CHEM 301: Homework assignment #5
|
|
- Brendan Hart
- 6 years ago
- Views:
Transcription
1 CHEM 30: Homework assignment #5 Solutions. A point mass rotates in a circle with l =. Calculate the magnitude of its angular momentum and all possible projections of the angular momentum on the z-axis. 0% The magnitude of the angular momentum is J = h l l + = J s + = J s. The projection of angular momentum onto the z-axis is determined by the quantum number m l. For l =, there are three possible values of m l :, 0, and +. The projections of the angular momentum for these three cases are: J z = hm l = m l = + : h; m l = 0 : 0; m l = : h. Schematically draw all angular momentum cones consistent with l =. 0% See Figure. Figure : The red cone corresponds to m l = +, the green cone actually, a circle in the xy-plane to m l = 0, and the blue cone to m l =. The length of the angular momentum vector J is J = h l l + = h since l = and is the same for all three cones. The projection of the angular momentum vector J onto the z-axis is J z = hm l and is the same for all vectors within the same cone. The angle θ is measured from the positive direction of the z-axis. Calculate the half-angles for each of the cones. 0% The angle θ is measured from the positive direction of the z-axis. For m l = +, its cosine can be written as: cos θ ml =+ = J m l=+ z J = h h =.
2 Thus, θ ml =+ = arccos =. It is evident that the cone for m l = is just a mirror reflection in the xy-plane of the cone for m l = +. Thus, it lies at an angle of, as measured from the negative direction of the z-axis, or θ ml = = = 3 as measured from the positive direction of the z-axis. Finally, for m l = 0 the angular momentum J is perpendicular to the z-axis since its projection onto the z-axis is J m l=0 z = 0. Thus, θ ml =0 =.. The force constant for the HI bond is k = 3 N/m. Calculate the vibrational frequency of an HI molecule. 0% Approximating the HI molecule by a harmonic oscillator, we can calculate its vibrational frequency using the formula ν = k µ, where µ is the reduced mass of the molecule: µ = m Hm I.008 a.u. 6.9 a.u. = m H + m I.008 a.u a.u. =.000 a.u. = kg and k = 3 N/m is the force constant. Thus, 3 N/m ν = kg = s. What is the average potential energy in the ground state of this molecule? 0% To calculate the average value s of any quantum mechanical observable s we can use one of the postulates of quantum mechanics: s = ψ x Ŝψ x dx, where Ŝ is the operator that corresponds to observable s and ψ x is the wavefunction of the system studied. We need to find the average potential energy in the vibrational ground state of a molecule that we describe as a harmonic oscillator. The potential energy operator is just the potential function V x, for the harmonic oscillator it is: ˆV = V x = kx.
3 From the lecture we know that the eigenfunctions for the harmonic oscillator are normalized products of Hermite polynomials and Gaussian functions: ψ v x = α v v! normalization constant H v α x Hermite polynomial exp αx. Gaussian function We are interested in the vibrational ground state, where v = 0: α ψ 0 x = H 0 0! 0 α x exp αx = = see lecture 5, slide = = α exp αx. Since ψ 0 x is a real-valued function, ψ 0 x = ψ 0 x. Plugging in the wavefunction in the ground state of a harmonic oscillator and the potential energy operator for the harmonic oscillator into the expression for the average value, we get: V = = k kx ψ0 x ψ 0 x dx = k x α exp αx dx = k α x ψ 0 x dx = = [look up the integral in a table of integrals] = k = k α. x exp αx dx = α α α = Here, kµ α = h. Plugging this into the expression for the average potential energy that we obtained results in: V = k α = k h km = h k m = h k m } {{} =ν = hν. The average potential energy is thus one half of the zero-point energy: E 0 = hν. Plugging in the numbers, we get: V = hν = J s s = J = 7 mev. 3
4 What is the wavelength that would excite this molecule into vibration? What part of the electromagnetic spectrum does this wavelength correspond to? 0% The energy levels of a harmonic oscillator are given by E = hν v +. In the context of this problem, to excite a molecule into vibration means to take it from the vibrational ground state v = 0 to the first vibrational excited state v =. The energy difference for these two states is: E = hν + hν 0 + = hν. This is the energy that a photon must supply to the molecule to facilitate this transition. The energy of a photon can also be written as E photon = hν photon. By comparing the expressions for E and E photon, we see that the frequency of the photon, ν photon, must be the same as the characteristic frequency of the harmonic oscillator, ν, that we calculated earlier. To convert the frequency to the wavelength, λ, use the relation c = λν, where c is the speed of light. Thus, λ = c ν = m s s = m =.33 µm. How would the vibrational frequency change if hydrogen were replaced by deuterium? 0% Since deuterium is approximately twice as heavy as hydrogen, but is still much lighter than iodine, the reduced mass for the DI molecule would be approximately a.u. instead of a.u. Since the frequency ν = k µ, then, assuming that k remains the same, increasing µ by roughly a factor of would reduce ν by roughly a factor of. 3. What manifestations of Heisenberg uncertainty principle have we seen in the lecture about the particle on a sphere and the harmonic oscillator? Give two examples, with brief explanations. 30%
