Solve Numerically : First normalize Then evaluate.

Slides:

Advertisements

Similar presentations

Rotations vs. Translations Translations Rotations.

Advertisements

Javier Junquera How to plot the radial part of the atomic orbitals.

CHAPTER 8 Hydrogen Atom 8.1 Spherical Coordinates

CHAPTER 7 The Hydrogen Atom

I.What to recall about motion in a central potential II.Example and solution of the radial equation for a particle trapped within radius “a” III.The spherical.

1 Cold molecules Mike Tarbutt. 2 Outline Lecture 1 – The electronic, vibrational and rotational structure of molecules. Lecture 2 – Transitions in molecules.

Chap 12 Quantum Theory: techniques and applications Objectives: Solve the Schrödinger equation for: Translational motion (Particle in a box) Vibrational.

Lecture 18: The Hydrogen Atom

Lecture 17: The Hydrogen Atom

1 7.1Application of the Schrödinger Equation to the Hydrogen Atom 7.2Solution of the Schrödinger Equation for Hydrogen 7.3Quantum Numbers 7.4Magnetic Effects.

QM in 3D Quantum Ch.4, Physical Systems, 24.Feb.2003 EJZ Schrödinger eqn in spherical coordinates Separation of variables (Prob.4.2 p.124) Angular equation.

Modern physics and Quantum Mechanics Physical Systems, 8 Mar.2007 EJZ More angular momentum and H atom Compare to Bohr atom Applications: Bohr magneton,

Quantum Ch.4 - continued Physical Systems, 27.Feb.2003 EJZ Recall solution to Schrödinger eqn in spherical coordinates with Coulomb potential (H atom)

Lecture 19: The Hydrogen Atom Reading: Zuhdahl Outline –The wavefunction for the H atom –Quantum numbers and nomenclature –Orbital shapes and.

Thermo & Stat Mech - Spring 2006 Class 19 1 Thermodynamics and Statistical Mechanics Partition Function.

Quantum Theory of Atoms The Bohr theory of Hydrogen(1913) cannot be extended to other atoms with more than one electron we have to solve the Schrödinger.

Wave mechanics in potentials Modern Ch.4, Physical Systems, 30.Jan.2003 EJZ Particle in a Box (Jason Russell), Prob.12 Overview of finite potentials Harmonic.

Quantum mechanics review. Reading for week of 1/28-2/1 – Chapters 1, 2, and 3.1,3.2 Reading for week of 2/4-2/8 – Chapter 4.

The Hydrogen Atom Quantum Physics 2002 Recommended Reading: Harris Chapter 6, Sections 3,4 Spherical coordinate system The Coulomb Potential Angular Momentum.

Physics 311 Classical Mechanics Welcome! Syllabus. Discussion of Classical Mechanics. Topics to be Covered. The Role of Classical Mechanics in Physics.

Semi-classics for non- integrable systems Lecture 8 of “Introduction to Quantum Chaos”

Ch 9 pages Lecture 23 – The Hydrogen Atom.

Physics 452 Quantum mechanics II Winter 2012 Karine Chesnel.

Sect. 3.4: The Virial Theorem Skim discussion. Read details on your own! Many particle system. Positions r i, momenta p i. Bounded. Define G  ∑ i r i.

Quantum mechanics unit 2

Physics 2170 – Spring Electron spin Homework is due Wednesday at 12:50pm Problem solving sessions M3-5.

Lecture VIII Hydrogen Atom and Many Electron Atoms dr hab. Ewa Popko.

Hydrogen Atom – Atomic Orbitals Lecture - 3 X Y Z x P (e - ) z Q y R.

An Electron Trapped in A Potential Well Probability densities for an infinite well Solve Schrödinger equation outside the well.

INTERFERENCE AND QUANTIZATION IN SEMICLASSICAL VIBRATIONAL RESPONSE FUNCTIONS Scott Gruenbaum Department of Chemistry and Chemical Biology Cornell University.

Physics 451 Quantum mechanics I Fall 2012 Nov 7, 2012 Karine Chesnel.

LECTURE 21 THE HYDROGEN AND HYDROGENIC ATOMS PHYSICS 420 SPRING 2006 Dennis Papadopoulos.

Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 18 A Quantum Mechanical Model for the Vibration and Rotation of Molecules.

Chapter 41 1D Wavefunctions. Topics: Schrödinger’s Equation: The Law of Psi Solving the Schrödinger Equation A Particle in a Rigid Box: Energies and Wave.

