Physics 443, Solutions to PS 8
|
|
- Lenard Lang
- 5 years ago
- Views:
Transcription
1 Physics 443, Solutions to PS 8.Griffiths 5.7. For notational simplicity, we define ψ. = ψa, ψ 2. = ψb, ψ 3. = ψ c. Then we can write: Distinguishable ψ({x i }) = Bosons ψ({x i }) = Fermions ψ({x i }) = 3 ψ i (x i ) i= 3 ψ a (x i )ψ b (x j )ψ c (x k )(ǫ ijk ) 2 i,j,k= 3 ψ a (x i )ψ b (x j )ψ c (x k )(ǫ ijk ) i,j,k= 2.Griffiths 5.. This question is solving a messy integral. Lets tackle the physics i.e. part(b) first. Since V ee = e2 4πǫ r r 2, using part (a) we can estimate that V ee = 5ǫ /2 = +34ev. Therefore an approximation to the energy of ground state of Helium would be 9+34 = 75ev, which compares nicely to the experimental value of 79ev. Now to the integral of part (a). Using the choice of coordinates recommended by Griffiths, we need to solve the following integral. I = ( ) 8 2 d 3 r d 3 exp( 4(r + r 2 )/a) r 2. πa r r2 2 2r r 2 cosθ 2 We proceed as follows. First we scale our variables r and r 2 to remove the length-scale a. Then we perform the integral over θ 2. Notice that as a function of θ 2, we simply have a square-root form of the integrand (since sin θ 2 dθ 2 = d(sin θ 2 )). Also the dφ 2 integral is trivial giving us a 2π. So we can rewrite our integral in the form ( ) 64 2π π 2 a d 3 r r 2 2 dr 2 exp( 4(r + r 2 )) ( r + r 2 r ) r 2. r r 2 r r 2 Notice that this last bit is different depending on which of r and r 2 is greater. In particular, if < r 2 < r it gives 2/r, and for r < r 2 < we have 2/r 2. This means that we can rearrange our integral as follows: ( ) 64 4π π 2 a d 3 r exp( 4r ) ( r r 2 ) 2 dr 2 e 4r 2 + dr 2 r 2 e 4r 2. r r
2 The dr 2 integrals can now be done. We turn to the d 3 r integral, where the angular part just gives us 4π. We are left with just the radial integral as follows: ( ) 32 I = dr (r e 4r 2r 2 a e 8r r e 8r ) = 5 4a 3.Griffiths 5.8. (a) Using ψ(x) = A sin(kx) + B cos(kx), ( < x < a) () and A sin(ka) = [e ika cos(ka)]b, (2) show that the wave function for a particle in the periodic delta function potnetial can be written in the form ψ(x) = C[sin(kx) + e ika sin k(a x)], ( x a). [We can solve Equation 2 for A and substitute into Equation to get B ( ψ(x) = (e ika cos(ka)) sin(kx) + cos(kx) sin(ka) ) sin(ka) = BeiKa ( sin(kx) + e ika (cos(kx) sin(ka) cos(ka) sin(kx) ) sin(ka) = BeiKa ( sin(kx) + e ika sin k(a x) ) sin(ka) = C ( sin(kx) + e ika sin k(a x) ) (b) We see from Equation 2 that if ka = = nπ then either B =, or e ika = cos(nπ) = ±. Then substitution into the equation matching the derivative of the wavefunction. ka e ika k[a cos(ka) B sin(ka)] = 2mαB/ h 2 (3) becomes ka e ika ka cos(nπ) = 2mαB/ h 2 (4) 2
