PH 451/551 Quantum Mechanics Capstone Winter 201x


 Lynette Atkinson
 1 years ago
 Views:
Transcription
1 These are the questions from the W7 exam presented as practice problems. The equation sheet is PH 45/55 Quantum Mechanics Capstone Winter x TOTAL POINTS: xx Weniger 6, time There are xx questions, for a total of xx points. They will surely present different levels of difficulty, so work quickly on the ones you find easy, leaving time for the ones you find harder. On average, budget about point per minute. Equations and integrals that may be useful: ψ Â φ = φ Â ψ * n n = ΔQ Q Q n ( ) π E n = n ml E = n n α mc E n = ( n + )ω (x) = L sin nπx L α = e 4πε c a = 4πε m e e ( ) = x a = V x p x + i x a = p x i x a n = n n a n = n + n + E n = ( n + )ω ϕ ( x) = π (x) = n/ n π 4 e x ϕ x /4 H n ( ) = x E () n = n ( ) H n ( ) E () n = π e x m n 4 n ( ) H m ( ) E n () E m () xe n ( ) ( ) = c nm m ( ) ( c ) nm = m ( ) H n ( ) H ' = d i Ε m n E () () n E m µ = g q S µ B = e µ N = e m m e m p d = qr S x S y i i S z S 4 x
2 These are the questions from the W7 exam presented as practice problems. The equation sheet is S x S y i i i i S z S J = j( j +) J z = m j J x, J y = ij z J ± = j j + ( ) ( ) m j m j ± JM = j j m m j j m m JM m m J + J = J + J J + J H hf = A SiI J ± = J x ± ij y ± H rel = p4 8m c H = e SO 8πε m c r LiS H Z = µ i B = µ BB ( g L z + g e S z ) ( E ) fs = α 4 mc n j + c f ( t) = i t ( ) l( l +) 4 ( )( l +) 4n E SO = 4 α j j + 4 mc n l l + E f E i t f H ( t ) i e i d t R i f = π H ψ = i t ψ H = E n e a x dx = a π xe x dx = x n e ax dx = n a n+ ψ t= = n V fi c n x e x dx = π 4 sinθ = sinθ cosθ cosθ = sin θ tanθ = tanθ tan θ sin mxsin nx dx = sin( m n) x sin( m + n) x, ( m ( m n) ( m + n) n ) cos mx cosnx dx = sin( m n)x + sin( m + n)x, ( m n ) m n m + n sin mx cosnx dx ( ) ( ) x ( m n) = cos m n ( ) ( ) x ( m + n) cos m + n, ( m n ) δ ( ω fi ω )
3 These are the questions from the W7 exam presented as practice problems. The equation sheet is s= F j = j = j m m s I = m I M F  m m j = j j = j j = m m j = m  m m m
4 These are the questions from the W7 exam presented as practice problems. The equation sheet is 4
5 These are the questions from the W7 exam presented as practice problems. The equation sheet is. [ points] Perturbed HO A particle is bound in the harmonic oscillator potential V x ( ) = x. A static perturbation H = γ x is applied to the system. (a) Calculate the firstorder corrections to (all) the energy eigenvalues using the perturbation formalism. (b) What is the exact energy? Explain why your answer is consistent with (a) above.. [ points] Degenerate perturbation theory A particle in a twodimensional harmonic oscillator potential U ( x, y) = ( x + y ) has energy eigenstates that are characterized by TWO quantum numbers n x and n y, each of which can have values,,,,. The energy eigenstates are products of the d energy eigenstates: n x,n y x ( x)y ( y) with corresponding energy eigenvalues E nx,n y = ( n x + )ω + ( n y + )ω = ( n x + n y +)ω (a) (5 points) What is the energy of the first excited state (the next state above the lowestenergy state)? Why is this energy level degenerate, and what is the degeneracy of the level? (b) (5 points) Calculate the new energy eigenvalues and eigenstates (to first order) of these degenerate states that comprise the first excited state when a small perturbation to the Hamiltonian of the form H '( x) = U ( ) is applied where δ ( x) is the Dirac delta δ x function. Proceed as follows: Construct the matrix that represents the perturbation H' in the degenerate subspace and use it to determine the firstorder changes, if any, to the energy eigenvalues and the eigenstates.. [ points] Fine structure In the simplest treatment of the hydrogen atom, the electron and proton experience only the Coulomb interaction, and the energy eigenstates are designated n l m l s m s. (a) What are the allowed values of these quantum numbers? (b) If the system is in the p state, what are the particular values of the quantum numbers and what is the (unperturbed) energy of the p state? What is its degeneracy? (c) In class we used degenerate perturbation theory to show that adding interactions to the Hamiltonian introduced fine structure to the Hatom spectrum and that the finestructure corrections are E ) fs ( = α 4 mc n j + 4n. Use this expression to show how the unperturbed p levels are changed: draw a diagram that shows the new level(s), correctly labeled, and find the size of the correction(s) in ev. (d) What wavelength resolution is necessary to resolve the fine structure in the s < > p transition? Give a fractional value rather than an absolute wavelength. 5
6 These are the questions from the W7 exam presented as practice problems. The equation sheet is 4. [ points] Two spin systems Two particles A and B form the quantum system of interest. Only the spin angular momentum is relevant in this problem. A is spin/ particle and B is a spin particle. (a) Write down all possible kets that describe the state of system in the (i) coupled, and (ii) uncoupled bases. (State what the numbers in the kets represent in both cases). (b) Use the ClebschGordan coefficients to write the coupled states as linear combinations of the uncoupled states. (c) If the system is in the state where the total angular momentum is given by j = /, and the zcomponent of the total angular momentum is m j = /, what is (and explain why) the probability that a measurement of the zcomponent of the angular momentum of B yields: (i), (ii), (iii)? 5. [ points] Identical particles Two indistinguishable, noninteracting, uncharged spin/ fermions are confined to a region of length L in onedimension. In your answers, explain your reasoning. (a) Find the energy eigenvalue(s) of the ground state. (b) Find the energy eigenstate(s) of the ground state. (c) Specify and discuss the degeneracy of the ground state. (d) Find the energy eigenvalue(s) of the first excited state. (e) Find the energy eigenstate(s) of the first excited state. (f) Specify and discuss the degeneracy of the first excited state. END OF EXAM 6
PHYSICS 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 informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of timeindependent 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 timeindependent
More informationQuantum Mechanics Solutions. λ i λ j v j v j v i v i.
Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =
More informationIntroduction 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 informationClassical Mechanics Comprehensive Exam
Name: Student ID: Classical Mechanics Comprehensive Exam Spring 2018 You may use any intermediate results in the textbook. No electronic devices (calculator, computer, cell phone etc) are allowed. For
More informationTime Independent Perturbation Theory Contd.
Time Independent Perturbation Theory Contd. A summary of the machinery for the Perturbation theory: H = H o + H p ; H 0 n >= E n n >; H Ψ n >= E n Ψ n > E n = E n + E n ; E n = < n H p n > + < m H p n
More informationSolution Set of Homework # 6 Monday, December 12, Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second Volume
Department of Physics Quantum II, 570 Temple University Instructor: Z.E. Meziani Solution Set of Homework # 6 Monday, December, 06 Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Laloë, Second
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 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 informationQuantum Mechanics Solutions
Quantum Mechanics Solutions (a (i f A and B are Hermitian, since (AB = B A = BA, operator AB is Hermitian if and only if A and B commute So, we know that [A,B] = 0, which means that the Hilbert space H
More informationQuantum Mechanics FKA081/FIM400 Final Exam 28 October 2015
Quantum Mechanics FKA081/FIM400 Final Exam 28 October 2015 Next review time for the exam: December 2nd between 14:0016:00 in my room. (This info is also available on the course homepage.) Examinator:
More information8.05 Quantum Physics II, Fall 2011 FINAL EXAM Thursday December 22, 9:00 am 12:00 You have 3 hours.
