Physics 443, Solutions to PS 1 1
|
|
- Milo Harris
- 6 years ago
- Views:
Transcription
1 Physics 443, Solutions to PS. Griffiths.9 For Φ(x, t A exp[ a( mx + it], we need that + h Φ(x, t dx. Using the known result of a Gaussian intergral + exp[ ax ]dx /a, we find that: am A h. ( The Schrödinger Equation is given by HΦ i h Φ, with H t ( h /m V (x. Plugging our Wavefunction into this Equation, we find: h m ( am + ( amx h h + V a h, a h a mx + V a h, V (x ma x. ( x + Being odd functions, x and p are zero. And using + x exp[ ax ]dx 0.5 /a 3, we have:. Griffiths.6 x h 4am, p ham, σ x x x h am, σ p p p am h, σ x σ p h. (3 d ψ dt ψ dx ( ψ t ψ + ψ ψ dx (4 t Courtesy Shaffique Adam
2 We use Schrodinger s equation and its complex conjugate to replace the time derivatives with space derivatives and the potential V (x. V (x is assumed real and independent of time. d ψ dt ψ dx ī h [ ( i h ψ h m x + V ψ ψ ī ( h h ψ m ( h ( ψ m x ψ ψ ψ dx x Integrate by parts and use the fact that normalizable wave functions are zero at infinity to drop the total derivative. Then ( d i h ( ψ ψ dt ψ dx ψ m x x ψ ψ dx x x 0 3. Griffiths.5 (a We have that Ψ(x, 0 A[ψ (x+ψ (x]. We know that the states ψ n (x sin k L nx, with k n n/l are normalized, orthogonal and real. Then L 0 L ( Ψ (x, 0Ψ(x, 0dx A ψ + ψ + ψ ψ dx 0 A ( A ] ψ x + V ψ dx (b Ψ(x, t ( ψ (xe iω t + ψ (xe iω t (5 where ω n hk n m. Then Ψ(x, t L [ ψ (x + ψ (x + ψ (xψ (x cos( ωt ] [ sin (k x + sin (k x + sin(k x sin(k x cos ωt ] where ω ω ω
3 (c (d x L 0 L x Ψ(x, t dx [ x sin (k x + x sin (k x + x sin(k x sin(k x cos ωt ] dx L 0 [ L + L + ( ] L (/L + cos ωt (3/L [ L 8 ] 9 cos ωt We got the first two terms in the integration by inspection since sin kx is symmetric about the midpoint of the well. We find the integral of the third term in a table. It is appended at the end of this solution set. The amplitude of the oscillation is p m d dt 8L 9 8L x ω sin ωt 9 (e An energy measurement would yield either E or E. The probability of measuring E is equivalent to the probablity that Ψ is in the state ψ, namely P ψ (xψ(x, tdx 4. Griffiths. H (E + E (a To normalize the wave function set Ψ(x, 0 dx where Ψ(x, 0 Ae a x 3
4 Then Ψ(x, 0 dx, ( A e ax + 0 ( A a A a 0 e ax dx (b The momentum distribution (c φ(k φ(k Ψ(x, 0e ikx dx a ( e a x e ikx dx a ( 0 e ax ikx + e ax ikx dx 0 ( a e ax ikx a ik 0 + eax ikx a ik 0 ( a a + ik + a ik a a a + k Ψ(x, t φ(ke i(kx ωkt dk a 3 a + k ei(kx ω kt dk and ω k hk m. (d In the limit where a, Ψ(x, t a e i(kx ω kt dk which looks like a δ(x at t 0 and will spread out in time due to the k dependence of ω. In the limit a 0, Ψ(x, t a3 4 k ei(kx ω kt dk
5 Small k, (long wavelenghts will dominate the distribution at t Griffiths. We have a Ψ(x, 0 exp( ax. The way to solve this problem is take the Fourier transform φ(k, since we know how to time evolve φ(k for a free particle. In particular, we have the following relations Ψ(x, t φ(k + dk φ(k exp( ikx i hk t m + dx Ψ(x, 0 exp( ikx. Putting these two equations together, one can solve for Ψ(x, t as Ψ(x, t ( a ( a 4 exp( ax exp( ikx exp(ikx exp( i hk t m 4 dydk exp( ay iky + ikx i hk t/m. This looks like a mess, but it is really just two integrals of the form given in the hint. Performing the integral over dy first, and then dk, we have Ψ(x, t ( a 4 dk ( a exp k 4a + ikx i hk t, m ( ( a 4 4a /4a + i ht/m exp ( ax exp + i hat/m + i hat/m ( a 4 x 4(/4a + i ht/m, dx dk, 5