3 and therefore B =. So the wave function at the top of the band is ψ(x) = A sin(kx) and it is ero at each delta function. 4.Griffiths 5.2. If there are delta function wells instead of spikes then V (x) = N i αδ(x ia), α <, and we have to consider solutions with E <. Then the general solution is ψ(x) = A sinh(κx) + B cosh(κx) ( < x < a) (5) where κ = 2mE/ h and the wave function to the left of the origin is Continuity at x = implies ψ(x a) = e ika [A sinh(κx) + B cosh(κx)] (6) e ika [A sinh(κa) + B cosh(κa)] = B (7) The boundary condition for the derivatives gives κa κe ika [A cosh(κa) + B sinh(κa)] = 2mα h 2 B (8) We solve Equation 7 for A and substitute into Equation 8 and get 2mα κ h 2 B sinh(κa) = ( B(eiKa cosh(κa)) e ika B(e ika cosh(κa)) cosh(κa) + B sinh 2 (κa) ) 2mα κ h 2 sinh(κa) = ( eika 2 cosh(κa) e ika ( cosh(κa)) cosh(κa) + sinh 2 (κa) ) = 2 coska 2 cosh(κa) coska = cosh(κa) + mα κ h 2 sinh(κa) f() = cosh() + β sinh() where = κa and β = mαa/ h 2. f() for the negative energy solutions with delta function wells, (β =.5) is plotted in Figure. f() for the positive energy states with delta function wells is shown in Figure 2. We see that the first band is has the same number of states as he subsequent bands. Some of these states are at energies less than ero, and the remainder of the band is filled by states with energies greater than ero. 3
4 .5 cosh() + β sinh() π π/2 Figure : Negative energy states, β =.5.5 β =.5 β =.5 cos() + β sin() π 2π 3π 4π 5π Figure 2: Positive energy states, β = ±.5. The red line is f() for delta function wells, and the blue dashed line for delta function spikes. 4
5 5.Griffiths (a) Because the energy E = 9/2 hω, and (n x + n y + n + 3/2) hω = E, we see that n x + n y + n = 3. For distinguishable particles, we have the following particle configurations: (, 3,,, ), d =, and (2,,,, ), d = 3, and (,,,, ), d = 6. Here d is the degeneracy of that configuration. Check that for each configuration, the sum of the entries is the number of particles N = 3. Also check that the sum of the degeneracy d = = is what we would expect from d 3 = (n + )(n + 2)/2 = (4)(5)/2 =. The probability of getting E = hω/2 is the sum of the contribution from the second configuration (3/)(2/3) plus the contribution from the third configuration (/3)(6/), giving P( hω/2) = (2/5). Similarly you can verify that P(3 hω/2) = 3/, P(5 hω/2) = /5, P(7 hω/2) = /. (b) For the case of Fermions, there is only one allowed configuration (,,, ). We have that with P = /3, each of the states 2E = hω, 3 hω, and 5 hω are possible. (c) For Bosons, the three configurations of part (a) are equally likely, giving us P( hω/2) = /3, P(3 hω/2) = 4/9, P(5 hω/2) = /9, P(7 hω/2) = /9. 6.Griffiths Evaluate the integrals N = E = V k 2 dk (9) 2π 2 e [( h2 k 2 /2m) µ]/k BT ± V h 2 k 4 dk () 2π 2 2m e [( h2 k 2 /2m) µ]/k BT ± for the case of identical fermions at absolute ero. [At T =, the chemical potential µ = E F. Then at T =, Equations 9 and become kf ρ = 2π 2 E = V h 2 2π 2 2m k 2 dk = 6π 2k3 F () kf k 4 dk = V π 2 h 2 2m k5 F (2) With the substitution k F = ( ) 2mE F 2, Equations and 2 become h 2 ρ = ( ) 3 2mEF 2 6π 2 h 2 EF = (6π 2 ρ) 2 h 2 3 2m 5
