the electron has spin. 27

11 What is spin? Where it does originate from? Bad question, we would not ask: why the electron has a charge? Properties of the electron: charge: 1 spin: 1/2 Spin is the intrinsic momentum of the electron. Spectroscopic application: Electron Spin Resonance (ESR): Summarized: The states of the hydrogen atom are quantized and are characterized by quantum numbers: n = 1, 2,... l = 0, 1,..., n 1 m = l, l + 1,..., l m s = 1 2, 1 2 The energy depends only on n: E n = 1 2 (E n 2 h ). There is a many-fold degeneracy! In magnetic field the energy splits up according to the magnetic quantum number m. 1 28

1.6. Quantum mechanical description of the hydrogen atom

1.6. Quantum mechanical description of the hydrogen atom 29.6. Quantum mechanical description of the hydrogen atom.6.. Hamiltonian for the hydrogen atom Atomic units To avoid dealing with very small numbers, let us introduce the so called atomic units : Quantity

More information

Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall Duration: 2h 30m

Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall Duration: 2h 30m Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. ------------------- Duration: 2h 30m Chapter 39 Quantum Mechanics of Atoms Units of Chapter 39 39-1 Quantum-Mechanical View of Atoms 39-2

More information

ECE440 Nanoelectronics. Lecture 07 Atomic Orbitals

ECE440 Nanoelectronics. Lecture 07 Atomic Orbitals ECE44 Nanoelectronics Lecture 7 Atomic Orbitals Atoms and atomic orbitals It is instructive to compare the simple model of a spherically symmetrical potential for r R V ( r) for r R and the simplest hydrogen

More information

The Hydrogen Atom. Dr. Sabry El-Taher 1. e 4. U U r

The Hydrogen Atom. Dr. Sabry El-Taher 1. e 4. U U r The Hydrogen Atom Atom is a 3D object, and the electron motion is three-dimensional. We ll start with the simplest case - The hydrogen atom. An electron and a proton (nucleus) are bound by the central-symmetric

More information

L z L L. Think of it as also affecting the angle

L z L L. Think of it as also affecting the angle Quantum Mechanics and Atomic Physics Lecture 19: Quantized Angular Momentum and Electron Spin http://www.physics.rutgers.edu/ugrad/361 h / d/361 Prof. Sean Oh Last time Raising/Lowering angular momentum

More information

The Hydrogen Atom. Thornton and Rex, Ch. 7

The Hydrogen Atom. Thornton and Rex, Ch. 7 The Hydrogen Atom Thornton and Rex, Ch. 7 Applying Schrodinger s Eqn to the Hydrogen Atom The potential: -1 e 2 V(r) = 4p e0 r Use spherical polar coordinates (with y(x,y,z) => y(r,q,f) ): 1 y 1 y ( r

More information

Introduction to Quantum Mechanics. and Quantum Numbers

Introduction to Quantum Mechanics. and Quantum Numbers Introduction to Quantum Mechanics and Quantum Numbers The Quantum Mechanical Model quantum mechanics: the application of quantum theory to explain the properties of matter, particularly electrons in atoms

More information

Potential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form

Potential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form Lecture 6 Page 1 Atoms L6.P1 Review of hydrogen atom Heavy proton (put at the origin), charge e and much lighter electron, charge -e. Potential energy, from Coulomb's law Potential is spherically symmetric.

More information

64-311/5: Atomic and Molecular Spectra

64-311/5: Atomic and Molecular Spectra 64-311-Questions.doc 64-311/5: Atomic and Molecular Spectra Dr T Reddish (Room 89-1 Essex Hall) SECTION 1: REVISION QUESTIONS FROM 64-310/14 ε ο = 8.854187817 x 10-1 Fm -1, h = 1.0545766 x 10-34 Js, e

More information

Atomic Structure and Atomic Spectra

Atomic Structure and Atomic Spectra Atomic Structure and Atomic Spectra Atomic Structure: Hydrogenic Atom Reading: Atkins, Ch. 10 (7 판 Ch. 13) The principles of quantum mechanics internal structure of atoms 1. Hydrogenic atom: one electron

More information

I. RADIAL PROBABILITY DISTRIBUTIONS (RPD) FOR S-ORBITALS

I. RADIAL PROBABILITY DISTRIBUTIONS (RPD) FOR S-ORBITALS 5. Lecture Summary #7 Readings for today: Section.0 (.9 in rd ed) Electron Spin, Section. (.0 in rd ed) The Electronic Structure of Hydrogen. Read for Lecture #8: Section. (. in rd ed) Orbital Energies

More information

Alkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split

Alkali metals show splitting of spectral lines in absence of magnetic field. s lines not split p, d lines split Electron Spin Electron spin hypothesis Solution to H atom problem gave three quantum numbers, n,, m. These apply to all atoms. Experiments show not complete description. Something missing. Alkali metals

More information

A more comprehensive theory was needed. 1925, Schrödinger and Heisenberg separately worked out a new theory Quantum Mechanics.

