Solution of Second Midterm Examination Thursday November 09, 2017
|
|
- Primrose Sanders
- 5 years ago
- Views:
Transcription
1 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 of three non interacting particles that are confined to move in a one dimensional infinite potential of length a: V (x = 0 for 0 < x < a and V (x = for other values of x. Determine the energy, degeneracy and wave function of the ground state and the first excited state when the three particles are Figure 1: Particle distribution among levels for the ground state (GS, the first excited state (FES for a system of three noninteracting identical Bosons (left and Fermions(right moving in an infinite well. Each state of the fermions system is -fold degenerate due to the two possible orientations of the isolated fermion a Identical particles of spin zero. Since the particles are non-interacting, the Hamiltonian of the three particle is the sum of the Hamiltonians of each particle moving in a one dimensional infinite potential. The total energy of the system of three particles is additive and is given by: and the wave function by E n1,n,n 3 = π ( n ma 1 + n + n 3 (1 ψ n1,n,n 3 (x 1, x, x 3 = 8 ( a 3 sin n1 π ( a x n π ( 1 sin a x n3 π sin a x 3 where n 1, n and n 3 are the possible quantum numbers that each particle can carry. ( 1
2 If all three particles are identical bosons, the ground state is non degenerate and will correspond to all particles in the lowest state n 1 = n = n 3 = 1 with the corresponding wave function ψ (0 = ψ 1 (x 1 ψ 1 (x ψ 1 (x 3 = since ψ n (x i = /a sin (nπx i /a. E (0 = E 1,1,1 = 3 π ma (3 8 ( π ( π ( π a 3 sin a x 1 sin a x sin a x 3 In the first excited state (which is also non-degenerate we have two particles in ψ 1 (each with energy ɛ 1 = π /(ma and one particle in ψ (with energy ɛ = π /(ma = ɛ 1 : E (1 = ɛ 1 + ɛ = ɛ 1 + ɛ 1 = 6ɛ 1 = 3 π ( ma (5 Since the particles are identical we can no longer say which particle is in which state only that two particles are in ψ 1 and one is in ψ. Starting from the general antisymmetrized expression of three identical particles but avoiding to double count since particle 1 and are described by ψ 1 and one is described by ψ. ψ (1 = b Identical particles of spin 1/.! 3! [ψ 1(x 1 ψ 1 (x ψ (x 3 + ψ 1 (x 1 ψ (x ψ 1 (x 3 + ψ (x 1 ψ 1 (x ψ 1 (x 3 ] (6 If the three particles are identical spin 1 fermions, the ground state corresponds to the case where two particles are in the lowest state ψ 1 (with one having a spin up + and the other spin down, while the third particle is in the next state ψ (its spin can be either up or down ±. The ground state energy is E (1 = ɛ 1 + ɛ = ɛ 1 + ɛ 1 = 6ɛ 1 = 3 π ma (7 The ground state wave function is antisymmetric and is given by the Slater determinant ψ (0 (x 1, x, x 3 = 1 ψ 1 (x 1 χ(s 1 ψ 1 (x χ(s ψ 1 (x 3 χ(s 3 3! ψ 1 (x 1 χ(s 1 ψ 1 (x χ(s ψ 1 (x 3 χ(s 3 ψ (x 1 χ(s 1 ψ (x χ(s ψ (x 3 χ(s 3 (8 This state is twofold degenerate, since there are four different ways of configuring the spins of the three fermions (the ground state (GS shown in Fig. 1 is just one of the two configurations. The first excited state corresponds to one particle in the lowest state ψ 1 (its spin can be either up or down and the other two particles in the state ψ ( the spin of one is up and down for the other. As in the ground state, there are also two different ways of configuring the spins of the three fermions in the first excited state (FES; The state shown in Fig. 1 is one of the four configurations. ψ (1 (x 1, x, x 3 = 1 3! ψ 1 (x 1 χ(s 1 ψ 1 (x χ(s ψ 1 (x 3 χ(s 3 ψ (x 1 χ(s 1 ψ (x χ(s ψ (x 3 χ(s 3 ψ (x 1 χ(s 1 ψ (x χ(s ψ (x 3 χ(s 3. (9