6 ) 5 2 E = V h 2 ( 2mEF π 2 2m h 2 = V h 2 ( 6π 2 ρ ) 5 3 π 2 2m h 2 ( = 6π 2 Nq ) 5 3 V 2/3 π 2 2m which agree with our analysis of free electrons in a solid where we found that E F = h2 2m (3ρπ2 ) 2/3 (3) E tot = h2 (3π 2 Nq) 5/3 π 2 m V 2/3 ] (4) (Note that for electrons there is an extra factor of 2 in Equations 3 and 4, to account for the spin degeneracy.) 7.Griffiths 5.29 (a) Show that for bosons the chemical potential must always be less than the minimum allowed energy. Hint : n(ǫ) cannot be negative. [For bosons, the most probable number of particles in a given state with energy ǫ is n(ǫ) = e (ǫ µ)/k BT We see that n(ǫ) < if µ > ǫ.] (b) In particular, for the ideal bose gas, µ(t) < for all T. Show that in this case µ(t) monotonically increases as T decreases, assuming N and V are held constant. Hint : Study Equation 9, with the minus sign. [Consider the exponent ( h 2 k 2 /2m µ)/k B T. If T, and µ <, then the exponent is infinite for all k, and N/V. The ratio N/V can only be preserved if µ(t) as T.] (c) A crisis (called Bose condensation) occurs when (as we lower T) µ(t) hits ero. Evaluate the integral, for µ =, and obtain the formula for the critical temperature T c at which this happens. Below the 6
7 critical temperature, the particles crowd into the ground state, and the calculational device of replacing the discrete sum N n = N, n= by a continuous integral (Equation 9) loses its validity. Hint : x s dx = Γ(s)ζ(s), e x where Γ is Euler s gamma function and ζ is the Riemann eta function. Look up the appropriate numerical values. [In the limit where µ =, Equation 9 for bosons is ρ = 2π 2 k 2 e [( h2 k 2 /2m)]/k BT dk Let x = ( h 2 k 2 /2mk B T) and the integral becomes ρ = ( ) 3/2 2mkB T x 3 2 4π 2 h 2 e x dx = 4π 2 = 4π 2 ( ) 3/2 2mkB T h 2 Γ( 3 2 )ζ(3 2 ) ( 2mkB T h 2 ) 3/2 π 2 (2.6238)] (d) Find the critical temperatiure for 4 He. Its density, at this temperature, is.5gm/cm 3. Comment : The experimental value of the critical temperature in 4 He is 2.7 K. [Solve for T = = ( 4π 2 ) 2/3 ρ ( h 2 π/2)(2.6238) 2mk B ( 4π 2 (5 kg/ m 3 )/( kg) ( π/2)(2.6238) ) 2/3 ( ) 2 ( J s) 2 2( kg)( m 2 kg s 2 K ) = 3.2 K] 7
PHYSICS DEPARTMENT, PRINCETON UNIVERSITY PHYSICS 505 MIDTERM EXAMINATION. October 25, 2012, 11:00am 12:20pm, Jadwin Hall A06 SOLUTIONS
PHYSICS DEPARTMENT, PRINCETON UNIVERSITY PHYSICS 505 MIDTERM EXAMINATION October 25, 2012, 11:00am 12:20pm, Jadwin Hall A06 SOLUTIONS This exam contains two problems. Work both problems. They count equally
More informationPhysics 505 Homework No. 4 Solutions S4-1
Physics 505 Homework No 4 s S4- From Prelims, January 2, 2007 Electron with effective mass An electron is moving in one dimension in a potential V (x) = 0 for x > 0 and V (x) = V 0 > 0 for x < 0 The region
More informationContents. 1 Solutions to Chapter 1 Exercises 3. 2 Solutions to Chapter 2 Exercises Solutions to Chapter 3 Exercises 57
Contents 1 Solutions to Chapter 1 Exercises 3 Solutions to Chapter Exercises 43 3 Solutions to Chapter 3 Exercises 57 4 Solutions to Chapter 4 Exercises 77 5 Solutions to Chapter 5 Exercises 89 6 Solutions
More informationApplied Nuclear Physics Homework #2
22.101 Applied Nuclear Physics Homework #2 Author: Lulu Li Professor: Bilge Yildiz, Paola Cappellaro, Ju Li, Sidney Yip Oct. 7, 2011 2 1. Answers: Refer to p16-17 on Krane, or 2.35 in Griffith. (a) x
More informationSupporting Information