5 Heisenberg s uncertainty principle says that if the operators Ŝ and ˆQ that correspond to observables s and q, respectively, commute, then s and q can be known simultaneously to arbitrary precision. However, if Ŝ and ˆQ do not commute then the uncertainties in the values of s and q, s and q, respectively, must satisfy s q h. For a particle on a sphere we have seen that although the angular momentum J and one of its projections can be known at the same time to arbitrary precision. However, once one projection of the angular momentum usually denoted J z is known, the other two projections usually denoted J x and J y must be completely indefinite. We can expect this to be a consequence of Heisenberg s uncertainty principle and may predict that the operator of the angular momentum projection Ĵz commutes with the operator of angular momentum Ĵ, but Ĵz does not commute with the operators of the two other projections of angular momentum, Ĵx and Ĵy. We have not written these operators out explicitly, but this, indeed, turns out to be the case. Similarly, both Ĵx and Ĵy will commute with Ĵ, but not with each other or with Ĵz. For a particle in a box we have seen that the existence of zero-point energy is a consequence of the position-momentum uncertainty principle a specific manifestation of Heisenberg s uncertainty principle, where the observable s is position x, and the observable q is momentum p. Let s recap that argument. Since a particle can only be found inside the box, its position x is not completely indefinite, and hence the uncertainty in position x. Since the operators of position and momentum do not commute, the uncertainty relation x p h must be satisfied. For x, this condition can only be satisfied if p 0. However, if p 0, we don t know exactly what the value of p is. We don t know what the value of p is exactly, but we definitely cannot say that p = 0 if this were the case, we would be certain about the value of p, and the uncertainty p would be 0. But if p 0, the kinetic energy E k = p m 0. Thus, the position-momentum uncertainty principle requires that there be a non-zero minimum energy that a particle that is confined to a box must possess: the zero-point energy. Zero-point energy also exists for the harmonic operator, and we can expect it to also be the consequence of the position-momentum uncertainty principle. Similar reasoning to the case of a particle in a box is possible. Although a particle can penetrate into the walls of a harmonic potential, it is still most likely to be found either inside the harmonic potential or just 5
6 outside its walls. Thus, the position of the particle is not completely indefinite: x. The position-momentum uncertainty principle requires that x p h, so if x, then p 0. Thus, we can t say that p or the kinetic energy that depends on it are exactly zero. There is a a non-zero minimum energy that a particle in a harmonic potential must have, the zero-point energy. 6
Physical Chemistry II Exam 2 Solutions
Chemistry 362 Spring 208 Dr Jean M Standard March 9, 208 Name KEY Physical Chemistry II Exam 2 Solutions ) (4 points) The harmonic vibrational frequency (in wavenumbers) of LiH is 4057 cm Based upon this
More informationProblems and Multiple Choice Questions
Problems and Multiple Choice Questions 1. A momentum operator in one dimension is 2. A position operator in 3 dimensions is 3. A kinetic energy operator in 1 dimension is 4. If two operator commute, a)
More informationA few principles of classical and quantum mechanics
A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 9, February 8, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer Lecture 9, February 8, 2006 The Harmonic Oscillator Consider a diatomic molecule. Such a molecule
More informationChem 452 Mega Practice Exam 1
Last Name: First Name: PSU ID #: Chem 45 Mega Practice Exam 1 Cover Sheet Closed Book, Notes, and NO Calculator The exam will consist of approximately 5 similar questions worth 4 points each. This mega-exam
More informationPhysical Chemistry II Exam 2 Solutions
Chemistry 362 Spring 2017 Dr Jean M Standard March 10, 2017 Name KEY Physical Chemistry II Exam 2 Solutions 1) (14 points) Use the potential energy and momentum operators for the harmonic oscillator to