3 The energy of these four states is given by E (1 = ɛ 1 + ɛ = ɛ 1 + 8ɛ 1 = 9ɛ 1 = 9 π ma (10 leading to an excitation energy of E (1 E (0 = 9ɛ 1 6ɛ 1 = 3 π /(ma. Problem. (10pts When a hydrogen-like atom is placed in a weak magnetic field B, the energy of interaction is described by the Zeeman Hamiltonian ĤZ Where g s is approximately equal to. Ĥ Z = µ B B ˆ L + gs µ B B ˆ S (11 a Derive an equation for the energy of the atom using first order perturbation theory. Assume that in the absense of the B, the wave functions for the atom are eigenfunctions of ˆL, Ŝ, Ĵ and Ĵz where ˆ J = ˆ L + ˆ S. If we choose the magnetic field along the Oz direction, the perturbing Hamiltonian may be written as: H int = µ B B(ˆL z + Ŝz = µ B B(Ĵz + Ŝz (1 where we used g s =. The first order correction to the energy is then given by: E (1 = L, S, J, M J µ B B(Ĵz + Ŝz L, S, J, M J (13 = l, 1/, l ± 1/, m µ B B(Ĵz + Ŝz l, 1/, l ± 1/, m (1 = µ B Bm + µ B B l, 1/, l ± 1/, m j Ŝz l, 1/, l ± 1/, m (15 The matrix element of Ŝz can be obtained by expressing the eigenfunctions common to ˆL, Ŝ, Ĵ and Ĵz in terms of the eigenfunctions common to ˆL, ˆL z, Ŝ, and Ŝz as given in the relation l ± m + 1/ l m + 1/ l ± 1/, m >= ± m 1/, 1/ > + m + 1/, 1/ > (16 l + 1 l + 1 giving the needed matrix element as: l, 1/, l±1/, m j Ŝz l, 1/, l±1/, m = Leading thus to 1 m (l±m+1/ l±m 1/ = ± (l + 1 (l + 1 (17 [ E (1 = µ B Bm 1 ± 1 ] = gµ B Bm (18 l + 1 For j = l ± 1/ The Landé-factor is thus found to be g = 1 ± [1/(l + 1] 3
4 b The valence electron of an alkali metal atom is excited to a p state. Into how many components is each of the levels split when a weak field B is applied? In the absence of the field there are two energy levels specified by (l = 1, s = 1/, j = 3/ and (l = 1, s = 1/, j = 1/. These are denoted by the spectroscopic symbols P 3/ and P 1/, respectively. When the field is applied the energy changes are given by [ E = µ B Bm j 1 ± 1 ] = gµ B Bm j (19 l + 1 The j = 3/ level ( P 3/ is split into four components since m j takes on four values. The j = 1/ level ( P 1/ is split into two components corresponding to m j = ±1/ c How large are the splittings in units of µ B B? The splittings depend on the value of the Landé-factor g(l, s, j. For the P 3/ level: For the P 1/ level: These splittings are shown in Fig. g = l l + 1 g = l l + 1 = 3 = 3 (0 (1 Figure : Splitting diagram You may use the following relation: l ± m + 1/ l m + 1/ l ± 1/, m >= ± m 1/, 1/ > + m + 1/, 1/ > ( l + 1 l + 1 Problem 3. (10pts
5 Consider two spin 1/ s S 1 and S, coupled by an interaction of the form a 0 e t τ S 1 S. At t =, the system is in the state +, (an eigenstate of S 1z and S z with the eigenvalues + / and / a Calculate P (+ + by using first-order time dependent perturbation theory. First, we write the state of the system at t = in the basis of Ŝ and Ŝz, namely { S, M } ψ( = +, = 1 [ 1, 0 + 0, 0 ] (3 where 1, 0 and 0, 0 are kets of the S, M basis. In 1 st -order time-dependent perturbation theory, we have P(+ + = 1 + e iωfit W fi (tdt (5 where ω fi and W fi (t and W fi (t = + W (t + We also express + in terms of { S, M in order for us to calculate the matrix element W fi (t, + = 1 [ 1, 0 0, 0 ] (6 This leads us to W fi (t = a 0e t τ = a 0e t τ = a 0e t τ [ ] ( 1, 0 0, 0 (S S1 S ( 1, 0 0, 0 ( (7 [ 1, 0 (S S 1 S 1, 0 0, 0 (S S 1 S 0, 0 ] (8 [ 3 ] ( 3 = a 0e t τ (9 P(+ + = 1 e iω fit Given that ω fi = 0 we can write the transition probability as ( a0 P(+ + = + a 0e t τ dt e t τ dt = a 0 πτ (30 (31 b Now assume that the two spins are also interacting in a static magnetic field B 0 parallel to Oz. The corresponding Zeeman Hamiltonian can be written H 0 = B 0 (γ 1 S 1z + γ S z (3 where γ 1 and γ are the gyromagnetic ratios of the two spins, assumed to be different. Assume that a(t = a 0 e t τ. Calculate P(+ + by first-order time-dependent perturbation theory. With fixed a 0 and τ, discuss the variation of P(+ + with respect to B 0. 5
6 Again, but now with ω fi 0 P(+ + = 1 + e iωfit W fi (tdt ω fi = E0 + E 0 + H 0 + = B 0 ( γ 1 H 0 + = B 0 ( γ 1 (33 (3 + γ + (35 γ + (36 As in a ω fi = B 0(γ 1 γ B 0( γ 1 + γ = B 0 (γ 1 γ (37 W fi (t = a 0e t τ (38 P(+ + = = = + e ib 0(γ 1 γ t a 0 e t τ dt a 0 + e ib 0(γ 1 γ t t τ dt ( a 0 B exp 0 (γ 1 γ πτ dt /τ (39 (0 (1 P(+ + = τ πa 0 ( B exp 0 (γ 1 γ τ ( We find a gaussian function describing the probability as a function of the magnetic field strength. Note that this expression for P(+ + in the limit B 0 0, agrees with part a. 6
7 Figure 3: Probability as a function of the magnetic field strength 7
