What is this? Jerry Gilfoyle The Hydrogen Atom 1 / 18
|
|
- Emery Parrish
- 5 years ago
- Views:
Transcription
1 What is this? Jey Gilfoyle The Hydogen Atom 1 / 18
2 What is this? The Hydogen Atom Jey Gilfoyle The Hydogen Atom 1 / 18
3 What is this? The Hydogen Atom Jey Gilfoyle The Hydogen Atom 1 / 18
4 What is this? The Hydogen Atom 1 ( 1 λ = R H n f 2 1 ) n i 2 R H - Rydbeg constant Jey Gilfoyle The Hydogen Atom 1 / 18
5 Hydogen Eigenvalues 13.6 ev E n = n 2 Quantitative compaison fo Balme seies hydogen in units of σ. Line My Results (Å) NIST Results (Å) Nomalized Pecent Diffeence Diffeence α 6.64 ± β 4.85 ± γ 4.39 ± α : n = 3 n = 2 β : n = 4 n = 2 γ : n = 5 n = 2 Jey Gilfoyle The Hydogen Atom 2 / 18
6 n = 8, l = 3, m = 1 Jey Gilfoyle The Hydogen Atom 3 / 18
7 How do we build the quantum model? 1 What is the mechanical enegy? Jey Gilfoyle The Hydogen Atom 4 / 18
8 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e Jey Gilfoyle The Hydogen Atom 4 / 18
9 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? Jey Gilfoyle The Hydogen Atom 4 / 18
10 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? ( 2 1 2µ sin θ 2 2µ 2 ϕ s ( ) e2 ϕ s( ) = Eϕ s ( ) θ sin θ θ sin 2 θ 2 2 φ ) ϕ s( ) e2 ϕs( ) = Eϕs( ) Jey Gilfoyle The Hydogen Atom 4 / 18
11 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? ( 2 1 2µ sin θ 2 2µ 2 ϕ s ( ) e2 ϕ s( ) = Eϕ s ( ) θ sin θ θ sin 2 θ 3 What do we know about the solution? 2 2 φ ) ϕ s( ) e2 ϕs( ) = Eϕs( ) Jey Gilfoyle The Hydogen Atom 4 / 18
12 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? ( 2 1 2µ sin θ 2 2µ 2 ϕ s ( ) e2 ϕ s( ) = Eϕ s ( ) θ sin θ θ sin 2 θ 3 What do we know about the solution? 2 2 φ ) ϕ s( ) e2 ϕs( ) = Eϕs( ) ϕ s ( ) = R()Θ(θ)Φ(φ) = R()Y m l (θ, φ) Jey Gilfoyle The Hydogen Atom 4 / 18
13 How do we build the quantum model? 1 What is the mechanical enegy? E = p2 2µ e2 = p2 2µ + L2 2µ 2 e2 µ = m pm e m p + m e m e 2 What is the Schoedinge equation? ( 2 1 2µ sin θ 2 2µ 2 ϕ s ( ) e2 ϕ s( ) = Eϕ s ( ) θ sin θ θ sin 2 θ 2 2 φ GO SOLVE IT! 3 What do we know about the solution? ) ϕ s( ) e2 ϕs( ) = Eϕs( ) ϕ s ( ) = R()Θ(θ)Φ(φ) = R()Y m l (θ, φ) Jey Gilfoyle The Hydogen Atom 4 / 18
14 Hydogen Bound State Eigenfunctions ϕ nlm (, θ, φ) = R nl ()Y m l (θ, φ) = (2κ) 3/2 A nl ρ l e ρ/2 F nl (ρ)y m l (θ, φ) Jey Gilfoyle The Hydogen Atom 5 / 18
15 Hydogen Bound State Eigenfunctions ϕ nlm (, θ, φ) = R nl ()Y m l (θ, φ) = (2κ) 3/2 A nl ρ l e ρ/2 F nl (ρ)y m l (θ, φ) F (ρ) = = a i ρ i (i + l + 1) λ a i+1 = (i + 1)(i + 2l + 2) a i a 0 = 1 i=0 E n = E ρ = 2κ κ = 2µ E F nl (ρ) = L 2l+1 n l 1 (ρ) A nl = 2 λ = Ze2 µ 2 E (n l 1)! 2n[(n + l)!] 3 Jey Gilfoyle The Hydogen Atom 5 / 18
16 Hydogen Eigenvalues (Enegy Levels) 5 Continuum States 0 E n = µ(e2 ) n 2 ev = 13.6 n 2 Enegy (ev) -5 Discete States Jey Gilfoyle The Hydogen Atom 6 / 18
17 Hydogen Bound State Eigenfunctions ψ Enlm (, θ, φ) = R nl ()Yl m (θ, φ) ( kmax ) = A nl ρ l e ρ b k ρ k Yl m (θ, φ) k=0 Jey Gilfoyle The Hydogen Atom 7 / 18