A more comprehensive theory was needed. 1925, Schrödinger and Heisenberg separately worked out a new theory Quantum Mechanics. Ch28 Quantum Mechanics of Atoms Bohr s model was very successful to explain line spectra and the ionization energy for hydrogen. However, it also had many limitations: It was not able to predict the line

More information

Ch. 1: Atoms: The Quantum World

Ch. 1: Atoms: The Quantum World Ch. 1: Atoms: The Quantum World CHEM 4A: General Chemistry with Quantitative Analysis Fall 2009 Instructor: Dr. Orlando E. Raola Santa Rosa Junior College Overview 1.1The nuclear atom 1.2 Characteristics

More information

MAGNETISM OF ATOMS QUANTUM-MECHANICAL BASICS. Janusz Adamowski AGH University of Science and Technology, Kraków, Poland

MAGNETISM OF ATOMS QUANTUM-MECHANICAL BASICS. Janusz Adamowski AGH University of Science and Technology, Kraków, Poland MAGNETISM OF ATOMS QUANTUM-MECHANICAL BASICS Janusz Adamowski AGH University of Science and Technology, Kraków, Poland 1 The magnetism of materials can be derived from the magnetic properties of atoms.

More information

Atomic Structure Ch , 9.6, 9.7

Atomic Structure Ch , 9.6, 9.7 Ch. 9.2-4, 9.6, 9.7 Magnetic moment of an orbiting electron: An electron orbiting a nucleus creates a current loop. A current loop behaves like a magnet with a magnetic moment µ:! µ =! µ B " L Bohr magneton:

More information

4/21/2010. Schrödinger Equation For Hydrogen Atom. Spherical Coordinates CHAPTER 8

4/21/2010. Schrödinger Equation For Hydrogen Atom. Spherical Coordinates CHAPTER 8 CHAPTER 8 Hydrogen Atom 8.1 Spherical Coordinates 8.2 Schrödinger's Equation in Spherical Coordinate 8.3 Separation of Variables 8.4 Three Quantum Numbers 8.5 Hydrogen Atom Wave Function 8.6 Electron Spin

More information

Sparks CH301. Quantum Mechanics. Waves? Particles? What and where are the electrons!? UNIT 2 Day 3. LM 14, 15 & 16 + HW due Friday, 8:45 am

Sparks CH301. Quantum Mechanics. Waves? Particles? What and where are the electrons!? UNIT 2 Day 3. LM 14, 15 & 16 + HW due Friday, 8:45 am Sparks CH301 Quantum Mechanics Waves? Particles? What and where are the electrons!? UNIT 2 Day 3 LM 14, 15 & 16 + HW due Friday, 8:45 am What are we going to learn today? The Simplest Atom - Hydrogen Relate

More information

(n, l, m l ) 3/2/2016. Quantum Numbers (QN) Plots of Energy Level. Roadmap for Exploring Hydrogen Atom

(n, l, m l ) 3/2/2016. Quantum Numbers (QN) Plots of Energy Level. Roadmap for Exploring Hydrogen Atom PHYS 34 Modern Physics Atom III: Angular Momentum and Spin Roadmap for Exploring Hydrogen Atom Today Contents: a) Orbital Angular Momentum and Magnetic Dipole Moment b) Electric Dipole Moment c) Stern

More information

Chemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron):

Chemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1-electron): April 6th, 24 Chemistry 2A 2nd Midterm. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (-electron): E n = m e Z 2 e 4 /2 2 n 2 = E Z 2 /n 2, n =, 2, 3,... where Ze is

More information

PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA)

PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 8 (ALKALI METAL SPECTRA) Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 8: Alkali metal spectra CHE_P8_M8 TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Multi-electron

More information

Line spectrum (contd.) Bohr s Planetary Atom

Line spectrum (contd.) Bohr s Planetary Atom Line spectrum (contd.) Hydrogen shows lines in the visible region of the spectrum (red, blue-green, blue and violet). The wavelengths of these lines can be calculated by an equation proposed by J. J. Balmer:

More information

Announcements. Lecture 20 Chapter. 7 QM in 3-dims & Hydrogen Atom. The Radial Part of Schrodinger Equation for Hydrogen Atom

Announcements. Lecture 20 Chapter. 7 QM in 3-dims & Hydrogen Atom. The Radial Part of Schrodinger Equation for Hydrogen Atom Announcements! HW7 : Chap.7 18, 20, 23, 32, 37, 38, 45, 47, 53, 57, 60! Physics Colloquium: Development in Electron Nuclear Dynamics Theory on Thursday @ 3:40pm! Quiz 2 (average: 9), Quiz 3: 4/19 *** Course

More information

CHAPTER 8 Atomic Physics

CHAPTER 8 Atomic Physics CHAPTER 8 Atomic Physics 8.1 Atomic Structure and the Periodic Table 8.2 Total Angular Momentum 8.3 Anomalous Zeeman Effect What distinguished Mendeleev was not only genius, but a passion for the elements.

More information

The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field:

The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field: SPIN 1/2 PARTICLE Stern-Gerlach experiment The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field: A silver atom

More information

Please read the following instructions:

Please read the following instructions: MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 11, 016 (8:30 a.m. - 9:45 a.m.) PLACE: RB 306 Only non-programmable calculators are allowed. Name: ID: Please read the following instructions:

More information

Final Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall.

Final Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall. Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Summary of Chapter 38 In Quantum Mechanics particles are represented by wave functions Ψ. The absolute square of the wave function Ψ 2

More information

Atomic Structure, Periodic Table, and Other Effects: Chapter 8 of Rex and T. Modern Physics

Atomic Structure, Periodic Table, and Other Effects: Chapter 8 of Rex and T. Modern Physics Atomic Structure, Periodic Table, and Other Effects: Chapter 8 of Rex and T Modern Physics 11/16 and 11/19/2018 1 Introduction In Chapter 7, we studied the hydrogen atom. What about other elements, e.g.,

More information

Properties of Elementary Particles

Properties of Elementary Particles and of Elementary s 01/11/2018 My Office Hours: Thursday 1:00-3:00 PM 212 Keen Building Outline 1 2 3 Consider the world at different scales... Cosmology - only gravity matters XXXXX Input: Mass distributions

More information

The Electronic Structure of Atoms

The Electronic Structure of Atoms The Electronic Structure of Atoms Classical Hydrogen-like atoms: Atomic Scale: 10-10 m or 1 Å + - Proton mass : Electron mass 1836 : 1 Problems with classical interpretation: - Should not be stable (electron

More information

( ). Expanding the square and keeping in mind that

( ). Expanding the square and keeping in mind that One-electron atom in a Magnetic Field When the atom is in a magnetic field the magnetic moment of the electron due to its orbital motion and its spin interacts with the field and the Schrodinger Hamiltonian

More information

Magnetic Moments and Spin

Magnetic Moments and Spin Magnetic Moments and Spin Still have several Homeworks to hand back Finish up comments about hydrogen atom and start on magnetic moment + spin. Eleventh Homework Set is due today and the last one has been

More information

Preliminary Quantum Questions

Preliminary Quantum Questions Preliminary Quantum Questions Thomas Ouldridge October 01 1. Certain quantities that appear in the theory of hydrogen have wider application in atomic physics: the Bohr radius a 0, the Rydberg constant

More information

Degeneracy & in particular to Hydrogen atom

Degeneracy & in particular to Hydrogen atom Degeneracy & in particular to Hydrogen atom In quantum mechanics, an energy level is said to be degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely,

More information

Chapter 10: Multi- Electron Atoms Optical Excitations

Chapter 10: Multi- Electron Atoms Optical Excitations Chapter 10: Multi- Electron Atoms Optical Excitations To describe the energy levels in multi-electron atoms, we need to include all forces. The strongest forces are the forces we already discussed in Chapter

More information

Many-Electron Atoms. Thornton and Rex, Ch. 8

Many-Electron Atoms. Thornton and Rex, Ch. 8 Many-Electron Atoms Thornton and Rex, Ch. 8 In principle, can now solve Sch. Eqn for any atom. In practice, -> Complicated! Goal-- To explain properties of elements from principles of quantum theory (without

More information

Introduction to Quantum Mechanics Prof. Manoj Kumar Harbola Department of Physics Indian Institute of Technology, Kanpur

Introduction to Quantum Mechanics Prof. Manoj Kumar Harbola Department of Physics Indian Institute of Technology, Kanpur Introduction to Quantum Mechanics Prof. Manoj Kumar Harbola Department of Physics Indian Institute of Technology, Kanpur Lecture - 04 Quantum conditions and atomic structure, electron spin and Pauli exclusion

More information

Multielectron Atoms.