Final 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 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 informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent Schrödinger equation
Lecture 17 Page 1 Lecture 17 L17.P1 Review Schrödinger equation The general solution of Schrödinger equation in three dimensions (if V does not depend on time) is where functions are solutions of time-independent
More 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 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 informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.
PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms
More information( ). Expanding the square and keeping in mind that
One-electron atom in a Magnetic Field When the atom is in a magnetic field the magnetic moment of the electron due to its orbital motion and its spin interacts with the field and the Schrodinger Hamiltonian
More informationPH 451/551 Quantum Mechanics Capstone Winter 201x
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
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 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 informationFor example, in one dimension if we had two particles in a one-dimensional 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 informationCHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM. (From Cohen-Tannoudji, Chapter XII)
CHAPTER 6: AN APPLICATION OF PERTURBATION THEORY THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM (From Cohen-Tannoudji, Chapter XII) We will now incorporate a weak relativistic effects as perturbation
More informationSommerfeld (1920) noted energy levels of Li deduced from spectroscopy looked like H, with slight adjustment of principal quantum number:
Spin. Historical Spectroscopy of Alkali atoms First expt. to suggest need for electron spin: observation of splitting of expected spectral lines for alkali atoms: i.e. expect one line based on analogy
More 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 informationIntroduction to Quantum Mechanics Physics Thursday February 21, Problem # 1 (10pts) We are given the operator U(m, n) defined by
Department of Physics Introduction to Quantum Mechanics Physics 5701 Temple University Z.-E. Meziani Thursday February 1, 017 Problem # 1 10pts We are given the operator Um, n defined by Ûm, n φ m >< φ
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 informationLecture #13 1. Incorporating a vector potential into the Hamiltonian 2. Spin postulates 3. Description of spin states 4. Identical particles in
Lecture #3. Incorporating a vector potential into the Hamiltonian. Spin postulates 3. Description of spin states 4. Identical particles in classical and QM 5. Exchange degeneracy - the fundamental problem
More 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 informationThe general solution of Schrödinger equation in three dimensions (if V does not depend on time) are solutions of time-independent 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 time-independent
More informationAngular Momentum set II
Angular Momentum set II PH - QM II Sem, 7-8 Problem : Using the commutation relations for the angular momentum operators, prove the Jacobi identity Problem : [ˆL x, [ˆL y, ˆL z ]] + [ˆL y, [ˆL z, ˆL x
More informationProb (solution by Michael Fisher) 1
Prob 975 (solution by Michael Fisher) We begin by expressing the initial state in a basis of the spherical harmonics, which will allow us to apply the operators ˆL and ˆL z θ, φ φ() = 4π sin θ sin φ =
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 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 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 information( ) dσ 1 dσ 2 + α * 2
Chemistry 36 Dr. Jean M. Standard Problem Set Solutions. The spin up and spin down eigenfunctions for each electron in a many-electron system are normalized and orthogonal as given by the relations, α
More informationQuantum and Atomic Physics: Questions Prof Andrew Steane 2005
Quantum and Atomic Physics: Questions Prof Andrew Steane 2005 Problem set. Some general quantum knowledge; Hydrogen gross structure, wavefunctions, quantum numbers, spectrum, scaling. General familiarity.