More informationIndicate if the statement is True (T) or False (F) by circling the letter (1 pt each):
Indicate if the statement is (T) or False (F) by circling the letter (1 pt each): False 1. In order to ensure that all observables are real valued, the eigenfunctions for an operator must also be real
More informationSIMPLE QUANTUM SYSTEMS
SIMPLE QUANTUM SYSTEMS Chapters 14, 18 "ceiiinosssttuu" (anagram in Latin which Hooke published in 1676 in his "Description of Helioscopes") and deciphered as "ut tensio sic vis" (elongation of any spring
More informationLecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor
Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor It turns out that the boundary condition of the wavefunction going to zero at infinity is sufficient to quantize the value of energy that
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 10, February 10, / 4
Chem 350/450 Physical Chemistry II (Quantum Mechanics 3 Credits Spring Semester 006 Christopher J. Cramer Lecture 10, February 10, 006 Solved Homework We are asked to find and for the first two
More informationCHM320 EXAM #2 USEFUL INFORMATION
CHM30 EXAM # USEFUL INFORMATION Constants mass of electron: m e = 9.11 10 31 kg. Rydberg constant: R H = 109737.35 cm 1 =.1798 10 18 J. speed of light: c = 3.00 10 8 m/s Planck constant: 6.66 10 34 Js
More informationQuantum mechanics (QM) deals with systems on atomic scale level, whose behaviours cannot be described by classical mechanics.
A 10-MINUTE RATHER QUICK INTRODUCTION TO QUANTUM MECHANICS 1. What is quantum mechanics (as opposed to classical mechanics)? Quantum mechanics (QM) deals with systems on atomic scale level, whose behaviours
More information1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: d 2 H = h2
15 Harmonic Oscillator 1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: d 2 H = h2 2mdx + 1 2 2 kx2 (15.1) where k is the force
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 informationChemistry 795T. NC State University. Lecture 4. Vibrational and Rotational Spectroscopy
Chemistry 795T Lecture 4 Vibrational and Rotational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule
More informationReading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom.
Chemistry 356 017: Problem set No. 6; Reading: Mathchapters F and G, MQ - Ch. 7-8, Lecture notes on hydrogen atom. The H atom involves spherical coordinates and angular momentum, which leads to the shapes
More informationA Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor
A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule
More informationVibrational motion. Harmonic oscillator ( 諧諧諧 ) - A particle undergoes harmonic motion. Parabolic ( 拋物線 ) (8.21) d 2 (8.23)
Vibrational motion Harmonic oscillator ( 諧諧諧 ) - A particle undergoes harmonic motion F == dv where k Parabolic V = 1 f k / dx = is Schrodinge h m d dx ψ f k f x the force constant x r + ( 拋物線 ) 1 equation
More information( ) = 9φ 1, ( ) = 4φ 2.
Chemistry 46 Dr Jean M Standard Homework Problem Set 6 Solutions The Hermitian operator A ˆ is associated with the physical observable A Two of the eigenfunctions of A ˆ are and These eigenfunctions are
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 6.2 6.3 6.4 6.5 6.6 6.7 The Schrödinger Wave Equation Expectation Values Infinite Square-Well Potential Finite Square-Well Potential Three-Dimensional Infinite-Potential
More informationPHYSICS 250 May 4, Final Exam - Solutions
Name: PHYSICS 250 May 4, 999 Final Exam - Solutions Instructions: Work all problems. You may use a calculator and two pages of notes you may have prepared. There are problems of varying length and difficulty.
More informationProblem Set 5 Solutions
Chemistry 362 Dr Jean M Standard Problem Set 5 Solutions ow many vibrational modes do the following molecules or ions possess? [int: Drawing Lewis structures may be useful in some cases] In all of the
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Chemistry 3502/4502 Final Exam Part I May 14, 2005 1. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle (e) The
More informationMomentum expectation Momentum expectation value value for for infinite square well
Quantum Mechanics and Atomic Physics Lecture 9: The Uncertainty Principle and Commutators http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Announcement Quiz in next class (Oct. 5): will cover Reed
More informationChemistry 881 Lecture Topics Fall 2001
Chemistry 881 Lecture Topics Fall 2001 Texts PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon MATHEMATICS for PHYSICAL CHEMISTRY, Mortimer i. Mathematics Review (M, Chapters 1,2,3 & 4; M&S,
More informationIf electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.