8.05 Quantum Physics II, Fall 0 FINAL EXAM Thursday December, 9:00 am :00 You have 3 hours. Answer all problems in the white books provided. Write YOUR NAME and YOUR SECTION on your white books. There
More informationGoal: find Lorentzviolating corrections to the spectrum of hydrogen including nonminimal effects
Goal: find Lorentzviolating corrections to the spectrum of hydrogen including nonminimal effects Method: RayleighSchrödinger Perturbation Theory Step 1: Find the eigenvectors ψ n and eigenvalues ε n
More informationFor example, in one dimension if we had two particles in a onedimensional infinite potential well described by the following two wave functions.
Identical particles In classical physics one can label particles in such a way as to leave the dynamics unaltered or follow the trajectory of the particles say by making a movie with a fast camera. Thus
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 informationPhysics 228 Today: Ch 41: 13: 3D quantum mechanics, hydrogen atom
Physics 228 Today: Ch 41: 13: 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 informationQuantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours.
Quantum Physics III (8.06) Spring 2007 FINAL EXAMINATION Monday May 21, 9:00 am You have 3 hours. There are 10 problems, totalling 180 points. Do all problems. Answer all problems in the white books provided.
More informationSolution of Second Midterm Examination Thursday November 09, 2017
Department of Physics Quantum Mechanics II, Physics 570 Temple University Instructor: Z.E. Meziani Solution of Second Midterm Examination Thursday November 09, 017 Problem 1. (10pts Consider a system
More informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of timeindependent Schrödinger equation
Lecture 27st Page 1 Lecture 27 L27.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 timeindependent
More informationLSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (07/2017) =!2 π 2 a cos π x
LSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (7/17) 1. For a particle trapped in the potential V(x) = for a x a and V(x) = otherwise, the ground state energy and eigenfunction
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 megaexam
More informationComparing and Improving Quark Models for the Triply Bottom Baryon Spectrum
Comparing and Improving Quark Models for the Triply Bottom Baryon Spectrum A thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science degree in Physics from 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 kineticenergy operator? (a) The free particle (e) The
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 informationCHAPTER 8 The Quantum Theory of Motion
I. Translational motion. CHAPTER 8 The Quantum Theory of Motion A. Single particle in free space, 1D. 1. Schrodinger eqn H ψ = Eψ! 2 2m d 2 dx 2 ψ = Eψ ; no boundary conditions 2. General solution: ψ
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 kineticenergy operator? (a) The free particle
More informationSolutions Final exam 633
Solutions Final exam 633 S.J. van Enk (Dated: June 9, 2008) (1) [25 points] You have a source that produces pairs of spin1/2 particles. With probability p they are in the singlet state, ( )/ 2, and with
More informationP3317 HW from Lecture 7+8 and Recitation 4
P3317 HW from Lecture 7+8 and Recitation 4 Due Friday Tuesday September 25 Problem 1. In class we argued that an ammonia atom in an electric field can be modeled by a twolevel system, described by a Schrodinger
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 information1. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants.
Sample final questions.. Estimate the lifetime of an excited state of hydrogen. Give your answer in terms of fundamental constants. 2. A onedimensional harmonic oscillator, originally in the ground state,
More informationKet space as a vector space over the complex numbers
Ket space as a vector space over the complex numbers kets ϕ> and complex numbers α with two operations Addition of two kets ϕ 1 >+ ϕ 2 > is also a ket ϕ 3 > Multiplication with complex numbers α ϕ 1 >
More informationψ s a ˆn a s b ˆn b ψ Hint: Because the state is spherically symmetric the answer can depend only on the angle between the two directions.