18 Hydogen Bound State Eigenfunctions ψ Enlm (, θ, φ) = R nl ()Yl m (θ, φ) ( kmax ) = A nl ρ l e ρ b k ρ k Yl m (θ, φ) k=0 ψ Enlm = 2(k + l + 1) λe2 b k+1 = (k + 1)(k + 2l + 2) b k b 0 = 1 2µW 2µ E n = W ρ = κ κ = λ = 2 2 W ( 2 na 0 a0 = 2 me 2 ) 3 ( ) l ( ) (n l 1)! 2n[(n + l)!] 3 e /na 0 2 (n + l)! L 2l+1 2 n l 1 Yl m (θ, φ) na 0 na 0 Jey Gilfoyle The Hydogen Atom 7 / 18
19 Recall the Solid Angle Jey Gilfoyle The Hydogen Atom 8 / 18
20 Spheical Diffeential Volume Element Jey Gilfoyle The Hydogen Atom 9 / 18
21 Hydogen Eigenfunctions Hydogen Pobability Density (n=4) Red - l=0 P Jey Gilfoyle The Hydogen Atom 10 / 18
22 Hydogen Eigenfunctions Hydogen Pobability Density (n=4) Red - l=0 Blue - l=1 P Jey Gilfoyle The Hydogen Atom 11 / 18
23 Hydogen Eigenfunctions Hydogen Pobability Density (n=4) Red - l=0 Blue - l=1 Geen - l=2 P Jey Gilfoyle The Hydogen Atom 12 / 18
24 Hydogen Eigenfunctions Hydogen Pobability Density (n=4) P Red - l=0 Blue - l=1 Geen - l=2 Gay - l= Jey Gilfoyle The Hydogen Atom 13 / 18
25 Do the peaks line up? Red : n=1, Blue: n= Pobability Density (angstoms) Jey Gilfoyle The Hydogen Atom 14 / 18
26 Old Obitals Jey Gilfoyle The Hydogen Atom 15 / 18
27 Old Obitals - New Obitals Jey Gilfoyle The Hydogen Atom 15 / 18
28 Old Obitals - New Obitals How ae these plots elated to what we know? Jey Gilfoyle The Hydogen Atom 15 / 18
29 Moe Hydogen Eigenfunctions Jey Gilfoyle The Hydogen Atom 16 / 18
30 Hydogen Eigenvalues 13.6 ev E n = n 2 Quantitative compaison fo Balme seies hydogen in units of σ. Line My Results (Å) NIST Results (Å) Nomalized Pecent Diffeence Diffeence α 6.64 ± β 4.85 ± γ 4.39 ± α : n = 3 n = 2 β : n = 4 n = 2 γ : n = 5 n = 2 Jey Gilfoyle The Hydogen Atom 17 / 18
31 Some Plots Jey Gilfoyle The Hydogen Atom 18 / 18
Lecture 7: Angular Momentum, Hydrogen Atom
Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z
More information3.012 Fund of Mat Sci: Bonding Lecture 5/6. Comic strip removed for copyright reasons.
3.12 Fund of Mat Sci: Bonding Lectue 5/6 THE HYDROGEN ATOM Comic stip emoved fo copyight easons. Last Time Metal sufaces and STM Diac notation Opeatos, commutatos, some postulates Homewok fo Mon Oct 3
More information( ) into above PDE. ( ), wherec = 1
xample of how to veify a Hydogen Solution The hydogen atom solution is pesented in section 7., equation 7.7, ψ nlml,θ,φ) R nl ) θ,φ ae shown in 7. and 7.. It is the solution of the patial diffeential equation
More information5.111 Lecture Summary #6 Monday, September 15, 2014
5.111 Lectue Summay #6 Monday, Septembe 15, 014 Readings fo today: Section 1.9 Atomic Obitals. Section 1.10 Electon Spin, Section 1.11 The Electonic Stuctue of Hydogen. (Same sections in 4 th ed.) Read
More informationPHYSICS 4E FINAL EXAM SPRING QUARTER 2010 PROF. HIRSCH JUNE 11 Formulas and constants: hc =12,400 ev A ; k B. = hf " #, # $ work function.