Supporting Information A: Calculation of radial distribution functions To get an effective propagator in one dimension, we first transform 1) into spherical coordinates: x a = ρ sin θ cos φ, y = ρ sin
More informationPhysics 218 Quantum Mechanics I Assignment 6
Physics 218 Quantum Mechanics I Assignment 6 Logan A. Morrison February 17, 2016 Problem 1 A non-relativistic beam of particles each with mass, m, and energy, E, which you can treat as a plane wave, is
More informationScattering in One Dimension
Chapter 4 The door beckoned, so he pushed through it. Even the street was better lit. He didn t know how, he just knew the patron seated at the corner table was his man. Spectacles, unkept hair, old sweater,
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 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 informationSchrödinger equation for the nuclear potential
Schrödinger equation for the nuclear potential Introduction to Nuclear Science Simon Fraser University Spring 2011 NUCS 342 January 24, 2011 NUCS 342 (Lecture 4) January 24, 2011 1 / 32 Outline 1 One-dimensional
More informationPhysics 505 Homework No. 12 Solutions S12-1
Physics 55 Homework No. 1 s S1-1 1. 1D ionization. This problem is from the January, 7, prelims. Consider a nonrelativistic mass m particle with coordinate x in one dimension that is subject to an attractive
More informationPhysics 280 Quantum Mechanics Lecture III
Summer 2016 1 1 Department of Physics Drexel University August 17, 2016 Announcements Homework: practice final online by Friday morning Announcements Homework: practice final online by Friday morning Two
More information8.04 Quantum Physics Lecture XV One-dimensional potentials: potential step Figure I: Potential step of height V 0. The particle is incident from the left with energy E. We analyze a time independent situation
More information1. Thegroundstatewavefunctionforahydrogenatomisψ 0 (r) =
CHEM 5: Examples for chapter 1. 1. Thegroundstatewavefunctionforahydrogenatomisψ (r = 1 e r/a. πa (a What is the probability for finding the electron within radius of a from the nucleus? (b Two excited
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 informationTentamen i Kvantfysik I
Karlstads Universitet Fysik Lösningar Tentamen i Kvantfysik I [ VT 7, FYGB7] Datum: 7-6-7 Tid: 4 9 Lärare: Jürgen Fuchs Tel: 54-7 87 Total poäng: 5 Godkänd / 3: 5 Väl godkänd: 375 4: 335 5: 4 Tentan består
More informationLecture 3. Energy bands in solids. 3.1 Bloch theorem
Lecture 3 Energy bands in solids In this lecture we will go beyond the free electron gas approximation by considering that in a real material electrons live inside the lattice formed by the ions (Fig.
More information1. The infinite square well
PHY3011 Wells and Barriers page 1 of 17 1. The infinite square well First we will revise the infinite square well which you did at level 2. Instead of the well extending from 0 to a, in all of the following
More informationNotes on Quantum Mechanics
Notes on Quantum Mechanics Kevin S. Huang Contents 1 The Wave Function 1 1.1 The Schrodinger Equation............................ 1 1. Probability.................................... 1.3 Normalization...................................