More information(relativistic effects kinetic energy & spin-orbit coupling) 3. Hyperfine structure: ) (spin-spin coupling of e & p + magnetic moments) 4.
4 Time-ind. Perturbation Theory II We said we solved the Hydrogen atom exactly, but we lied. There are a number of physical effects our solution of the Hamiltonian H = p /m e /r left out. We already said
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 informationwhere A and α are real constants. 1a) Determine A Solution: We must normalize the solution, which in spherical coordinates means
Midterm #, Physics 5C, Spring 8. Write your responses below, on the back, or on the extra pages. Show your work, and take care to explain what you are doing; partial credit will be given for incomplete
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 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 information4πε. me 1,2,3,... 1 n. H atom 4. in a.u. atomic units. energy: 1 a.u. = ev distance 1 a.u. = Å
H atom 4 E a me =, n=,,3,... 8ε 0 0 π me e e 0 hn ε h = = 0.59Å E = me (4 πε ) 4 e 0 n n in a.u. atomic units E = r = Z n nao Z = e = me = 4πε = 0 energy: a.u. = 7. ev distance a.u. = 0.59 Å General results
More informationExercises for Quantum Mechanics (TFFY54)
Exercises for Quantum Mechanics (TFFY54) Johan Henriksson and Patrick Norman Department of Physics, Chemistry and Biology, Linköping University, SE-581 83 Linköping, Sweden Spring Term 007 1 For a Hermitian
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 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 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 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 information2.4. Quantum Mechanical description of hydrogen atom
2.4. Quantum Mechanical description of hydrogen atom Atomic units Quantity Atomic unit SI Conversion Ang. mom. h [J s] h = 1, 05459 10 34 Js Mass m e [kg] m e = 9, 1094 10 31 kg Charge e [C] e = 1, 6022
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 spin-1/2 particles. With probability p they are in the singlet state, ( )/ 2, and with
More informationLecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of
More informationEnergy Level Energy Level Diagrams for Diagrams for Simple Hydrogen Model
Quantum Mechanics and Atomic Physics Lecture 20: Real Hydrogen Atom /Identical particles http://www.physics.rutgers.edu/ugrad/361 physics edu/ugrad/361 Prof. Sean Oh Last time Hydrogen atom: electron in
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 informationα β β α β β 0 β α 0 0 0
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Chemistry 5.6 Quantum Mechanics Fall 03 Problem Set #7 Reading Assignment: McQuarrie 9.-9.5, 0.-0.5,Matlab and Linear Algebra Handouts ( = Easier = More
More informationQUALIFYING EXAMINATION, Part 1. 2:00 PM 5:00 PM, Thursday September 3, 2009
QUALIFYING EXAMINATION, Part 1 2:00 PM 5:00 PM, Thursday September 3, 2009 Attempt all parts of all four problems. Please begin your answer to each problem on a separate sheet, write your 3 digit code
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 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 informationAre these states normalized? A) Yes
QMII-. Consider two kets and their corresponding column vectors: Ψ = φ = Are these two state orthogonal? Is ψ φ = 0? A) Yes ) No Answer: A Are these states normalized? A) Yes ) No Answer: (each state has
More informationQuantum Physics III (8.06) Spring 2016 Assignment 3
Quantum Physics III (8.6) Spring 6 Assignment 3 Readings Griffiths Chapter 9 on time-dependent perturbation theory Shankar Chapter 8 Cohen-Tannoudji, Chapter XIII. Problem Set 3. Semi-classical approximation
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 spin-half 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 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 informationFirst Problem Set for Physics 847 (Statistical Physics II)
First Problem Set for Physics 847 (Statistical Physics II) Important dates: Feb 0 0:30am-:8pm midterm exam, Mar 6 9:30am-:8am final exam Due date: Tuesday, Jan 3. Review 0 points Let us start by reviewing
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 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 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 informationSt Hugh s 2 nd Year: Quantum Mechanics II. Reading. Topics. The following sources are recommended for this tutorial:
St Hugh s 2 nd Year: Quantum Mechanics II Reading The following sources are recommended for this tutorial: The key text (especially here in Oxford) is Molecular Quantum Mechanics, P. W. Atkins and R. S.