PHYSICS 4E FINAL EXAM SPRING QUARTER 1 Fomulas and constants: hc =1,4 ev A ; k B =1/11,6 ev/k ; ke =14.4eVA ; m e c =.511"1 6 ev ; m p /m e =1836 Relativistic enegy - momentum elation E = m c 4 + p c ;
More information3.23 Electrical, Optical, and Magnetic Properties of Materials
MIT OpenCouseWae http://ocw.mit.edu 3.23 Electical, Optical, and Magnetic Popeties of Mateials Fall 27 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems. 3.23 Fall
More information= e2. = 2e2. = 3e2. V = Ze2. where Z is the atomic numnber. Thus, we take as the Hamiltonian for a hydrogenic. H = p2 r. (19.4)
Chapte 9 Hydogen Atom I What is H int? That depends on the physical system and the accuacy with which it is descibed. A natual stating point is the fom H int = p + V, (9.) µ which descibes a two-paticle
More informationThree-dimensional systems with spherical symmetry
Thee-dimensiona systems with spheica symmety Thee-dimensiona systems with spheica symmety 006 Quantum Mechanics Pof. Y. F. Chen Thee-dimensiona systems with spheica symmety We conside a patice moving in
More information1 r 2 sin 2 θ. This must be the case as we can see by the following argument + L2
PHYS 4 3. The momentum operator in three dimensions is p = i Therefore the momentum-squared operator is [ p 2 = 2 2 = 2 r 2 ) + r 2 r r r 2 sin θ We notice that this can be written as sin θ ) + θ θ r 2
More informationThe Hydrogen Atom. Chapter 18. P. J. Grandinetti. Nov 6, Chem P. J. Grandinetti (Chem. 4300) The Hydrogen Atom Nov 6, / 41
The Hydrogen Atom Chapter 18 P. J. Grandinetti Chem. 4300 Nov 6, 2017 P. J. Grandinetti (Chem. 4300) The Hydrogen Atom Nov 6, 2017 1 / 41 The Hydrogen Atom Hydrogen atom is simplest atomic system where
More informationPhysics 505 Homework No. 9 Solutions S9-1
Physics 505 Homewok No 9 s S9-1 1 As pomised, hee is the tick fo summing the matix elements fo the Stak effect fo the gound state of the hydogen atom Recall, we need to calculate the coection to the gound
More informationAnyone who can contemplate quantum mechanics without getting dizzy hasn t understood it. --Niels Bohr. Lecture 17, p 1
Anyone who can contemplate quantum mechanics without getting dizzy hasn t undestood it. --Niels Boh Lectue 17, p 1 Special (Optional) Lectue Quantum Infomation One of the most moden applications of QM
More information3.23 Electrical, Optical, and Magnetic Properties of Materials
MIT OpenCouseWae http://ocw.mit.edu 3.3 Electical, Optical, and Magnetic Popeties of Mateials Fall 7 Fo infomation about citing these mateials o ou Tems of Use, visit: http://ocw.mit.edu/tems. 3.3 Fall
More informationQuantum theory of angular momentum
Quantum theoy of angula momentum Igo Mazets igo.mazets+e141@tuwien.ac.at (Atominstitut TU Wien, Stadionallee 2, 1020 Wien Time: Fiday, 13:00 14:30 Place: Feihaus, Sem.R. DA gün 06B (exception date 18 Nov.:
More informationPreliminary Exam: Quantum Physics 1/14/2011, 9:00-3:00
Peliminay Exam: Quantum Physics /4/ 9:-: Answe a total of SIX questions of which at least TWO ae fom section A and at least THREE ae fom section B Fo you answes you can use eithe the blue books o individual
More informationEnergy Levels Of Hydrogen Atom Using Ladder Operators. Ava Khamseh Supervisor: Dr. Brian Pendleton The University of Edinburgh August 2011