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 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 informationModern Physics for Scientists and Engineers Instructors Solutions. Joseph N. Burchett
Modern Physics for Scientists and Engineers Instructors Solutions Joseph N. Burchett January 1, 011 Introduction - Solutions 1. We are given a spherical distribution of charge and asked to calculate the
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 informationPhysics 342 Lecture 17. Midterm I Recap. Lecture 17. Physics 342 Quantum Mechanics I
Physics 342 Lecture 17 Midterm I Recap Lecture 17 Physics 342 Quantum Mechanics I Monday, March 1th, 28 17.1 Introduction In the context of the first midterm, there are a few points I d like to make about
More informationSOLUTIONS. PROBLEM 1. The Hamiltonian of the particle in the gravitational field can be written as, x 0, + U(x), U(x) =
SOLUTIONS PROBLEM 1. The Hailtonian of the particle in the gravitational field can be written as { Ĥ = ˆp2, x 0, + U(x), U(x) = (1) 2 gx, x > 0. The siplest estiate coes fro the uncertainty relation. If
More informationElectrons in a periodic potential
Chapter 3 Electrons in a periodic potential 3.1 Bloch s theorem. We consider in this chapter electrons under the influence of a static, periodic potential V (x), i.e. such that it fulfills V (x) = V (x
More informationTopological insulator part II: Berry Phase and Topological index
Phys60.nb 11 3 Topological insulator part II: Berry Phase and Topological index 3.1. Last chapter Topological insulator: an insulator in the bulk and a metal near the boundary (surface or edge) Quantum
More informationQUANTUM MECHANICS A (SPA 5319) The Finite Square Well
QUANTUM MECHANICS A (SPA 5319) The Finite Square Well We have already solved the problem of the infinite square well. Let us now solve the more realistic finite square well problem. Consider the potential
More informationPH 253 Final Exam: Solution
PH 53 Final Exam: Solution 1. A particle of mass m is confined to a one-dimensional box of width L, that is, the potential energy of the particle is infinite everywhere except in the interval 0
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 informationIntroduction to Quantum Mechanics
Introduction to Quantum Mechanics INEL 5209 - Solid State Devices - Spring 2012 Manuel Toledo January 23, 2012 Manuel Toledo Intro to QM 1/ 26 Outline 1 Review Time dependent Schrödinger equation Momentum
More informationPH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5
PH575 Spring 2009 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 Spring 2009 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 Spring 2009 POP QUIZ
More informationStatistical Mechanics Notes. Ryan D. Reece
Statistical Mechanics Notes Ryan D. Reece August 11, 2006 Contents 1 Thermodynamics 3 1.1 State Variables.......................... 3 1.2 Inexact Differentials....................... 5 1.3 Work and Heat..........................
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 informationElectrons in metals PHYS208. revised Go to Topics Autumn 2010
Go to Topics Autumn 010 Electrons in metals revised 0.1.010 PHYS08 Topics 0.1.010 Classical Models The Drude Theory of metals Conductivity - static electric field Thermal conductivity Fourier Law Wiedemann-Franz
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 informationApplications of Quantum Theory to Some Simple Systems
Applications of Quantum Theory to Some Simple Systems Arbitrariness in the value of total energy. We will use classical mechanics, and for simplicity of the discussion, consider a particle of mass m moving
More informationMagnets, 1D quantum system, and quantum Phase transitions
134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider
More informationdoes not change the dynamics of the system, i.e. that it leaves the Schrödinger equation invariant,
FYST5 Quantum Mechanics II 9..212 1. intermediate eam (1. välikoe): 4 problems, 4 hours 1. As you remember, the Hamilton operator for a charged particle interacting with an electromagentic field can be
More informationliquid He
8.333: Statistical Mechanics I Problem Set # 6 Due: 12/6/13 @ mid-night According to MIT regulations, no problem set can have a due date later than 12/6/13, and I have extended the due date to the last
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 informationThe Stark Effect. a. Evaluate a matrix element. = ee. = ee 200 r cos θ 210. r 2 dr. Angular part use Griffiths page 139 for Yl m (θ, φ)
The Stark Effect a. Evaluate a matrix element. H ee z ee r cos θ ee r dr Angular part use Griffiths page 9 for Yl m θ, φ) Use the famous substitution to find that the angular part is dφ Y θ, φ) cos θ Y
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 informationLecture 4: Resonant Scattering
Lecture 4: Resonant Scattering Sep 16, 2008 Fall 2008 8.513 Quantum Transport Analyticity properties of S-matrix Poles and zeros in a complex plane Isolated resonances; Breit-Wigner theory Quasi-stationary
More informationPhysics 200 Lecture 4. Integration. Lecture 4. Physics 200 Laboratory
Physics 2 Lecture 4 Integration Lecture 4 Physics 2 Laboratory Monday, February 21st, 211 Integration is the flip-side of differentiation in fact, it is often possible to write a differential equation
More information7.4. Why we have two different types of materials: conductors and insulators?