Multielectron Atoms. Multielectron Atoms. Chem 639. Spectroscopy. Spring 00 S.Smirnov Atomic Units Mass m e 1 9.109 10 31 kg Charge e 1.60 10 19 C Angular momentum 1 1.055 10 34 J s Permittivity 4 0 1 1.113 10 10 C J 1 m 1

More information

Chem What is the difference between an orbit (Bohr model) and an orbital (quantum mechanical model)?

Chem What is the difference between an orbit (Bohr model) and an orbital (quantum mechanical model)? Reading: sections 6.5-6.6 As you read this material, ask yourself the following questions: What are wave functions and orbitals, how do orbitals differ from orbits? What can we learn about an electron

More information

Probability and Normalization

Probability and Normalization Probability and Normalization Although we don t know exactly where the particle might be inside the box, we know that it has to be in the box. This means that, ψ ( x) dx = 1 (normalization condition) L

More information

Physics 43 Exam 2 Spring 2018

Physics 43 Exam 2 Spring 2018 Physics 43 Exam 2 Spring 2018 Print Name: Conceptual Circle the best answer. (2 points each) 1. Quantum physics agrees with the classical physics limit when a. the total angular momentum is a small multiple

More information

Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas. Chapter 8: Quantum Theory: Techniques and Applications

Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas. Chapter 8: Quantum Theory: Techniques and Applications Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas Chapter 8: Quantum Theory: Techniques and Applications TRANSLATIONAL MOTION wavefunction of free particle, ψ k = Ae ikx + Be ikx,

More information

4πε. me 1,2,3,... 1 n. H atom 4. in a.u. atomic units. energy: 1 a.u. = ev distance 1 a.u. = Å

4πε. me 1,2,3,... 1 n. H atom 4. in a.u. atomic units. energy: 1 a.u. = ev distance 1 a.u. = Å H atom 4 E a me =, n=,,3,... 8ε 0 0 π me e e 0 hn ε h = = 0.59Å E = me (4 πε ) 4 e 0 n n in a.u. atomic units E = r = Z n nao Z = e = me = 4πε = 0 energy: a.u. = 7. ev distance a.u. = 0.59 Å General results

More information

Physics and Chemistry of the Interstellar Medium

Physics and Chemistry of the Interstellar Medium 0. Physics and Chemistry of the Interstellar Medium Pierre Hily-Blant Master2 Lecture, LAOG pierre.hilyblant@obs.ujf-grenoble.fr, Office 53 2012-2013 Pierre Hily-Blant (Master2) The ISM 2012-2013 1 / 220

More information

Chapter 7 The Quantum-Mechanical Model of the Atom

Chapter 7 The Quantum-Mechanical Model of the Atom Chapter 7 The Quantum-Mechanical Model of the Atom Electron Energy electron energy and position are complimentary because KE = ½mv 2 for an electron with a given energy, the best we can do is describe

More information

Many-Electron Atoms. Thornton and Rex, Ch. 8

Many-Electron Atoms. Thornton and Rex, Ch. 8 Many-Electron Atoms Thornton and Rex, Ch. 8 In principle, can now solve Sch. Eqn for any atom. In practice, -> Complicated! Goal-- To explain properties of elements from principles of quantum theory (without

More information

Chapter Electron Spin. * Fine structure:many spectral lines consist of two separate. lines that are very close to each other.

Chapter Electron Spin. * Fine structure:many spectral lines consist of two separate. lines that are very close to each other. Chapter 7 7. Electron Spin * Fine structure:many spectral lines consist of two separate lines that are very close to each other. ex. H atom, first line of Balmer series n = 3 n = => 656.3nm in reality,

More information

Chapter 28. Atomic Physics

Chapter 28. Atomic Physics Chapter 28 Atomic Physics Bohr s Correspondence Principle Bohr s Correspondence Principle states that quantum mechanics is in agreement with classical physics when the energy differences between quantized

More information

atoms and light. Chapter Goal: To understand the structure and properties of atoms.

atoms and light. Chapter Goal: To understand the structure and properties of atoms. Quantum mechanics provides us with an understanding of atomic structure and atomic properties. Lasers are one of the most important applications of the quantummechanical properties of atoms and light.