Enegy Levels Of Hydogen Atom Using Ladde Opeatos Ava Khamseh Supeviso: D. Bian Pendleton The Univesity of Edinbugh August 11 1 Abstact The aim of this pape is to fist use the Schödinge wavefunction methods
More informationQuantum Mechanics: The Hydrogen Atom
Quantum Mechanics: The Hydrogen Atom 4th April 9 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen
More information20th Century Atomic Theory - Hydrogen Atom
0th Centuy Atomic Theoy - Hydogen Atom Ruthefod s scatteing expeiments (Section.5, pp. 53-55) in 1910 led to a nuclea model of the atom whee all the positive chage and most of the mass wee concentated
More informationThe Postulates. What is a postulate? Jerry Gilfoyle The Rules of the Quantum Game 1 / 21
The Postulates What is a postulate? Jerry Gilfoyle The Rules of the Quantum Game 1 / 21 The Postulates What is a postulate? 1 suggest or assume the existence, fact, or truth of (something) as a basis for
More informationDoublet structure of Alkali spectra:
Doublet stuctue of : Caeful examination of the specta of alkali metals shows that each membe of some of the seies ae closed doublets. Fo example, sodium yellow line, coesponding to 3p 3s tansition, is
More information2m r2 (~r )+V (~r ) (~r )=E (~r )
Review of the Hydrogen Atom The Schrodinger equation (for 1D, 2D, or 3D) can be expressed as: ~ 2 2m r2 (~r, t )+V (~r ) (~r, t )=i~ @ @t The Laplacian is the divergence of the gradient: r 2 =r r The time-independent
More informationGraduate Quantum Mechanics I: Prelims and Solutions (Fall 2015)
Graduate Quantum Mechanics I: Prelims and Solutions (Fall 015 Problem 1 (0 points Suppose A and B are two two-level systems represented by the Pauli-matrices σx A,B σ x = ( 0 1 ;σ 1 0 y = ( ( 0 i 1 0 ;σ
More informationOutlines of Quantum Physics
Duality S. Eq Hydrogen Outlines of 1 Wave-Particle Duality 2 The Schrödinger Equation 3 The Hydrogen Atom Schrödinger Eq. of the Hydrogen Atom Noninteracting Particles and Separation of Variables The One-Particle
More informationc n ψ n (r)e ient/ h (2) where E n = 1 mc 2 α 2 Z 2 ψ(r) = c n ψ n (r) = c n = ψn(r)ψ(r)d 3 x e 2r/a0 1 πa e 3r/a0 r 2 dr c 1 2 = 2 9 /3 6 = 0.
Poblem {a} Fo t : Ψ(, t ψ(e iet/ h ( whee E mc α (α /7 ψ( e /a πa Hee we have used the gound state wavefunction fo Z. Fo t, Ψ(, t can be witten as a supeposition of Z hydogenic wavefunctions ψ n (: Ψ(,
More information1.4 Solution of the Hydrogen Atom
The phase of α can freely be chosen to be real so that α = h (l m)(l + m + 1). Then L + l m = h (l m)(l + m + 1) l m + 1 (1.24) L l m = h (l + m)(l m + 1) l m 1 (1.25) Since m is bounded, it follow that
More informationElectric fields : Stark effect, dipole & quadrupole polarizability.
Electric fields : Stark effect, dipole & quadrupole polarizability. We are often interested in the effect of an external electric field on the energy levels and wavefunction of H and other one-electron
More informationf(k) e p 2 (k) e iax 2 (k a) r 2 e a x a a 2 + k 2 e a2 x 1 2 H(x) ik p (k) 4 r 3 cos Y 2 = 4
Fouie tansfom pais: f(x) 1 f(k) e p 2 (k) p e iax 2 (k a) 2 e a x a a 2 + k 2 e a2 x 1 2, a > 0 a p k2 /4a2 e 2 1 H(x) ik p 2 + 2 (k) The fist few Y m Y 0 0 = Y 0 1 = Y ±1 1 = l : 1 Y2 0 = 4 3 ±1 cos Y
More informationPhysics 862: Atoms, Nuclei, and Elementary Particles
Physics 86: Atoms, Nuclei, and Elementay Paticles Bian Bockelman Septembe 11, 008 Contents 1 Cental Field Poblems 1.1 Classical Teatment......................... 1. Quantum Teatment.........................