Phys463.nb 55 7.3.5. Folding, Reduced Brillouin zone and extended Brillouin zone for free particles without lattices In the presence of a lattice, we can also unfold the extended Brillouin zone to get
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationHomework 1. 0 for z L. Ae ikz + Be ikz for z 0 Ce κz + De κz for 0 < z < L Fe ikz
Homework. Problem. Consider the tunneling problem with the potential barrier: for z (z) = > for < z < L for z L. Write the wavefunction as: ψ(z) = Ae ikz + Be ikz for z Ce κz + De κz for < z < L Fe ikz
More informationChapter 14. Ideal Bose gas Equation of state
Chapter 14 Ideal Bose gas In this chapter, we shall study the thermodynamic properties of a gas of non-interacting bosons. We will show that the symmetrization of the wavefunction due to the indistinguishability
More informationBrief review of Quantum Mechanics (QM)
Brief review of Quantum Mechanics (QM) Note: This is a collection of several formulae and facts that we will use throughout the course. It is by no means a complete discussion of QM, nor will I attempt
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 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 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 three-dimensional harmonic oscillator potential is Ĥ = h m + mω
More informationQuantization of gravity, giants and sound waves p.1/12
Quantization of gravity, giants and sound waves Gautam Mandal ISM06 December 14, 2006 Quantization of gravity, giants and sound waves p.1/12 Based on... GM 0502104 A.Dhar, GM, N.Suryanarayana 0509164 A.Dhar,
More informationSolutions for Homework 4
Solutions for Homework 4 October 6, 2006 1 Kittel 3.8 - Young s modulus and Poison ratio As shown in the figure stretching a cubic crystal in the x direction with a stress Xx causes a strain e xx = δl/l
More informationPhysics 221A Fall 1996 Notes 16 Bloch s Theorem and Band Structure in One Dimension
Physics 221A Fall 1996 Notes 16 Bloch s Theorem and Band Structure in One Dimension In these notes we examine Bloch s theorem and band structure in problems with periodic potentials, as a part of our survey
More informationLectures 21 and 22: Hydrogen Atom. 1 The Hydrogen Atom 1. 2 Hydrogen atom spectrum 4
Lectures and : Hydrogen Atom B. Zwiebach May 4, 06 Contents The Hydrogen Atom Hydrogen atom spectrum 4 The Hydrogen Atom Our goal here is to show that the two-body quantum mechanical problem of the hydrogen
More informationAmmonia molecule, from Chapter 9 of the Feynman Lectures, Vol 3. Example of a 2-state system, with a small energy difference between the symmetric
Ammonia molecule, from Chapter 9 of the Feynman Lectures, Vol 3. Eample of a -state system, with a small energy difference between the symmetric and antisymmetric combinations of states and. This energy
More information. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω. Friday April 1 ± ǁ
. α β γ δ ε ζ η θ ι κ λ μ Aμ ν(x) ξ ο π ρ ς σ τ υ φ χ ψ ω. Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Friday April 1 ± ǁ 1 Chapter 5. Photons: Covariant Theory 5.1. The classical fields 5.2. Covariant
More informationP3317 HW from Lecture 15 and Recitation 8
P3317 HW from Lecture 15 and Recitation 8 Due Oct 23, 218 Problem 1. Variational Energy of Helium Here we will estimate the ground state energy of Helium. Helium has two electrons circling around a nucleus
More informationPhysics 112 The Classical Ideal Gas
Physics 112 The Classical Ideal Gas Peter Young (Dated: February 6, 2012) We will obtain the equation of state and other properties, such as energy and entropy, of the classical ideal gas. We will start
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 informationLecture 6 Scattering theory Partial Wave Analysis. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 6 Scattering theory Partial Wave Analysis SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 The Born approximation for the differential cross section is valid if the interaction
More informationPhysics 607 Exam 2. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physics 607 Exam Please be well-organized, and show all significant steps clearly in all problems. You are graded on your work, so please do not just write down answers with no explanation! Do all your
More informationRelativity Problem Set 9 - Solutions
Relativity Problem Set 9 - Solutions Prof. J. Gerton October 3, 011 Problem 1 (10 pts.) The quantum harmonic oscillator (a) The Schroedinger equation for the ground state of the 1D QHO is ) ( m x + mω
More informationPAPER 71 COSMOLOGY. Attempt THREE questions There are seven questions in total The questions carry equal weight