6 Using the definition of ω answer as Ψ(x, t a/[ + ( hat/m ], we can rewrite our x ωe ω. For large t, we have that ω is inversely proportional to time, so the wavefunction is of the same form but with its rms spread proportional to t. We can also notice that when written in this form, that x, p 0. For x, we integrate directly and find /4ω. p involves integration by parts as follows p h Ψ Ψ, + h Ψ Ψ, ( ( h ω ax ax exp( ω x, i hat/m + i hat/m h 4 a ω3 x exp( ω x, h a. (6 We see that σ x σ p σ x σ p h/. h a/(ω, which when t 0, ω a, and 6. Current Vector Find the current density carried by a plane wave Ae ikx in one dimension, showing that it is in fact what one would expect from the formula ρv, and verify that it satisfies the equation of continuity. [For ψ A exp(ikx, We know that the current density is j i h ( ψ ψ ψ ψ, m x x i h m ( ika ika. (7 Therefore, j ( hk/m(a (v(ρ. The continuity equation t ρ.j is trivially satisfied.] 6
7 7. Commutators Prove the following: where a is a constant number. [Â, ˆB] [ ˆB, Â] [Â + ˆB, Ĉ] [Â, Ĉ] + [ ˆB, Ĉ] [a, Â] 0 [aâ, ˆB] a[â, ˆB] [Â ˆB, Ĉ] Â[ ˆB, Ĉ] + [Â, Ĉ] ˆB [Using the definition that [A, B] AB BA, it is straight forward to show these relations. We just do one of them here. [AB, C] ABC CAB ABC + ( ACB + ACB + ( ACB + ACB CAB 8. Sum Rule A[B, C] + (ACB ACB + [A, C]B A[B, C] + [A, C]B (8 (a Consider the commutator [H, x] [H, x] [ p + V (x, x] m m [p, x] Then (p[p, x] + [p, x]p (where we have used the result of problem 7. m i h m p ψ n [H, x] ψ n ψ n [Hx xh] ψ n i h m ψ n p ψ n ψ n (E n x xe n ψ n (E n E n ψ n x ψ n ψ n p ψ n i m h (E n E n ψ n x ψ n 7
8 (b ψn p m ψ n ψn [H, x] ψ n h m h m h m h m h m h ψn [H, x] ψ n (9 n ψ n [H, x] ψ n ψ n [H, x] ψ n (0 n (E n E n ψ n x ψ n (E n E n ψ n x ψ n n (E n E n ψ n x ψ n ψ n x ψ n n (E n E n ψ n x ψ n In going from Equation 9 to 0 we use the fact the ψ n is a complete set. Integrals x sin(ax sin(bxdx ( cos(a bx cos(a bx x sin(a bx + (a b (a + b a b x sin(a + bx a + b 8
Physics 443, Solutions to PS 2
. Griffiths.. Physics 443, Solutions to PS The raising and lowering operators are a ± mω ( iˆp + mωˆx) where ˆp and ˆx are momentum and position operators. Then ˆx mω (a + + a ) mω ˆp i (a + a ) The expectation
More informationWave Packets. Eef van Beveren Centro de Física Teórica Departamento de Física da Faculdade de Ciências e Tecnologia Universidade de Coimbra (Portugal)
Wave Packets Eef van Beveren Centro de Física Teórica Departamento de Física da Faculdade de Ciências e Tecnologia Universidade de Coimbra (Portugal) http://cft.fis.uc.pt/eef de Setembro de 009 Contents
More informationSOLUTIONS for Homework #2. 1. The given wave function can be normalized to the total probability equal to 1, ψ(x) = Ne λ x.