More informationThe Schrödinger Equation in Three Dimensions
The Schödinge Equation in Thee Dimensions Paticle in a Rigid Thee-Dimensional Box (Catesian Coodinates) To illustate the solution of the time-independent Schödinge equation (TISE) in thee dimensions, we
More informationAngular momentum. Quantum mechanics. Orbital angular momentum
Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular
More informationMore On Carbon Monoxide
More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations of CO 1 / 26 More On Carbon Monoxide E = 0.25 ± 0.05 ev Electron beam results Jerry Gilfoyle The Configurations
More informationChapter 8: Spherical Coordinates
6 Chapte 8: Spheical Coodinates Tiple Integals We've seen that Mathematica can compute integals in Catesian coodinates (x, y, z). Howeve, atoms ae bette descibed using spheical coodinates (, q, f). Hee
More informationQUANTUM MECHANICS A (SPA 5319) The Finite Square Well
QUANTUM MECHANICS A (SPA 5319) The Finite Square Well We have already solved the problem of the infinite square well. Let us now solve the more realistic finite square well problem. Consider the potential
More informationKey Concepts for this section
Key Concepts fo this section 1: Loentz foce law, Field, Maxwell s equation : Ion Tanspot, Nenst-Planck equation 3: (Quasi)electostatics, potential function, 4: Laplace s equation, Uniqueness 5: Debye laye,
More informationQuantum Mechanics and Stellar Spectroscopy
Quantum Mechanics and Stella Spectoscopy http://apod.nasa.gov/apod/ Recall the electic foce. Like gavity it is a 1/ 2 foce/ That is: F elec = Z 1Z 2 e 2 2 whee Z 1 and Z 2 ae the (intege) numbes of electonic
More informationPhysics 216 Spring The Variational Computation of the Ground State Energy of Helium
Physics 26 Spring 22 The Variational Computation of the Ground State Energy of Helium I. Introduction to the variational computation where The Hamiltonian for the two-electron system of the helium atom
More informationWe 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 informationChemistry 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 informationNuclear size corrections to the energy levels of single-electron atoms
Nuclea size coections to the enegy levels of single-electon atoms Babak Nadii Nii a eseach Institute fo Astonomy and Astophysics of Maagha (IAAM IAN P. O. Box: 554-44. Abstact A study is made of nuclea
More informationPhysics 2203, 2011: Equation sheet for second midterm. General properties of Schrödinger s Equation: Quantum Mechanics. Ψ + UΨ = i t.
General properties of Schrödinger s Equation: Quantum Mechanics Schrödinger Equation (time dependent) m Standing wave Ψ(x,t) = Ψ(x)e iωt Schrödinger Equation (time independent) Ψ x m Ψ x Ψ + UΨ = i t +UΨ
More informationQuantum Mechanics I - Session 5
Quantum Mechanics I - Session 5 Apil 7, 015 1 Commuting opeatos - an example Remine: You saw in class that Â, ˆB ae commuting opeatos iff they have a complete set of commuting obsevables. In aition you
More informationMechanics Physics 151
Mechanics Physics 151 Lectue 5 Cental Foce Poblem (Chapte 3) What We Did Last Time Intoduced Hamilton s Pinciple Action integal is stationay fo the actual path Deived Lagange s Equations Used calculus
More informationPOISSON S EQUATION 2 V 0
POISSON S EQUATION We have seen how to solve the equation but geneally we have V V4k We now look at a vey geneal way of attacking this poblem though Geen s Functions. It tuns out that this poblem has applications
More informationQUASI-STATIONARY ELECTRON STATES IN SPHERICAL ANTI-DOT WITH DONOR IMPURITY * 1. INTRODUCTION
ATOMIC PHYSICS QUASI-STATIONARY ELECTRON STATES IN SPHERICAL ANTI-DOT ITH DONOR IMPURITY * V. HOLOVATSKY, O. MAKHANETS, I. FRANKIV Chenivtsi National Univesity, Chenivtsi, 581, Ukaine, E-mail: ktf@chnu.edu.ua
More informationKey Questions. ECE 340 Lecture 4 : Bonding Forces and Energy Bands 1/28/14. Class Outline: v Crystal Diffraction Crystal Bonding
ECE 340 Lectue 4 : onding Foces and Enegy ands v Cystal Diffaction Class Outline: Things you should know when you leave Key Questions Why is the oh model useful? What is the Schödinge equation? What is
More information( ) ( ) 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 Couse evaluations Tuesday, Dec. 9 in class Deceasing paticle size Quantum dots paticle in box) Optional exta class: eview of mateial since Exam
More information( n x ( ) Last Time Exam 3 results. Question. 3-D particle in box: summary. Modified Bohr model. 3-D Hydrogen atom. r n. = n 2 a o
Last Time Exam 3 esults Quantum tunneling 3-dimensional wave functions Deceasing paticle size Quantum dots paticle in box) This week s honos lectue: Pof. ad histian, Positon Emission Tomogaphy Tue. Dec.