CHEM 2060 Lecture 18: Particle in a Box L18-1 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk
More informationSimple Harmonic Oscillator
Classical harmonic oscillator Linear force acting on a particle (Hooke s law): F =!kx From Newton s law: F = ma = m d x dt =!kx " d x dt + # x = 0, # = k / m Position and momentum solutions oscillate in
More information5.1 Classical Harmonic Oscillator
Chapter 5 Harmonic Oscillator 5.1 Classical Harmonic Oscillator m l o l Hooke s Law give the force exerting on the mass as: f = k(l l o ) where l o is the equilibrium length of the spring and k is the
More information1.3 Harmonic Oscillator
1.3 Harmonic Oscillator 1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: H = h2 d 2 2mdx + 1 2 2 kx2 (1.3.1) where k is the force
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationQuantum Physics 130A. April 1, 2006
Quantum Physics 130A April 1, 2006 2 1 HOMEWORK 1: Due Friday, Apr. 14 1. A polished silver plate is hit by beams of photons of known energy. It is measured that the maximum electron energy is 3.1 ± 0.11
More informationProblem Set 3 Solutions
Chemistry 36 Dr Jean M Standard Problem Set 3 Solutions 1 Verify for the particle in a one-dimensional box by explicit integration that the wavefunction ψ x) = π x ' sin ) is normalized To verify that
More informationQuantum Chemistry Exam 2 Solutions
Chemistry 46 Fall 17 Dr. Jean M. Standard November 8, 17 Name KEY Quantum Chemistry Exam Solutions 1.) ( points) Answer the following questions by selecting the correct answer from the choices provided.
More information( ) 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 informationCHAPTER 8 The Quantum Theory of Motion
I. Translational motion. CHAPTER 8 The Quantum Theory of Motion A. Single particle in free space, 1-D. 1. Schrodinger eqn H ψ = Eψ! 2 2m d 2 dx 2 ψ = Eψ ; no boundary conditions 2. General solution: ψ
More informationPhysics 217 Problem Set 1 Due: Friday, Aug 29th, 2008
Problem Set 1 Due: Friday, Aug 29th, 2008 Course page: http://www.physics.wustl.edu/~alford/p217/ Review of complex numbers. See appendix K of the textbook. 1. Consider complex numbers z = 1.5 + 0.5i and
More informationPY 351 Modern Physics - Lecture notes, 3
PY 351 Modern Physics - Lecture notes, 3 Copyright by Claudio Rebbi, Boston University, October 2016. These notes cannot be duplicated and distributed without explicit permission of the author. Time dependence
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationChemistry 3502/4502. Final Exam Part I. May 14, 2005
Advocacy chit Chemistry 350/450 Final Exam Part I May 4, 005. For which of the below systems is = where H is the Hamiltonian operator and T is the kinetic-energy operator? (a) The free particle
More informationChemistry 483 Lecture Topics Fall 2009
Chemistry 483 Lecture Topics Fall 2009 Text PHYSICAL CHEMISTRY A Molecular Approach McQuarrie and Simon A. Background (M&S,Chapter 1) Blackbody Radiation Photoelectric effect DeBroglie Wavelength Atomic
More informationThe one and three-dimensional particle in a box are prototypes of bound systems. As we
6 Lecture 10 The one and three-dimensional particle in a box are prototypes of bound systems. As we move on in our study of quantum chemistry, we'll be considering bound systems that are more and more
More informationElectromagnetic Radiation. Chapter 12: Phenomena. Chapter 12: Quantum Mechanics and Atomic Theory. Quantum Theory. Electromagnetic Radiation
Chapter 12: Phenomena Phenomena: Different wavelengths of electromagnetic radiation were directed onto two different metal sample (see picture). Scientists then recorded if any particles were ejected and
More informationQuantum Physics (PHY-4215)
Quantum Physics (PHY-4215) Gabriele Travaglini March 31, 2012 1 From classical physics to quantum physics 1.1 Brief introduction to the course The end of classical physics: 1. Planck s quantum hypothesis
More informationQuantum Mechanics I Physics 5701
Quantum Mechanics I Physics 5701 Z. E. Meziani 02/24//2017 Physics 5701 Lecture Commutation of Observables and First Consequences of the Postulates Outline 1 Commutation Relations 2 Uncertainty Relations