More information

Wave Nature of Matter. Wave Nature of Matter. Wave Nature of Matter. Light has wave-like and particle-like properties

Wave Nature of Matter. Wave Nature of Matter. Wave Nature of Matter. Light has wave-like and particle-like properties Wave Nature of Matter Light has wave-like and particle-like properties Can matter have wave and particle properties? de Broglie s hypothesis: matter has wave-like properties in addition to the expected

More information

Welcome back to PHYS 3305

Welcome back to PHYS 3305 Welcome back to PHYS 3305 Otto Stern 1888-1969 Walther Gerlach 1889-1979 Today s Lecture: Angular Momentum Quantization Stern-Gerlach Experiment AnNouncements Reading Assignment for Nov 14th: Harris 8.2-8.5.

More information

Welcome back to PHY 3305

Welcome back to PHY 3305 Welcome back to PHY 3305 Today s Lecture: Hydrogen Atom Part I John von Neumann 1903-1957 One-Dimensional Atom To analyze the hydrogen atom, we must solve the Schrodinger equation for the Coulomb potential

More information

Sommerfeld (1920) noted energy levels of Li deduced from spectroscopy looked like H, with slight adjustment of principal quantum number:

Sommerfeld (1920) noted energy levels of Li deduced from spectroscopy looked like H, with slight adjustment of principal quantum number: Spin. Historical Spectroscopy of Alkali atoms First expt. to suggest need for electron spin: observation of splitting of expected spectral lines for alkali atoms: i.e. expect one line based on analogy

More information

QUANTUM MECHANICS AND ATOMIC STRUCTURE

QUANTUM MECHANICS AND ATOMIC STRUCTURE 5 CHAPTER QUANTUM MECHANICS AND ATOMIC STRUCTURE 5.1 The Hydrogen Atom 5.2 Shell Model for Many-Electron Atoms 5.3 Aufbau Principle and Electron Configurations 5.4 Shells and the Periodic Table: Photoelectron

More information

Lecture #13 1. Incorporating a vector potential into the Hamiltonian 2. Spin postulates 3. Description of spin states 4. Identical particles in

Lecture #13 1. Incorporating a vector potential into the Hamiltonian 2. Spin postulates 3. Description of spin states 4. Identical particles in Lecture #3. Incorporating a vector potential into the Hamiltonian. Spin postulates 3. Description of spin states 4. Identical particles in classical and QM 5. Exchange degeneracy - the fundamental problem

More information

Atomic Term Symbols and Energy Splitting. λ=5890 Å

Atomic Term Symbols and Energy Splitting. λ=5890 Å Chemistry 362 Spring 2018 Dr. Jean M. Standard April 18, 2018 Atomic Term Symbols and Energy Splitting 1. Atomic Term Symbols and the Sodium D-Line The sodium D-line is responsible for the familiar orange

More information

Stern-Gerlach Experiment and Spin

Stern-Gerlach Experiment and Spin Stern-Gerlach Experiment and Spin 1 Abstract Vedat Tanrıverdi Physics Department, METU tvedat@metu.edu.tr The historical development of spin and Stern-Gerlach experiment are summarized. Then some questions

More information

Mendeleev s Periodic Law

Mendeleev s Periodic Law Mendeleev s Periodic Law Periodic Law When the elements are arranged in order of increasing atomic mass, certain sets of properties recur periodically. Mendeleev s Periodic Law allows us to predict what

More information

Physics 1C Lecture 29B

Physics 1C Lecture 29B Physics 1C Lecture 29B Emission Spectra! The easiest gas to analyze is hydrogen gas.! Four prominent visible lines were observed, as well as several ultraviolet lines.! In 1885, Johann Balmer, found a

More information

(b) The wavelength of the radiation that corresponds to this energy is 6

(b) The wavelength of the radiation that corresponds to this energy is 6 Chapter 7 Problem Solutions 1. A beam of electrons enters a uniform 1.0-T magnetic field. (a) Find the energy difference between electrons whose spins are parallel and antiparallel to the field. (b) Find

More information

6.1 Nondegenerate Perturbation Theory

6.1 Nondegenerate Perturbation Theory 6.1 Nondegenerate Perturbation Theory Analytic solutions to the Schrödinger equation have not been found for many interesting systems. Fortunately, it is often possible to find expressions which are analytic