More informationPES 3950/PHYS 6950: Homework Assignment 6
PES 3950/PHYS 6950: Homewok Assignment 6 Handed out: Monday Apil 7 Due in: Wednesday May 6, at the stat of class at 3:05 pm shap Show all woking and easoning to eceive full points. Question 1 [5 points]
More informationOrbital Angular Momentum Eigenfunctions
Obital Angula Moentu Eigenfunctions Michael Fowle 1/11/08 Intoduction In the last lectue we established that the opeatos J Jz have a coon set of eigenkets j J j = j( j+ 1 ) j Jz j = j whee j ae integes
More informationb Ψ Ψ Principles of Organic Chemistry lecture 22, page 1
Pinciples of Oganic Chemisty lectue, page. Basis fo LCAO and Hückel MO Theoy.. Souces... Hypephysics online. http://hypephysics.phy-ast.gsu.edu/hbase/quantum/qm.html#c... Zimmeman, H. E., Quantum Mechanics
More informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationQuantum Mechanics and Stellar Spectroscopy
Quantum Mechanics and Stella Spectoscopy Recall the electic foce. Like gavity it is a 1/ 2 foce/ That is: e = 4.803 10 10 esu e 2 = 2.307 10 19 dyne cm 2 F elec = Z 1 Z 2 e2 2 whee Z 1 and Z 2 ae the (intege)
More informationOne-electron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation:
One-electron Atom The atomic orbitals of hydrogen-like atoms are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's
More informationLecture 10. Central potential
Lecture 10 Central potential 89 90 LECTURE 10. CENTRAL POTENTIAL 10.1 Introduction We are now ready to study a generic class of three-dimensional physical systems. They are the systems that have a central
More informationThe Central Force Problem: Hydrogen Atom
The Central Force Problem: Hydrogen Atom B. Ramachandran Separation of Variables The Schrödinger equation for an atomic system with Z protons in the nucleus and one electron outside is h µ Ze ψ = Eψ, r
More informationA Relativistic Electron in a Coulomb Potential
A Relativistic Electon in a Coulomb Potential Alfed Whitehead Physics 518, Fall 009 The Poblem Solve the Diac Equation fo an electon in a Coulomb potential. Identify the conseved quantum numbes. Specify
More informationLectures 21 and 22: Hydrogen Atom. 1 The Hydrogen Atom 1. 2 Hydrogen atom spectrum 4
Lectures and : Hydrogen Atom B. Zwiebach May 4, 06 Contents The Hydrogen Atom Hydrogen atom spectrum 4 The Hydrogen Atom Our goal here is to show that the two-body quantum mechanical problem of the hydrogen
More informationr 2 dr h2 α = 8m2 q 4 Substituting we find that variational estimate for the energy is m e q 4 E G = 4
Variational calculations for Hydrogen and Helium Recall the variational principle See Chapter 16 of the textbook The variational theorem states that for a Hermitian operator H with the smallest eigenvalue
More informationLecture 8: Radial Distribution Function, Electron Spin, Helium Atom
Lecture 8: Radial Distribution Function, Electron Spin, Helium Atom Radial Distribution Function The interpretation of the square of the wavefunction is the probability density at r, θ, φ. This function
More informationPhysics 506 Winter 2006 Homework Assignment #9 Solutions
Physics 506 Winte 2006 Homewok Assignment #9 Solutions Textbook poblems: Ch. 12: 12.2, 12.9, 12.13, 12.14 12.2 a) Show fom Hamilton s pinciple that Lagangians that diffe only by a total time deivative
More information2 Lecture 2: The Bohr atom (1913) and the Schrödinger equation (1925)
1 Lectue 1: The beginnings of quantum physics 1. The Sten-Gelach expeiment. Atomic clocks 3. Planck 1900, blackbody adiation, and E ω 4. Photoelectic effect 5. Electon diffaction though cystals, de Boglie
More informationPhysics 4617/5617: Quantum Physics Course Lecture Notes
Physics 467/567: Quantum Physics Course Lecture Notes Dr. Donald G. Luttermoser East Tennessee State University Edition 5. Abstract These class notes are designed for use of the instructor and students
More informationIntroduction to Quantum Physics and Models of Hydrogen Atom
Introduction to Quantum Physics and Models of Hydrogen Atom Tien-Tsan Shieh Department of Applied Math National Chiao-Tung University November 7, 2012 Physics and Models of Hydrogen November Atom 7, 2012
More informationThe 3 dimensional Schrödinger Equation
Chapter 6 The 3 dimensional Schrödinger Equation 6.1 Angular Momentum To study how angular momentum is represented in quantum mechanics we start by reviewing the classical vector of orbital angular momentum
More informationF(r) = r f (r) 4.8. Central forces The most interesting problems in classical mechanics are about central forces.