MATHEMATICAL TRIPOS Part III Friday 31 May 00 9 to 1 PAPER 71 COSMOLOGY Attempt THREE questions There are seven questions in total The questions carry equal weight You may make free use of the information
More information1 r 2 sin 2 θ. This must be the case as we can see by the following argument + L2
PHYS 4 3. The momentum operator in three dimensions is p = i Therefore the momentum-squared operator is [ p 2 = 2 2 = 2 r 2 ) + r 2 r r r 2 sin θ We notice that this can be written as sin θ ) + θ θ r 2
More informationMATH 131P: PRACTICE FINAL SOLUTIONS DECEMBER 12, 2012
MATH 3P: PRACTICE FINAL SOLUTIONS DECEMBER, This is a closed ook, closed notes, no calculators/computers exam. There are 6 prolems. Write your solutions to Prolems -3 in lue ook #, and your solutions to
More information8.04 Spring 2013 April 09, 2013 Problem 1. (15 points) Mathematical Preliminaries: Facts about Unitary Operators. Uφ u = uφ u
Problem Set 7 Solutions 8.4 Spring 13 April 9, 13 Problem 1. (15 points) Mathematical Preliminaries: Facts about Unitary Operators (a) (3 points) Suppose φ u is an eigenfunction of U with eigenvalue u,
More informationPH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5
PH575 Spring 2014 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 POP QUIZ Phonon dispersion relation:
More information1 Schroenger s Equation for the Hydrogen Atom
Schroenger s Equation for the Hydrogen Atom Here is the Schroedinger equation in D in spherical polar coordinates. Note that the definitions of θ and φ are the exact reverse of what they are in mathematics.
More informationA space probe to Jupiter
Problem 3 Page 1 Problem 3 A space probe to Jupiter We consider in this problem a method frequently used to accelerate space probes in the desired direction. The space probe flies by a planet, and can
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 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 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 informationPHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101
PHY 114 A General Physics II 11 AM-1:15 PM TR Olin 101 Plan for Lecture 3 (Chapter 40-4): Some topics in Quantum Theory 1. Particle behaviors of electromagnetic waves. Wave behaviors of particles 3. Quantized
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 informationPHY 5246: Theoretical Dynamics, Fall September 28 th, 2015 Midterm Exam # 1
Name: SOLUTIONS PHY 5246: Theoretical Dynamics, Fall 2015 September 28 th, 2015 Mierm Exam # 1 Always remember to write full work for what you do. This will help your grade in case of incomplete or wrong
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 informationLecture 10 - Moment of Inertia
Lecture 10 - oment of Inertia A Puzzle... Question For any object, there are typically many ways to calculate the moment of inertia I = r 2 dm, usually by doing the integration by considering different
More informationLecture 18: 3D Review, Examples
Lecture 18: 3D Review, Examples A real (2D) quantum dot http://pages.unibas.ch/physmeso/pictures/pictures.html Lecture 18, p 1 Lect. 16: Particle in a 3D Box (3) The energy eigenstates and energy values
More informationConformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.
Lent 29 COMPLEX METHODS G. Taylor A star means optional and not necessarily harder. Conformal maps. (i) Let f(z) = az + b, with ad bc. Where in C is f conformal? cz + d (ii) Let f(z) = z +. What are the
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 one-dimensional box of width.2 nm. (a) Find the lowest
More informationSemiconductor Physics
1 Semiconductor Physics 1.1 Introduction 2 1.2 The Band Theory of Solids 2 1.3 The Kronig Penney Model 3 1.4 The Bragg Model 8 1.5 Effective Mass 8 1.6 Number of States in a Band 10 1.7 Band Filling 12
More informationPart II - Electronic Properties of Solids Lecture 12: The Electron Gas (Kittel Ch. 6) Physics 460 F 2006 Lect 12 1
Part II - Electronic Properties of Solids Lecture 12: The Electron Gas (Kittel Ch. 6) Physics 460 F 2006 Lect 12 1 Outline Overview - role of electrons in solids The starting point for understanding electrons
More informationGraduate Quantum Mechanics I: Prelims and Solutions (Fall 2015)
Graduate Quantum Mechanics I: Prelims and Solutions (Fall 015 Problem 1 (0 points Suppose A and B are two two-level systems represented by the Pauli-matrices σx A,B σ x = ( 0 1 ;σ 1 0 y = ( ( 0 i 1 0 ;σ
More information9.1 Introduction to bifurcation of equilibrium and structural
Module 9 Stability and Buckling Readings: BC Ch 14 earning Objectives Understand the basic concept of structural instability and bifurcation of equilibrium. Derive the basic buckling load of beams subject
More informationPHYSICS 219 Homework 2 Due in class, Wednesday May 3. Makeup lectures on Friday May 12 and 19, usual time. Location will be ISB 231 or 235.