More information

CHAPTER STRUCTURE OF ATOM

CHAPTER STRUCTURE OF ATOM 12 CHAPTER STRUCTURE OF ATOM 1. The spectrum of He is expected to be similar to that [1988] H Li + Na He + 2. The number of spherical nodes in 3p orbitals are [1988] one three none two 3. If r is the radius

More information

Atomic Spectra in Astrophysics

Atomic Spectra in Astrophysics Atomic Spectra in Astrophysics Potsdam University : Wi 2016-17 : Dr. Lidia Oskinova lida@astro.physik.uni-potsdam.de Complex Atoms Non-relativistic Schrödinger Equation 02 [ N i=1 ( ) 2 2m e 2 i Ze2 4πǫ

More information

( ( ; R H = 109,677 cm -1

( ( ; R H = 109,677 cm -1 CHAPTER 9 Atomic Structure and Spectra I. The Hydrogenic Atoms (one electron species). H, He +1, Li 2+, A. Clues from Line Spectra. Reminder: fundamental equations of spectroscopy: ε Photon = hν relation

More information

Looks aren't everything... (Let's take a spin).

Looks aren't everything... (Let's take a spin). Looks aren't everything... (Let's take a spin). Bohr correctly deduced the energies for the hydrogen atom, but he didn t know about the Schrödinger Eq. and he didn t know about wavefunctions. So his picture

More information

Select/Special Topics in Atomic Physics Prof. P.C. Deshmukh Department Of Physics Indian Institute of Technology, Madras

Select/Special Topics in Atomic Physics Prof. P.C. Deshmukh Department Of Physics Indian Institute of Technology, Madras Select/Special Topics in Atomic Physics Prof. P.C. Deshmukh Department Of Physics Indian Institute of Technology, Madras Lecture - 37 Stark - Zeeman Spectroscopy Well, let us continue our discussion on

More information

H!!!! = E! Lecture 7 - Atomic Structure. Chem 103, Section F0F Unit II - Quantum Theory and Atomic Structure Lecture 7. Lecture 7 - Introduction

H!!!! = E! Lecture 7 - Atomic Structure. Chem 103, Section F0F Unit II - Quantum Theory and Atomic Structure Lecture 7. Lecture 7 - Introduction Chem 103, Section F0F Unit II - Quantum Theory and Atomic Structure Lecture 7 Lecture 7 - Atomic Structure Reading in Silberberg - Chapter 7, Section 4 The Qunatum-Mechanical Model of the Atom The Quantum

More information

e L 2m e the Bohr magneton

e L 2m e the Bohr magneton e L μl = L = μb 2m with : μ B e e 2m e the Bohr magneton Classical interation of magnetic moment and B field: (Young and Freedman, Ch. 27) E = potential energy = μ i B = μbcosθ τ = torque = μ B, perpendicular

More information

Luigi Paolasini

Luigi Paolasini Luigi Paolasini paolasini@esrf.fr LECTURE 2: LONELY ATOMS - Systems of electrons - Spin-orbit interaction and LS coupling - Fine structure - Hund s rules - Magnetic susceptibilities Reference books: -

More information

Problem Set 8 Solutions

Problem Set 8 Solutions University of Alabama Department of Physics and Astronomy PH 253 / LeClair Spring 21 Problem Set 8 Solutions 1. Multiplicity of atomic magnetic moments. Calculate the magnetic moments that are possible

More information

ONE AND MANY ELECTRON ATOMS Chapter 15

ONE AND MANY ELECTRON ATOMS Chapter 15 See Week 8 lecture notes. This is exactly the same as the Hamiltonian for nonrigid rotation. In Week 8 lecture notes it was shown that this is the operator for Lˆ 2, the square of the angular momentum.