4.8. Cental foces The most inteesting poblems in classical mechanics ae about cental foces. Definition of a cental foce: (i) the diection of the foce F() is paallel o antipaallel to ; in othe wods, fo
More informationSchrödinger equation for central potentials
Chapter 2 Schrödinger equation for central potentials In this chapter we will extend the concepts and methods introduced in the previous chapter for a one-dimensional problem to a specific and very important
More informationSchrödinger equation for central potentials
Chapter 2 Schrödinger equation for central potentials In this chapter we will extend the concepts and methods introduced in the previous chapter ifor a one-dimenional problem to a specific and very important
More informationLecture 1. time, say t=0, to find the wavefunction at any subsequent time t. This can be carried out by
Lectue The Schödinge equation In quantum mechanics, the fundamenta quantity that descibes both the patice-ike and waveike chaacteistics of patices is wavefunction, Ψ(. The pobabiity of finding a patice
More informationThe Hydrogen atom. Chapter The Schrödinger Equation. 2.2 Angular momentum
Chapter 2 The Hydrogen atom In the previous chapter we gave a quick overview of the Bohr model, which is only really valid in the semiclassical limit. cf. section 1.7.) We now begin our task in earnest
More informationPhysical Chemistry II (Chapter 4 1) Rigid Rotor Models and Angular Momentum Eigenstates
Physical Chemisty II (Chapte 4 ) Rigid Roto Models and Angula Momentum Eigenstates Tae Kyu Kim Depatment of Chemisty Rm. 30 (tkkim@pusan.ac.k) http://cafe.nave.com/moneo76 SUMMAR CHAPTER 3 A simple QM
More informationAddition of Angular Momentum
Addition of Angula Moentu We ve leaned tat angula oentu i ipotant in quantu ecanic Obital angula oentu L Spin angula oentu S Fo ultielecton ato, we need to lean to add angula oentu Multiple electon, eac
More informationUNIVERSITY OF MARYLAND Department of Physics College Park, Maryland. PHYSICS Ph.D. QUALIFYING EXAMINATION PART II
UNIVERSITY OF MARYLAND Department of Physics College Park, Maryland PHYSICS Ph.D. QUALIFYING EXAMINATION PART II January 22, 2016 9:00 a.m. 1:00 p.m. Do any four problems. Each problem is worth 25 points.
More information11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below.
Fall 2007 Qualifie Pat II 12 minute questions 11) A thin, unifom od of mass M is suppoted by two vetical stings, as shown below. Find the tension in the emaining sting immediately afte one of the stings
More informationApproximation Methods in QM
Chapter 3 Approximation Methods in QM Contents 3.1 Time independent PT (nondegenerate)............... 5 3. Degenerate perturbation theory (PT)................. 59 3.3 Time dependent PT and Fermi s golden
More informationMany Electron Atoms. Electrons can be put into approximate orbitals and the properties of the many electron systems can be catalogued
Many Electon Atoms The many body poblem cannot be solved analytically. We content ouselves with developing appoximate methods that can yield quite accuate esults (but usually equie a compute). The electons
More informationChemical Engineering 412
Chemical Engineeing 41 Intoductoy Nuclea Engineeing Lectue 16 Nuclea eacto Theoy III Neuton Tanspot 1 One-goup eacto Equation Mono-enegetic neutons (Neuton Balance) DD φφ aa φφ + ss 1 vv vv is neuton speed
More informationObjectives. We will also get to know about the wavefunction and its use in developing the concept of the structure of atoms.
Modue "Atomic physics and atomic stuctue" Lectue 7 Quantum Mechanica teatment of One-eecton atoms Page 1 Objectives In this ectue, we wi appy the Schodinge Equation to the simpe system Hydogen and compae
More informationQ. Obtain the Hamiltonian for a one electron atom in the presence of an external magnetic field.
Syed Ashad Hussain Lectue Deatment of Physics Tiua Univesity www.sahussaintu.wodess.com Q. Obtain the Hamiltonian fo a one electon atom in the esence of an extenal magnetic field. To have an idea about
More informationQuantum Physics Lecture 8
Quantum Physics ecture 8 Steady state Schroedinger Equation (SSSE): eigenvalue & eigenfunction particle in a box re-visited Wavefunctions and energy states normalisation probability density Expectation
More informationScattering in Three Dimensions
Scatteing in Thee Dimensions Scatteing expeiments ae an impotant souce of infomation about quantum systems, anging in enegy fom vey low enegy chemical eactions to the highest possible enegies at the LHC.
More informationPhysics 115C Homework 2
Physics 5C Homework Problem Our full Hamiltonian is H = p m + mω x +βx 4 = H +H where the unperturbed Hamiltonian is our usual and the perturbation is H = p m + mω x H = βx 4 Assuming β is small, the perturbation
More informationQuantum Theory of Angular Momentum and Atomic Structure
Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the
More informationIV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance
IV. Electronic Spectroscopy, Angular Momentum, and Magnetic Resonance The foundation of electronic spectroscopy is the exact solution of the time-independent Schrodinger equation for the hydrogen atom.