1. Quantum Mechanics (Fall 2004) Two spinhalf particles are in a state with total spin zero. Let ˆn a and ˆn b be unit vectors in two arbitrary directions. Calculate the expectation value of the product
More informationOneelectron Atom. (in spherical coordinates), where Y lm. are spherical harmonics, we arrive at the following Schrödinger equation:
Oneelectron Atom The atomic orbitals of hydrogenlike 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 informationDepartment of Physics, Princeton University. Graduate Preliminary Examination Part II. Friday, May 10, :00 am  12:00 noon
Department of Physics, Princeton University Graduate Preliminary Examination Part II Friday, May 10, 2013 9:00 am  12:00 noon Answer TWO out of the THREE questions in Section A (Quantum Mechanics) and
More informationChemistry 120A 2nd Midterm. 1. (36 pts) For this question, recall the energy levels of the Hydrogenic Hamiltonian (1electron):
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 informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 14, 015 1:00PM to 3:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of this
More informationPhysics 221A Fall 1996 Notes 21 Hyperfine Structure in Hydrogen and Alkali Atoms
Physics 221A Fall 1996 Notes 21 Hyperfine Structure in Hydrogen and Alkali Atoms Hyperfine effects in atomic physics are due to the interaction of the atomic electrons with the electric and magnetic multipole
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 informationEach problem is worth 34 points. 1. Harmonic Oscillator Consider the Hamiltonian for a simple harmonic oscillator. 2ml 2 0. d 2
Physics 443 Prelim # with solutions March 7, 8 Each problem is worth 34 points.. Harmonic Oscillator Consider the Hamiltonian for a simple harmonic oscillator H p m + mω x (a Use dimensional analysis to
More informationAngular Momentum. Andreas Wacker Mathematical Physics Lund University
Angular Momentum Andreas Wacker Mathematical Physics Lund University Commutation relations of (orbital) angular momentum Angular momentum in analogy with classical case L= r p satisfies commutation relations
More informationChemistry 532 Practice Final Exam Fall 2012 Solutions
Chemistry 53 Practice Final Exam Fall Solutions x e ax dx π a 3/ ; π sin 3 xdx 4 3 π cos nx dx π; sin θ cos θ + K x n e ax dx n! a n+ ; r r r r ˆL h r ˆL z h i φ ˆL x i hsin φ + cot θ cos φ θ φ ) ˆLy i
More informationQuantum Mechanics: A Paradigms Approach David H. McIntyre. 27 July Corrections to the 1 st printing
Quantum Mechanics: A Paradigms Approach David H. McIntyre 7 July 6 Corrections to the st printing page xxi, last paragraph before "The End", nd line: Change "become" to "became". page, first line after
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 informationFinal Examination. Tuesday December 15, :30 am 12:30 pm. particles that are in the same spin state 1 2, + 1 2
Department of Physics Quantum Mechanics I, Physics 57 Temple University Instructor: Z.E. Meziani Final Examination Tuesday December 5, 5 :3 am :3 pm Problem. pts) Consider a system of three non interacting,
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 informationQuantum Physics II (8.05) Fall 2002 Outline
Quantum Physics II (8.05) Fall 2002 Outline 1. General structure of quantum mechanics. 8.04 was based primarily on wave mechanics. We review that foundation with the intent to build a more formal basis
More informationUniversity of Michigan Physics Department Graduate Qualifying Examination
Name: University of Michigan Physics Department Graduate Qualifying Examination Part II: Modern Physics Saturday 17 May 2014 9:30 am 2:30 pm Exam Number: This is a closed book exam, but a number of useful
More informationPhysics 115C Homework 3
Physics 115C Homework 3 Problem 1 In this problem, it will be convenient to introduce the Einstein summation convention. Note that we can write S = i S i i where the sum is over i = x,y,z. In the Einstein
More informationQuantum Mechanics in One Dimension. Solutions of Selected Problems
Chapter 6 Quantum Mechanics in One Dimension. Solutions of Selected Problems 6.1 Problem 6.13 (In the text book) A proton is confined to moving in a onedimensional box of width.2 nm. (a) Find the lowest
More information( ). Expanding the square and keeping in mind that
Oneelectron 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( ) in the interaction picture arises only
Physics 606, Quantum Mechanics, Final Exam NAME 1 Atomic transitions due to timedependent electric field Consider a hydrogen atom which is in its ground state for t < 0 For t > 0 it is subjected to a
More information( r) = 1 Z. e Zr/a 0. + n +1δ n', n+1 ). dt ' e i ( ε n ε i )t'/! a n ( t) = n ψ t = 1 i! e iε n t/! n' x n = Physics 624, Quantum II  Exam 1
Physics 624, Quantum II  Exam 1 Please show all your work on the separate sheets provided (and be sure to include your name) You are graded on your work on those pages, with partial credit where it is
More informationTotal Angular Momentum for Hydrogen
Physics 4 Lecture 7 Total Angular Momentum for Hydrogen Lecture 7 Physics 4 Quantum Mechanics I Friday, April th, 008 We have the Hydrogen Hamiltonian for central potential φ(r), we can write: H r = p
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 information1 Commutators (10 pts)
Final Exam Solutions 37A Fall 0 I. Siddiqi / E. Dodds Commutators 0 pts) ) Consider the operator Â = Ĵx Ĵ y + ĴyĴx where J i represents the total angular momentum in the ith direction. a) Express both
More informationLecture 7. More dimensions
Lecture 7 More dimensions 67 68 LECTURE 7. MORE DIMENSIONS 7.1 Introduction In this lecture we generalize the concepts introduced so far to systems that evolve in more than one spatial dimension. While
More informationChemistry 3502/4502. Exam III. All Hallows Eve/Samhain, ) This is a multiple choice exam. Circle the correct answer.