More information

Chm 331 Fall 2015, Exercise Set 4 NMR Review Problems

Chm 331 Fall 2015, Exercise Set 4 NMR Review Problems Chm 331 Fall 015, Exercise Set 4 NMR Review Problems Mr. Linck Version.0. Compiled December 1, 015 at 11:04:44 4.1 Diagonal Matrix Elements for the nmr H 0 Find the diagonal matrix elements for H 0 (the

More information

Magnetism of Atoms and Ions. Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D Karlsruhe

Magnetism of Atoms and Ions. Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D Karlsruhe Magnetism of Atoms and Ions Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D-76131 Karlsruhe 1 0. Overview Literature J.M.D. Coey, Magnetism and

More information

Quantum Mechanics & Atomic Structure (Chapter 11)

Quantum Mechanics & Atomic Structure (Chapter 11) Quantum Mechanics & Atomic Structure (Chapter 11) Quantum mechanics: Microscopic theory of light & matter at molecular scale and smaller. Atoms and radiation (light) have both wave-like and particlelike

More information

( ) ( ) Last Time. 3-D particle in box: summary. Modified Bohr model. 3-dimensional Hydrogen atom. Orbital magnetic dipole moment

( ) ( ) Last Time. 3-D particle in box: summary. Modified Bohr model. 3-dimensional Hydrogen atom. Orbital magnetic dipole moment Last Time 3-dimensional quantum states and wave functions Decreasing particle size Quantum dots (particle in box) Course evaluations Thursday, Dec. 10 in class Last HW assignment available : for practice

More information

Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom

Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228 Happy April Fools Day Example / Worked Problems What is the ratio of the

More information

i = cos 2 0i + ei sin 2 1i

i = cos 2 0i + ei sin 2 1i Chapter 10 Spin 10.1 Spin 1 as a Qubit In this chapter we will explore quantum spin, which exhibits behavior that is intrinsically quantum mechanical. For our purposes the most important particles are

More information

5.111 Lecture Summary #6

5.111 Lecture Summary #6 5.111 Lecture Summary #6 Readings for today: Section 1.9 (1.8 in 3 rd ed) Atomic Orbitals. Read for Lecture #7: Section 1.10 (1.9 in 3 rd ed) Electron Spin, Section 1.11 (1.10 in 3 rd ed) The Electronic

More information

Complete nomenclature for electron orbitals

Complete nomenclature for electron orbitals Complete nomenclature for electron orbitals Bohr s model worked but it lacked a satisfactory reason why. De Broglie suggested that all particles have a wave nature. u l=h/p Enter de Broglie again It was

More information

We now turn to our first quantum mechanical problems that represent real, as

We now turn to our first quantum mechanical problems that represent real, as 84 Lectures 16-17 We now turn to our first quantum mechanical problems that represent real, as opposed to idealized, systems. These problems are the structures of atoms. We will begin first with hydrogen-like

More information

20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R

20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R 20 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 2. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian

More information

Introduction to Quantum Mechanics PVK - Solutions. Nicolas Lanzetti

Introduction to Quantum Mechanics PVK - Solutions. Nicolas Lanzetti Introduction to Quantum Mechanics PVK - Solutions Nicolas Lanzetti lnicolas@student.ethz.ch 1 Contents 1 The Wave Function and the Schrödinger Equation 3 1.1 Quick Checks......................................

More information

Chapter 9: Multi- Electron Atoms Ground States and X- ray Excitation

Chapter 9: Multi- Electron Atoms Ground States and X- ray Excitation Chapter 9: Multi- Electron Atoms Ground States and X- ray Excitation Up to now we have considered one-electron atoms. Almost all atoms are multiple-electron atoms and their description is more complicated

More information

St Hugh s 2 nd Year: Quantum Mechanics II. Reading. Topics. The following sources are recommended for this tutorial:

St Hugh s 2 nd Year: Quantum Mechanics II. Reading. Topics. The following sources are recommended for this tutorial: St Hugh s 2 nd Year: Quantum Mechanics II Reading The following sources are recommended for this tutorial: The key text (especially here in Oxford) is Molecular Quantum Mechanics, P. W. Atkins and R. S.

More information

Multielectron Atoms and Periodic Table

Multielectron Atoms and Periodic Table GRE Question Multielectron Atoms and Periodic Table Helium Atom 2 2m e ( 2 1 + 2 2) + 2ke 2 2ke 2 + ke2 r 1 r 2 r 2 r 1 Electron-electron repulsion term destroys spherical symmetry. No analytic solution

More information

Fine structure in hydrogen - relativistic effects

Fine structure in hydrogen - relativistic effects LNPhysiqueAtomique016 Fine structure in hydrogen - relativistic effects Electron spin ; relativistic effects In a spectrum from H (or from an alkali), one finds that spectral lines appears in pairs. take

More information

Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid

Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Quantum Physics II (8.05) Fall 2002 Assignment 12 and Study Aid Announcement This handout includes 9 problems. The first 5 are the problem set due. The last 4 cover material from the final few lectures

More information

H atom solution. 1 Introduction 2. 2 Coordinate system 2. 3 Variable separation 4

H atom solution. 1 Introduction 2. 2 Coordinate system 2. 3 Variable separation 4 H atom solution Contents 1 Introduction 2 2 Coordinate system 2 3 Variable separation 4 4 Wavefunction solutions 6 4.1 Solution for Φ........................... 6 4.2 Solution for Θ...........................