More informationMagnetism of Atoms and Ions. Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D Karlsruhe
Magnetism of Atoms and Ions Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D-76131 Karlsruhe 1 0. Overview Literature J.M.D. Coey, Magnetism and
More informationINTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM
INTRODUCTION TO QUANTUM ELECTRODYNAMICS by Lawrence R. Mead, Prof. Physics, USM I. The interaction of electromagnetic fields with matter. The Lagrangian for the charge q in electromagnetic potentials V
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 informationPhysics 139B Solutions to Homework Set 4 Fall 2009
Physics 139B Solutions to Homework Set 4 Fall 9 1. Liboff, problem 1.16 on page 594 595. Consider an atom whose electrons are L S coupled so that the good quantum numbers are j l s m j and eigenstates
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 one-dimensional harmonic oscillator, originally in the ground state,
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 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 information14. Structure of Nuclei
14. Structure of Nuclei Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 14. Structure of Nuclei 1 In this section... Magic Numbers The Nuclear Shell Model Excited States Dr. Tina Potter 14.
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 information1.1 Quantum mechanics of one particle
1 Second quantization 1.1 Quantum mechanics of one particle In quantum mechanics the physical state of a particle is described in terms of a ket Ψ. This ket belongs to a Hilbert space which is nothing
More informationLøsningsforslag Eksamen 18. desember 2003 TFY4250 Atom- og molekylfysikk og FY2045 Innføring i kvantemekanikk
Eksamen TFY450 18. desember 003 - løsningsforslag 1 Oppgave 1 Løsningsforslag Eksamen 18. desember 003 TFY450 Atom- og molekylfysikk og FY045 Innføring i kvantemekanikk a. With Ĥ = ˆK + V = h + V (x),
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 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 information16.1. PROBLEM SET I 197
6.. PROBLEM SET I 97 Answers: Problem set I. a In one dimension, the current operator is specified by ĵ = m ψ ˆpψ + ψˆpψ. Applied to the left hand side of the system outside the region of the potential,
More informationSection 10: Many Particle Quantum Mechanics Solutions
Physics 143a: Quantum Mechanics I Section 10: Many Particle Quantum Mechanics Solutions Spring 015, Harvard Here is a summary of the most important points from this week (with a few of my own tidbits),
More informationQualification Exam: Quantum Mechanics
Qualification Exam: Quantum Mechanics Name:, QEID#76977605: October, 2017 Qualification Exam QEID#76977605 2 1 Undergraduate level Problem 1. 1983-Fall-QM-U-1 ID:QM-U-2 Consider two spin 1/2 particles
More informationMAGNETISM OF ATOMS QUANTUM-MECHANICAL BASICS. Janusz Adamowski AGH University of Science and Technology, Kraków, Poland
MAGNETISM OF ATOMS QUANTUM-MECHANICAL BASICS Janusz Adamowski AGH University of Science and Technology, Kraków, Poland 1 The magnetism of materials can be derived from the magnetic properties of atoms.
More informationLecture 3: Helium Readings: Foot Chapter 3
Lecture 3: Helium Readings: Foot Chapter 3 Last Week: the hydrogen atom, eigenstate wave functions, and the gross and fine energy structure for hydrogen-like single-electron atoms E n Z n = hcr Zα / µ
More informationAddition of Angular Momenta
Addition of Angular Momenta What we have so far considered to be an exact solution for the many electron problem, should really be called exact non-relativistic solution. A relativistic treatment is needed
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 informationPlease read the following instructions:
MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 16, 17 (8:3 a.m. - 9:45 a.m.) PLACE: RB 11 Only non-programmable calculators are allowed. Name: ID: Please read the following instructions: This
More informationIn the following, we investigate the time-dependent two-component wave function ψ(t) = ( )
Ph.D. Qualifier, Quantum mechanics DO ONLY 3 OF THE 4 QUESTIONS Note the additional material for questions 1 and 3 at the end. PROBLEM 1. In the presence of a magnetic field B = (B x, B y, B z ), the dynamics
More information0 belonging to the unperturbed Hamiltonian H 0 are known
Time Independent Perturbation Theory D Perturbation theory is used in two qualitatively different contexts in quantum chemistry. It allows one to estimate (because perturbation theory is usually employed
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 information(1.1) In particular, ψ( q 1, m 1 ; ; q N, m N ) 2 is the probability to find the first particle
Chapter 1 Identical particles 1.1 Distinguishable particles The Hilbert space of N has to be a subspace H = N n=1h n. Observables Ân of the n-th particle are self-adjoint operators of the form 1 1 1 1
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 14:00-16:00 in my room. (This info is also available on the course homepage.) Examinator:
More informationPotential energy, from Coulomb's law. Potential is spherically symmetric. Therefore, solutions must have form
Lecture 6 Page 1 Atoms L6.P1 Review of hydrogen atom Heavy proton (put at the origin), charge e and much lighter electron, charge -e. Potential energy, from Coulomb's law Potential is spherically symmetric.