More information1 Commutators (10 pts)
Final Exam Solutions 37A Fall 0 I. Siddiqi / E. Dodds Commutators 0 pts) ) Consider the operator  = Ĵx Ĵ y + ĴyĴx where J i represents the total angular momentum in the ith direction. a) Express both
More information( )( s 1
Chemistry 362 Dr Jean M Standard Homework Problem Set 6 Solutions l Calculate the reduced mass in kg for the OH radical The reduced mass for OH is m O m H m O + m H To properly calculate the reduced mass
More informationLecture 12: Particle in 1D boxes & Simple Harmonic Oscillator
Lecture 12: Particle in 1D boxes & Simple Harmonic Oscillator U(x) E Dx y(x) x Dx Lecture 12, p 1 Properties of Bound States Several trends exhibited by the particle-in-box states are generic to bound
More informationLast Name or Student ID
12/05/18, Chem433 Final Exam Last Name or Student ID 1. (2 pts) 12. (3 pts) 2. (6 pts) 13. (3 pts) 3. (3 pts) 14. (2 pts) 4. (3 pts) 15. (3 pts) 5. (4 pts) 16. (3 pts) 6. (2 pts) 17. (15 pts) 7. (9 pts)
More informationHarmonic Oscillator Eigenvalues and Eigenfunctions
Chemistry 46 Fall 217 Dr. Jean M. Standard October 4, 217 Harmonic Oscillator Eigenvalues and Eigenfunctions The Quantum Mechanical Harmonic Oscillator The quantum mechanical harmonic oscillator in one
More informationREVIEW: The Matching Method Algorithm
Lecture 26: Numerov Algorithm for Solving the Time-Independent Schrödinger Equation 1 REVIEW: The Matching Method Algorithm Need for a more general method The shooting method for solving the time-independent
More informationLecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators
Lecture Spin, orbital, and total angular momentum 70.00 Mechanics Very brief background MATH-GA In 9, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform
More informationProblem 1: Step Potential (10 points)
Problem 1: Step Potential (10 points) 1 Consider the potential V (x). V (x) = { 0, x 0 V, x > 0 A particle of mass m and kinetic energy E approaches the step from x < 0. a) Write the solution to Schrodinger
More informationECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline:
ECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline: Time-Dependent Schrödinger Equation Solutions to thetime-dependent Schrödinger Equation Expansion of Energy Eigenstates Things you
More informationP3317 HW from Lecture and Recitation 10
P3317 HW from Lecture 18+19 and Recitation 10 Due Nov 6, 2018 Problem 1. Equipartition Note: This is a problem from classical statistical mechanics. We will need the answer for the next few problems, and
More informationUnderstand the basic principles of spectroscopy using selection rules and the energy levels. Derive Hund s Rule from the symmetrization postulate.
CHEM 5314: Advanced Physical Chemistry Overall Goals: Use quantum mechanics to understand that molecules have quantized translational, rotational, vibrational, and electronic energy levels. In a large
More informationGeneral Physical Chemistry II
General Physical Chemistry II Lecture 3 Aleksey Kocherzhenko September 2, 2014" Last time " The time-independent Schrödinger equation" Erwin Schrödinger " ~ 2 2m d 2 (x) dx 2 The wavefunction:" (x) The
More informationCHAPTER 13 LECTURE NOTES
CHAPTER 13 LECTURE NOTES Spectroscopy is concerned with the measurement of (a) the wavelengths (or frequencies) at which molecules absorb/emit energy, and (b) the amount of radiation absorbed at these
More informationChapter 12: Phenomena
Chapter 12: Phenomena K Fe Phenomena: Different wavelengths of electromagnetic radiation were directed onto two different metal sample (see picture). Scientists then recorded if any particles were ejected
More informationBasic Quantum Mechanics
Frederick Lanni 10feb'12 Basic Quantum Mechanics Part I. Where Schrodinger's equation comes from. A. Planck's quantum hypothesis, formulated in 1900, was that exchange of energy between an electromagnetic
More informationC. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.
Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is
More informationPHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep
Problem set 1 Due Sep 15 2005 1. Let V be the set of all complex valued functions of a real variable θ, that are periodic with period 2π. That is u(θ + 2π) = u(θ), for all u V. (1) (i) Show that this V
More information(Refer Slide Time: 1:20) (Refer Slide Time: 1:24 min)
Engineering Chemistry - 1 Prof. K. Mangala Sunder Department of Chemistry Indian Institute of Technology, Madras Lecture - 5 Module 1: Atoms and Molecules Harmonic Oscillator (Continued) (Refer Slide Time:
More information8.04 Spring 2013 March 12, 2013 Problem 1. (10 points) The Probability Current
Prolem Set 5 Solutions 8.04 Spring 03 March, 03 Prolem. (0 points) The Proaility Current We wish to prove that dp a = J(a, t) J(, t). () dt Since P a (t) is the proaility of finding the particle in the
More informationExercises
Exercises. 1.1 The power delivered to a photodetector which collects 8.0 10 7 photons in 3.8 ms from monochromatic light is 0.72 microwatt. What is the frequency of the light? 1.2 The speed of a proton
More information1.6. 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= ( Prove the nonexistence of electron in the nucleus on the basis of uncertainty principle.
Worked out examples (Quantum mechanics). A microscope, using photons, is employed to locate an electron in an atom within a distance of. Å. What is the uncertainty in the momentum of the electron located
More informationQualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!
Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics
More informationQuantum 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 informationPHYS 3313 Section 001 Lecture #20
PHYS 3313 Section 001 ecture #0 Monday, April 10, 017 Dr. Amir Farbin Infinite Square-well Potential Finite Square Well Potential Penetration Depth Degeneracy Simple Harmonic Oscillator 1 Announcements
More informationInfrared Spectroscopy
Infrared Spectroscopy The Interaction of Light with Matter Electric fields apply forces to charges, according to F = qe In an electric field, a positive charge will experience a force, but a negative charge
More informationCHAPTER NUMBER 7: Quantum Theory: Introduction and Principles
CHAPTER NUMBER 7: Quantum Theory: Introduction and Principles Art PowerPoints Peter Atkins & Julio De Paula 2010 1 mm 1000 m 100 m 10 m 1000 nm 100 nm 10 nm 1 nm 10 Å 1 Å Quantum phenomena 7.1 Energy quantization
More informationIn this lecture we will go through the method of coupling of angular momentum.
Lecture 3 : Title : Coupling of angular momentum Page-0 In this lecture we will go through the method of coupling of angular momentum. We will start with the need for this coupling and then develop the
More informationINTRODUCTION TO QUANTUM MECHANICS
4 CHAPTER INTRODUCTION TO QUANTUM MECHANICS 4.1 Preliminaries: Wave Motion and Light 4.2 Evidence for Energy Quantization in Atoms 4.3 The Bohr Model: Predicting Discrete Energy Levels in Atoms 4.4 Evidence
More informationS.E. of H.O., con con t
Quantum Mechanics and Atomic Physics Lecture 11: The Harmonic Oscillator: Part I http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh The Classical Harmonic Oscillator Classical mechanics examples Mass
More informationd 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5)
LECTURE 12 Maxwell Velocity Distribution Suppose we have a dilute gas of molecules, each with mass m. If the gas is dilute enough, we can ignore the interactions between the molecules and the energy will
More informationQuantum 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 informationOpinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6. Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- 6.6 Simple Harmonic
More informationeigenvalues eigenfunctions
Born-Oppenheimer Approximation Atoms and molecules consist of heavy nuclei and light electrons. Consider (for simplicity) a diatomic molecule (e.g. HCl). Clamp/freeze the nuclei in space, a distance r
More information3. Quantum Mechanics in 3D
3. Quantum Mechanics in 3D 3.1 Introduction Last time, we derived the time dependent Schrödinger equation, starting from three basic postulates: 1) The time evolution of a state can be expressed as a unitary
More informationPHYS 3313 Section 001 Lecture # 22
PHYS 3313 Section 001 Lecture # 22 Dr. Barry Spurlock Simple Harmonic Oscillator Barriers and Tunneling Alpha Particle Decay Schrodinger Equation on Hydrogen Atom Solutions for Schrodinger Equation for
More informationPart One: Light Waves, Photons, and Bohr Theory. 2. Beyond that, nothing was known of arrangement of the electrons.