PHYSICS 219 Homework 2 Due in class, Wednesday May 3 Note: Makeup lectures on Friday May 12 and 19, usual time. Location will be ISB 231 or 235. No lecture: May 8 (I m away at a meeting) and May 29 (holiday).
More informationINDIAN INSTITUTE OF TECHNOLOGY GUWAHATI Department of Physics MID SEMESTER EXAMINATION Statistical Mechanics: PH704 Solution
INDIAN INSTITUTE OF TECHNOLOGY GUWAHATI Department of Physics MID SEMESTER EXAMINATION Statistical Mechanics: PH74 Solution. There are two possible point defects in the crystal structure, Schottky and
More informationIdentical Particles. Bosons and Fermions
Identical Particles Bosons and Fermions In Quantum Mechanics there is no difference between particles and fields. The objects which we refer to as fields in classical physics (electromagnetic field, field
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 informationLecture 4. 1 de Broglie wavelength and Galilean transformations 1. 2 Phase and Group Velocities 4. 3 Choosing the wavefunction for a free particle 6
Lecture 4 B. Zwiebach February 18, 2016 Contents 1 de Broglie wavelength and Galilean transformations 1 2 Phase and Group Velocities 4 3 Choosing the wavefunction for a free particle 6 1 de Broglie wavelength
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 informationQuantum Physics I (8.04) Spring 2016 Assignment 6
Quantum Physics I (8.04) Spring 016 Assignment 6 MIT Physics Department Due Friday April 1, 016 March 17, 016 1:00 noon Reading: Griffiths section.6. For the following week sections.5 and.3. Problem Set
More informationQualifying Exam for Ph.D. Candidacy Department of Physics October 11, 2014 Part I
Qualifying Exam for Ph.D. Candidacy Department of Physics October 11, 214 Part I Instructions: The following problems are intended to probe your understanding of basic physical principles. When answering
More informationPhysics GRE: Quantum Mechanics. G. J. Loges 1. University of Rochester Dept. of Physics & Astronomy. xkcd.com/849/
Physics GRE: Quantum Mechanics G. J. Loges University of Rochester Dept. of Physics & Astronomy xkcd.com/849/ c Gregory Loges, 06 Contents Introduction. History............................................
More information5.62 Physical Chemistry II Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.62 Spring 2008 Lecture
More informationXI. INTRODUCTION TO QUANTUM MECHANICS. C. Cohen-Tannoudji et al., Quantum Mechanics I, Wiley. Outline: Electromagnetic waves and photons
XI. INTRODUCTION TO QUANTUM MECHANICS C. Cohen-Tannoudji et al., Quantum Mechanics I, Wiley. Outline: Electromagnetic waves and photons Material particles and matter waves Quantum description of a particle:
More informationDavid J. Starling Penn State Hazleton PHYS 214
Not all chemicals are bad. Without chemicals such as hydrogen and oxygen, for example, there would be no way to make water, a vital ingredient in beer. -Dave Barry David J. Starling Penn State Hazleton
More informationExact Solutions of the Einstein Equations
Notes from phz 6607, Special and General Relativity University of Florida, Fall 2004, Detweiler Exact Solutions of the Einstein Equations These notes are not a substitute in any manner for class lectures.
More information