More information

only two orbitals, and therefore only two combinations to worry about, but things get

only two orbitals, and therefore only two combinations to worry about, but things get 131 Lecture 1 It is fairly easy to write down an antisymmetric wavefunction for helium since there are only two orbitals, and therefore only two combinations to worry about, but things get complicated

More information

An Introduction to Hyperfine Structure and Its G-factor

An Introduction to Hyperfine Structure and Its G-factor An Introduction to Hyperfine Structure and Its G-factor Xiqiao Wang East Tennessee State University April 25, 2012 1 1. Introduction In a book chapter entitled Model Calculations of Radiation Induced Damage

More information

Chemistry 3502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2003 Christopher J. Cramer. Lecture 15, October 8, 2003

Chemistry 3502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2003 Christopher J. Cramer. Lecture 15, October 8, 2003 Chemistry 350 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 003 Christopher J. Cramer Lecture 15, October 8, 003 Solved Homework Given that for a hydrogenic atom H = T + V 1 1 = Zr

More information

Angular Momentum Quantization: Physical Manifestations and Chemical Consequences

Angular Momentum Quantization: Physical Manifestations and Chemical Consequences Angular Momentum Quantization: Physical Manifestations and Chemical Consequences Michael Fowler, University of Virginia 7/7/07 The Stern-Gerlach Experiment We ve established that for the hydrogen atom,

More information

Chapter 4 Section 2 Notes

Chapter 4 Section 2 Notes Chapter 4 Section 2 Notes Vocabulary Heisenberg Uncertainty Principle- states that it is impossible to determine simultaneously both the position and velocity of an electron or any other particle. Quantum

More information

Chapter 8: Electrons in Atoms Electromagnetic Radiation

Chapter 8: Electrons in Atoms Electromagnetic Radiation Chapter 8: Electrons in Atoms Electromagnetic Radiation Electromagnetic (EM) radiation is a form of energy transmission modeled as waves moving through space. (see below left) Electromagnetic Radiation

More information

THE NATURE OF THE ATOM. alpha particle source

THE NATURE OF THE ATOM. alpha particle source chapter THE NATURE OF THE ATOM www.tutor-homework.com (for tutoring, homework help, or help with online classes) Section 30.1 Rutherford Scattering and the Nuclear Atom 1. Which model of atomic structure

More information

Hydrogen Atom Energies. Selection Rules. Hydrogen Atom Spectroscopy. Where is the Electron?

Hydrogen Atom Energies. Selection Rules. Hydrogen Atom Spectroscopy. Where is the Electron? Hydrogen Atom Energies Selection Rules Transitions can only occur between states that differ in l by 1, i.e. Δl = ± 1 E n = {-me 4 / 32π 2 ε 02 ħ 2 } {1/n 2 } = -13.6 ev / n 2 07/09/2011 PHYS255: Quantum

More information

( ) electron gives S = 1/2 and L = l 1

( ) electron gives S = 1/2 and L = l 1 Practice Modern Physics II, W018, Set 1 Question 1 Energy Level Diagram of Boron ion B + For neutral B, Z = 5 (A) Draw the fine-structure diagram of B + that includes all n = 3 states Label the states

More information

Electronic structure the number of electrons in an atom as well as the distribution of electrons around the nucleus and their energies

Electronic structure the number of electrons in an atom as well as the distribution of electrons around the nucleus and their energies Chemistry: The Central Science Chapter 6: Electronic Structure of Atoms Electronic structure the number of electrons in an atom as well as the distribution of electrons around the nucleus and their energies

More information

Electronic Structure of Atoms. Chapter 6

Electronic Structure of Atoms. Chapter 6 Electronic Structure of Atoms Chapter 6 Electronic Structure of Atoms 1. The Wave Nature of Light All waves have: a) characteristic wavelength, λ b) amplitude, A Electronic Structure of Atoms 1. The Wave

More information

The general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation

The general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation Lecture 17 Page 1 Lecture 17 L17.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent

More information