CHAPTER SEVEN: QUANTUM THEORY AND THE ATOM Part One: Light Waves, Photons, and Bohr Theory A. The Wave Nature of Light (Section 7.1) 1. Structure of atom had been established as cloud of electrons around
More informationSummary: angular momentum derivation
Summary: angular momentum derivation L = r p L x = yp z zp y, etc. [x, p y ] = 0, etc. (-) (-) (-3) Angular momentum commutation relations [L x, L y ] = i hl z (-4) [L i, L j ] = i hɛ ijk L k (-5) Levi-Civita
More informationIntroduction to Vibrational Spectroscopy
Introduction to Vibrational Spectroscopy Harmonic oscillators The classical harmonic oscillator The uantum mechanical harmonic oscillator Harmonic approximations in molecular vibrations Vibrational spectroscopy
More informationChem 1A, Fall 2015, Midterm Exam 1. Version A September 21, 2015 (Prof. Head-Gordon) 2
Chem 1A, Fall 2015, Midterm Exam 1. Version A September 21, 2015 (Prof. Head-Gordon) 2 Name: Student ID: TA: Contents: 9 pages A. Multiple choice (7 points) B. Stoichiometry (10 points) C. Photoelectric
More informationLecture 12. The harmonic oscillator
Lecture 12 The harmonic oscillator 107 108 LECTURE 12. THE HARMONIC OSCILLATOR 12.1 Introduction In this chapter, we are going to find explicitly the eigenfunctions and eigenvalues for the time-independent
More informationPHY413 Quantum Mechanics B Duration: 2 hours 30 minutes
BSc/MSci Examination by Course Unit Thursday nd May 4 : - :3 PHY43 Quantum Mechanics B Duration: hours 3 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO
More informationThe wavefunction ψ for an electron confined to move within a box of linear size L = m, is a standing wave as shown.
1. This question is about quantum aspects of the electron. The wavefunction ψ for an electron confined to move within a box of linear size L = 1.0 10 10 m, is a standing wave as shown. State what is meant
More informationAppendix A. The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System
Appendix A The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System Real quantum mechanical systems have the tendency to become mathematically
More information(3.1) Module 1 : Atomic Structure Lecture 3 : Angular Momentum. Objectives In this Lecture you will learn the following
Module 1 : Atomic Structure Lecture 3 : Angular Momentum Objectives In this Lecture you will learn the following Define angular momentum and obtain the operators for angular momentum. Solve the problem
More informationQuantum Theory of the Atom
The Wave Nature of Light Quantum Theory of the Atom Electromagnetic radiation carries energy = radiant energy some forms are visible light, x rays, and radio waves Wavelength ( λ) is the distance between
More informationCh. 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 informationdt r r r V(x,t) = F(x,t)dx
Quantum Mechanics and Atomic Physics Lecture 3: Schroedinger s Equation: Part I http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Announcement First homework due on Wednesday Sept 14 at the beginning
More informationLecture 2: simple QM problems
Reminder: http://www.star.le.ac.uk/nrt3/qm/ Lecture : simple QM problems Quantum mechanics describes physical particles as waves of probability. We shall see how this works in some simple applications,
More informationIntroduction to particle physics Lecture 3: Quantum Mechanics
Introduction to particle physics Lecture 3: Quantum Mechanics Frank Krauss IPPP Durham U Durham, Epiphany term 2010 Outline 1 Planck s hypothesis 2 Substantiating Planck s claim 3 More quantisation: Bohr
More informationWe also find the development of famous Schrodinger equation to describe the quantization of energy levels of atoms.
Lecture 4 TITLE: Quantization of radiation and matter: Wave-Particle duality Objectives In this lecture, we will discuss the development of quantization of matter and light. We will understand the need
More informationWave nature of particles
Wave nature of particles We have thus far developed a model of atomic structure based on the particle nature of matter: Atoms have a dense nucleus of positive charge with electrons orbiting the nucleus
More informationLecture 6 Quantum Mechanical Systems and Measurements
Lecture 6 Quantum Mechanical Systems and Measurements Today s Program: 1. Simple Harmonic Oscillator (SHO). Principle of spectral decomposition. 3. Predicting the results of measurements, fourth postulate
More informationThe Sommerfeld Polynomial Method: Harmonic Oscillator Example
Chemistry 460 Fall 2017 Dr. Jean M. Standard October 2, 2017 The Sommerfeld Polynomial Method: Harmonic Oscillator Example Scaling the Harmonic Oscillator Equation Recall the basic definitions of the harmonic
More informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More information