More informationPHYSICS 721/821 - Spring Semester ODU. Graduate Quantum Mechanics II Midterm Exam - Solution
PHYSICS 72/82 - Spring Semester 2 - ODU Graduate Quantum Mechanics II Midterm Exam - Solution Problem ) An electron (mass 5, ev/c 2 ) is in a one-dimensional potential well as sketched to the right (the
More informationChapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found.
Chapter 2 Approximation Methods Can be Used When Exact Solutions to the Schrödinger Equation Can Not be Found. In applying quantum mechanics to 'real' chemical problems, one is usually faced with a Schrödinger
More informationA.1 Alkaline atoms in magnetic fields
164 Appendix the Kohn, virial and Bertrand s theorem, with an original approach. Annex A.4 summarizes elements of the elastic collisions theory required to address scattering problems. Eventually, annex
More informationIntermission: Let s review the essentials of the Helium Atom
PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the
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 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 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 informationSolutions to chapter 4 problems
Chapter 9 Solutions to chapter 4 problems Solution to Exercise 47 For example, the x component of the angular momentum is defined as ˆL x ŷˆp z ẑ ˆp y The position and momentum observables are Hermitian;
More information0 + E (1) and the first correction to the ground state energy is given by
1 Problem set 9 Handout: 1/24 Due date: 1/31 Problem 1 Prove that the energy to first order for the lowest-energy state of a perturbed system is an upper bound for the exact energy of the lowest-energy
More information(a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron, but neglecting spin-orbit interactions.
1. Quantum Mechanics (Spring 2007) Consider a hydrogen atom in a weak uniform magnetic field B = Bê z. (a) Write down the total Hamiltonian of this system, including the spin degree of freedom of the electron,
More informationParticle in one-dimensional box
Particle in the box Particle in one-dimensional box V(x) -a 0 a +~ An example of a situation in which only bound states exist in a quantum system. We consider the stationary states of a particle confined
More informationQuantum Physics 2006/07
Quantum Physics 6/7 Lecture 7: More on the Dirac Equation In the last lecture we showed that the Dirac equation for a free particle i h t ψr, t = i hc α + β mc ψr, t has plane wave solutions ψr, t = exp
More informationAtomic and molecular physics Revision lecture
Atomic and molecular physics Revision lecture Answer all questions Angular momentum J`2 ` J z j,m = j j+1 j,m j,m =m j,m Allowed values of mgo from j to +jin integer steps If there is no external field,
More informationHomework Solution Set # 3. Thursday, September 22, Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Lalo, Second Volume Complement G X
Deparmen of Physics Quanum Mechanics II, 570 Temple Universiy Insrucor: Z.-E. Meziani Homework Soluion Se # 3 Thursday, Sepember, 06 Texbook: Claude Cohen Tannoudji, Bernard Diu and Franck Lalo, Second
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 informationMolecular Term Symbols
Molecular Term Symbols A molecular configuration is a specification of the occupied molecular orbitals in a molecule. For example, N : σ gσ uπ 4 uσ g A given configuration may have several different states
More informationChem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.
Chem 44 Review for Exam Hydrogenic atoms: The Coulomb energy between two point charges Ze and e: V r Ze r Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative
More informationAn introduction to magnetism in three parts
An introduction to magnetism in three parts Wulf Wulfhekel Physikalisches Institut, Karlsruhe Institute of Technology (KIT) Wolfgang Gaede Str. 1, D-76131 Karlsruhe 0. Overview Chapters of the three lectures
More information