More informationA Hartree-Fock Example Using Helium
Univesity of Connecticut DigitalCommons@UConn Chemisty Education Mateials Depatment of Chemisty June 6 A Hatee-Fock Example Using Helium Cal W. David Univesity of Connecticut, Cal.David@uconn.edu Follow
More informationForce of gravity and its potential function
F. W. Phs0 E:\Ecel files\ch gavitational foce and potential.doc page of 6 0/0/005 8:9 PM Last pinted 0/0/005 8:9:00 PM Foce of gavit and its potential function (.) Let us calculate the potential function
More informationTight-Binding Model of Electronic Structures
Tight-Binding Model of Electronic Structures Consider a collection of N atoms. The electronic structure of this system refers to its electronic wave function and the description of how it is related to
More informationA Lattice Energy Calculation for LiH
A Lattice Enegy Calculation fo LiH Fank Riou Lithium hyie is a white cystalline soli with the face-centee cubic cystal stuctue (see lattice shown below). The moel fo LiH(s) popose in this stuy constists
More informationAPPENDIX. For the 2 lectures of Claude Cohen-Tannoudji on Atom-Atom Interactions in Ultracold Quantum Gases
APPENDIX Fo the lectues of Claude Cohen-Tannoudji on Atom-Atom Inteactions in Ultacold Quantum Gases Pupose of this Appendix Demonstate the othonomalization elation(ϕ ϕ = δ k k δ δ )k - The wave function
More informationCHEM1101 Worksheet 3: The Energy Levels Of Electrons
CHEM1101 Woksheet 3: The Enegy Levels Of Electons Model 1: Two chaged Paticles Sepaated by a Distance Accoding to Coulomb, the potential enegy of two stationay paticles with chages q 1 and q 2 sepaated
More informationJerry Gilfoyle The Hydrogen Optical Spectrum 1 / 15
Jerry Gilfoyle The Hydrogen Optical Spectrum 1 / 15 What holds atoms together? Jerry Gilfoyle The Hydrogen Optical Spectrum 1 / 15 What holds atoms together? How do we know? Jerry Gilfoyle The Hydrogen
More informationQuestion Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if
Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7
More informationExercises : Questions
Exercises 18.05.2017: Questions Problem 1 where Calculate the following commutators: a) [ Ĥ, ˆp ], b) [ Ĥ, ˆr ], Ĥ = 1 2m ˆp2 + V ˆr), 1) ˆp 2 = ˆp 2 x + ˆp 2 y + ˆp 2 z and V ˆr) = V ˆx, ŷ, ẑ) is an arbitrary
More informationIntroduction to Spherical Harmonics
Introduction to Spherical Harmonics Lawrence Liu 3 June 4 Possibly useful information. Legendre polynomials. Rodrigues formula:. Generating function: d n P n x = x n n! dx n n. wx, t = xt t = P n xt n,
More informationSolutions to exam : 1FA352 Quantum Mechanics 10 hp 1
Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem (4 p): (a) Define the concept of unitary operator and show that the operator e ipa/ is unitary (p is the momentum operator in one dimension) (b)
More information1.2 Differential cross section
.2. DIFFERENTIAL CROSS SECTION Febuay 9, 205 Lectue VIII.2 Diffeential coss section We found that the solution to the Schodinge equation has the fom e ik x ψ 2π 3/2 fk, k + e ik x and that fk, k = 2 m
More informationThis gives rise to the separable equation dr/r = 2 cot θ dθ which may be integrated to yield r(θ) = R sin 2 θ (3)
Physics 506 Winte 2008 Homewok Assignment #10 Solutions Textbook poblems: Ch. 12: 12.10, 12.13, 12.16, 12.19 12.10 A chaged paticle finds itself instantaneously in the equatoial plane of the eath s magnetic
More informationProblem 1. Part b. Part a. Wayne Witzke ProblemSet #1 PHY 361. Calculate x, the expected value of x, defined by
Poblem Pat a The nomal distibution Gaussian distibution o bell cuve has the fom f Ce µ Calculate the nomalization facto C by equiing the distibution to be nomalized f Substituting in f, defined above,
More informationIn the previous section we considered problems where the
5.4 Hydodynamically Fully Developed and Themally Developing Lamina Flow In the pevious section we consideed poblems whee the velocity and tempeatue pofile wee fully developed, so that the heat tansfe coefficient
More informationUnit 6 Practice Test. Which vector diagram correctly shows the change in velocity Δv of the mass during this time? (1) (1) A. Energy KE.
Unit 6 actice Test 1. Which one of the following gaphs best epesents the aiation of the kinetic enegy, KE, and of the gaitational potential enegy, GE, of an obiting satellite with its distance fom the
More information