B Chemistry 3502/4502 Exam III All Hallows Eve/Samhain, 2003 1) This is a multiple choice exam. Circle the correct answer. 2) There is one correct answer to every problem. There is no partial credit. 3)
More information6.1 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 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 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 informationLecture #1. Review. Postulates of quantum mechanics (13) Postulate 1
L1.P1 Lecture #1 Review Postulates of quantum mechanics (13) Postulate 1 The state of a system at any instant of time may be represented by a wave function which is continuous and differentiable. Specifically,
More informationChm 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 information26 Group Theory Basics
26 Group Theory Basics 1. Reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian.
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 informationSolution to Problem Set No. 6: Time Independent Perturbation Theory
Solution to Problem Set No. 6: Time Independent Perturbation Theory Simon Lin December, 17 1 The Anharmonic Oscillator 1.1 As a first step we invert the definitions of creation and annihilation operators
More informationM04M.1 Particles on a Line
Part I Mechanics M04M.1 Particles on a Line M04M.1 Particles on a Line Two elastic spherical particles with masses m and M (m M) are constrained to move along a straight line with an elastically reflecting
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 electronnucleus interaction H LS = α L S, also known as the spinorbit
More informationProblem 1: A 3D Spherical Well(10 Points)
Problem : A 3D Spherical Well( Points) For this problem, consider a particle of mass m in a threedimensional spherical potential well, V (r), given as, V = r a/2 V = W r > a/2. with W >. All of the following
More informationProbability 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 informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More information2 Canonical quantization
Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.
More informationRotations in Quantum Mechanics
Rotations in Quantum Mechanics We have seen that physical transformations are represented in quantum mechanics by unitary operators acting on the Hilbert space. In this section, we ll think about the specific
More informationQuantum Mechanics Final Exam Solutions Fall 2015
171.303 Quantum Mechanics Final Exam Solutions Fall 015 Problem 1 (a) For convenience, let θ be a real number between 0 and π so that a sinθ and b cosθ, which is possible since a +b 1. Then the operator
More informationSpring /2/ pts 1 point per minute
Physics 519 MIDTERM Name: Spring 014 6//14 80 pts 1 point per minute Exam procedures. Please write your name above. Please sit away from other students. If you have a question about the exam, please ask.
More informationPreliminary Examination  Day 1 Thursday, August 9, 2018
UNL  Department of Physics and Astronomy Preliminary Examination  Day Thursday, August 9, 8 This test covers the topics of Thermodynamics and Statistical Mechanics (Topic ) and Quantum Mechanics (Topic
More informationONE 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 informationQuantum Physics III (8.06) Spring 2008 Final Exam Solutions
Quantum Physics III (8.6) Spring 8 Final Exam Solutions May 19, 8 1. Short answer questions (35 points) (a) ( points) α 4 mc (b) ( points) µ B B, where µ B = e m (c) (3 points) In the variational ansatz,
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 August 23, 208 9:00 a.m. :00 p.m. Do any four problems. Each problem is worth 25 points.
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 informationPhysics 742 Graduate Quantum Mechanics 2 Solutions to Second Exam, Spring 2017
Physics 74 Graduate Quantum Mechanics Solutions to Second Exam Spring 17 The points for each question are marked. Each question is worth points. Some possibly useful formulas appear at the end of the test.