Wednesday, Nov. 13, 2013 PHYS , Fall 2013 Dr. Jaehoon Yu 1 PHYS 3313 – Section 001 Lecture #18 Wednesday, Nov. 13, 2013 Dr. Jaehoon Yu Solutions.

Hydrogen Atom and QM in 3-D 1. HW 8, problem 6.32 and A review of the hydrogen atom 2. Quiz Topics in this chapter:  The hydrogen atom  The.

Physics Lecture 14 3/22/ Andrew Brandt Monday March 22, 2010 Dr. Andrew Brandt 1.Hydrogen Atom 2.HW 6 on Ch. 7 to be assigned Weds 3/24.

Quantum mechanics unit 2

Physics 361 Principles of Modern Physics Lecture 22.

Quantum Chemistry: Our Agenda (along with Engel)

The Bohr Model and The principal Quantum Number Physics 12 Adv.

MS310 Quantum Physical Chemistry

Physics 451 Quantum mechanics I Fall 2012 Nov 20, 2012 Karine Chesnel.

A QUANTUM LEAP IN MATH By Michael Smith.

Intermediate Quantum Mechanics PHYS307 Professor Scott Heinekamp Goals of the course by speculating on possible analogies between waves moving in a uniform.

Standing Waves Reminder Confined waves can interfere with their reflections Easy to see in one and two dimensions –Spring and slinky –Water surface –Membrane.

IR Spectroscopy Wave length ~ 100 mm to 1 mm

Hydrogen Atom PHY Outline  review of L z operator, eigenfunction, eigenvalues rotational kinetic energy traveling and standing waves.

2.1Application of the Schrödinger Equation to the Hydrogen Atom 2.2Solution of the Schrödinger Equation for Hydrogen 2.3Quantum Numbers 2.4Magnetic Effects.

Quantum Chemistry: Our Agenda Postulates in quantum mechanics (Ch. 3) Schrödinger equation (Ch. 2) Simple examples of V(r) Particle in a box (Ch. 4-5)

The Hydrogen Atom The only atom that can be solved exactly.

Phonons Packets of sound found present in the lattice as it vibrates … but the lattice vibration cannot be heard. Unlike static lattice model , which.

CHAPTER 7 The Hydrogen Atom

Quantum Theory of Hydrogen Atom

Population distribution of vibrational levels in the 21 state of NaLi

Ground State of the He Atom – 1s State

The Hydrogen Atom The only atom that can be solved exactly.

Preamble to the Constant Alpha

Derivation of the Rydberg Constant

Diatomic molecules

Central Potential Another important problem in quantum mechanics is the central potential problem This means V = V(r) only This means angular momentum.

QM2 Concept Test 2.1 Which one of the following pictures represents the surface of constant

Paramagnetism – semiclassical

Bohr Model In addition to the atomic line spectra of single electron atoms, there were other successes of the Bohr Model X-ray spectra Frank-Hertz experiment.

Quantum Theory of Hydrogen Atom

6: Barrier Tunneling and Atomic Physics

Indices Practice questions

The Shell Model of the Nucleus 2. The primitive model

CHAPTER 7 The Hydrogen Atom

Presentation transcript:

Solve Numerically : First normalize Then evaluate

Average Values of Powers of the Coordinate

Kepler Coulomb : V(r ) = - k/r Isotropic Harmonic Oscillator : V(r ) = Kr2/2

Numerical Solution: Hydrogen atom (3p) E = - Ry / 9 ; L = 3ħ/2 ; V(r ) = - Ry a0 /r Radial momentum ( Units : [energy] = Ry ; [length] = a0 Limits: A- = 1.2085 ; A+ = 16.794 Normalization : C = 84.82 Moments: r = 12.375 r -1 = 0.1111 r -2 = 0.02469 r -3 = 0.01097

Numerical Solution: Isotropic SHO (nr=0 ; l=3) E = 9ħ/2 ; L = 7ħ/2 ; V(r ) = m2 r2 / 2 Radial momentum ( Units : [energy] = ħ ; [length] = (ħ/m)1/2 Limits: A- = 1.2929 ; A+ = 2.7071 Normalization : C = 2.215 Moments: r 2 = 4.500 r 4 = 24.25 r -2 = 0.2825 r -4 = 0.1050

Einstein-Brillouin-Keller Action Quantization (1917) (1926) (1958) Bohr-Sommerfeld-Wilson quantization used fuzzy math, neglecting caustics at turning points in librations. The correct semiclassical action quantization condition is: where i = 0 (rotations) Topological Maslov Index = 2 (librations) It yields astonishingly accurate results !!!