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More informationAtkins & 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 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 information1. Electricity and Magnetism (Fall 1995, Part 1) A metal sphere has a radius R and a charge Q.
1. Electricity and Magnetism (Fall 1995, Part 1) A metal sphere has a radius R and a charge Q. (a) Compute the electric part of the Maxwell stress tensor T ij (r) = 1 {E i E j 12 } 4π E2 δ ij both inside
More informationJ10M.1  Rod on a Rail (M93M.2)
Part I  Mechanics J10M.1  Rod on a Rail (M93M.2) J10M.1  Rod on a Rail (M93M.2) s α l θ g z x A uniform rod of length l and mass m moves in the xz plane. One end of the rod is suspended from a straight
More informationUGC ACADEMY LEADING INSTITUE FOR CSIRJRF/NET, GATE & JAM PH 05 PHYSICAL SCIENCE TEST SERIES # 1. Quantum, Statistical & Thermal Physics
UGC ACADEMY LEADING INSTITUE FOR CSIRJRF/NET, GATE & JAM BOOKLET CODE SUBJECT CODE PH 05 PHYSICAL SCIENCE TEST SERIES # Quantum, Statistical & Thermal Physics Timing: 3: H M.M: 00 Instructions. This test
More informationDepartment of Physics and Astronomy University of Georgia
Department of Physics and Astronomy University of Georgia August 2007 Written Comprehensive Exam Day 1 This is a closedbook, closednote exam. You may use a calculator, but only for arithmetic functions
More information8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Friday May 24, Final Exam
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Friday May 24, 2012 Final Exam Last Name: First Name: Check Recitation Instructor Time R01 Barton Zwiebach 10:00 R02
More informationMATH325  QUANTUM MECHANICS  SOLUTION SHEET 11
MATH35  QUANTUM MECHANICS  SOLUTION SHEET. The Hamiltonian for a particle of mass m moving in three dimensions under the influence of a threedimensional harmonic oscillator potential is Ĥ = h m + mω
More informationPerturbation Theory. Andreas Wacker Mathematical Physics Lund University
Perturbation Theory Andreas Wacker Mathematical Physics Lund University General starting point Hamiltonian ^H (t) has typically noanalytic solution of Ψ(t) Decompose Ĥ (t )=Ĥ 0 + V (t) known eigenstates
More informationMath Questions for the 2011 PhD Qualifier Exam 1. Evaluate the following definite integral 3" 4 where! ( x) is the Dirac!  function. # " 4 [ ( )] dx x 2! cos x 2. Consider the differential equation dx
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 informationQuantum Mechanics: Fundamentals
Kurt Gottfried TungMow Yan Quantum Mechanics: Fundamentals Second Edition With 75 Figures Springer Preface vii Fundamental Concepts 1 1.1 Complementarity and Uncertainty 1 (a) Complementarity 2 (b) The
More informationDegeneracy & 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 informationPhysics 828 Problem Set 7 Due Wednesday 02/24/2010
Physics 88 Problem Set 7 Due Wednesday /4/ 7)a)Consider the proton to be a uniformly charged sphere of radius f m Determine the correction to the s ground state energy 4 points) This is a standard problem
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 20, 2017 9:00 a.m. 1:00 p.m. Do any four problems. Each problem is worth 25 points.
More informationFurther Quantum Mechanics Problem Set
CWPP 212 Further Quantum Mechanics Problem Set 1 Further Quantum Mechanics Christopher Palmer 212 Problem Set There are three problem sets, suitable for use at the end of Hilary Term, beginning of Trinity
More information5.61 Physical Chemistry Final Exam 12/16/09. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Chemistry Physical Chemistry
5.6 Physical Chemistry Final Exam 2/6/09 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry Chemistry  5.6 Physical Chemistry Final Examination () PRINT your name on the cover page. (2) It
More informationSECOND PUBLIC EXAMINATION. Honour School of Physics Part C: 4 Year Course. Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS
A047W SECOND PUBLIC EXAMINATION Honour School of Physics Part C: 4 Year Course Honour School of Physics and Philosophy Part C C4: PARTICLE PHYSICS TRINITY TERM 05 Thursday, 8 June,.30 pm 5.45 pm 5 minutes
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 information