SOLUTIONS for Homework #. The given wave function can be normalized to the total probability equal to, ψ(x) = Ne λ x. () To get we choose dx ψ(x) = N dx e λx =, () 0 N = λ. (3) This state corresponds to
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 informationChemistry 532 Problem Set 7 Spring 2012 Solutions
Chemistry 53 Problem Set 7 Spring 01 Solutions 1. The study of the time-independent Schrödinger equation for a one-dimensional particle subject to the potential function leads to the differential equation
More informationt [Ψ (x, t)ψ(x, t)]dx = 0 Ψ (x, t) Ψ(x, t) + Ψ Ψ(x, t) (x, t) = 0 t Ψ(HΨ) dx = Ψ (HΨ)dx
.6 Reality check... The next few sections take a bodyswerve into more formal language. But they make some very important points, so its worth while to do them! One thing that might worry us is whether
More informationLinear second-order differential equations with constant coefficients and nonzero right-hand side
Linear second-order differential equations with constant coefficients and nonzero right-hand side We return to the damped, driven simple harmonic oscillator d 2 y dy + 2b dt2 dt + ω2 0y = F sin ωt We note
More informationLecture 7. 1 Wavepackets and Uncertainty 1. 2 Wavepacket Shape Changes 4. 3 Time evolution of a free wave packet 6. 1 Φ(k)e ikx dk. (1.
Lecture 7 B. Zwiebach February 8, 06 Contents Wavepackets and Uncertainty Wavepacket Shape Changes 4 3 Time evolution of a free wave packet 6 Wavepackets and Uncertainty A wavepacket is a superposition
More informationPHY 396 K. Problem set #5. Due October 9, 2008.
PHY 396 K. Problem set #5. Due October 9, 2008.. First, an exercise in bosonic commutation relations [â α, â β = 0, [â α, â β = 0, [â α, â β = δ αβ. ( (a Calculate the commutators [â αâ β, â γ, [â αâ β,
More informationPhysics 137A Quantum Mechanics Fall 2012 Midterm II - Solutions
Physics 37A Quantum Mechanics Fall 0 Midterm II - Solutions These are the solutions to the exam given to Lecture Problem [5 points] Consider a particle with mass m charge q in a simple harmonic oscillator
More information8.04 Spring 2013 March 12, 2013 Problem 1. (10 points) The Probability Current
Prolem Set 5 Solutions 8.04 Spring 03 March, 03 Prolem. (0 points) The Proaility Current We wish to prove that dp a = J(a, t) J(, t). () dt Since P a (t) is the proaility of finding the particle in the
More information1 Schrödinger s Equation
Physical Electronics Quantum Mechanics Handout April 10, 003 1 Schrödinger s Equation One-Dimensional, Time-Dependent version Schrödinger s equation originates from conservation of energy. h Ψ m x + V
More informationContinuous quantum states, Particle on a line and Uncertainty relations
Continuous quantum states, Particle on a line and Uncertainty relations So far we have considered k-level (discrete) quantum systems. Now we turn our attention to continuous quantum systems, such as a
More informationto the potential V to get V + V 0 0Ψ. Let Ψ ( x,t ) =ψ x dx 2
Physics 0 Homework # Spring 017 Due Wednesday, 4/1/17 1. Griffith s 1.8 We start with by adding V 0 to the potential V to get V + V 0. The Schrödinger equation reads: i! dψ dt =! d Ψ m dx + VΨ + V 0Ψ.
More informationPhysics 215b: Problem Set 5
Physics 25b: Problem Set 5 Prof. Matthew Fisher Solutions prepared by: James Sully April 3, 203 Please let me know if you encounter any typos in the solutions. Problem 20 Let us write the wavefunction
More informationLecture 8. 1 Uncovering momentum space 1. 2 Expectation Values of Operators 4. 3 Time dependence of expectation values 6
Lecture 8 B. Zwiebach February 29, 206 Contents Uncovering momentum space 2 Expectation Values of Operators 4 Time dependence of expectation values 6 Uncovering momentum space We now begin a series of
More informationProblem Set 3 Solutions
Chemistry 36 Dr Jean M Standard Problem Set 3 Solutions 1 Verify for the particle in a one-dimensional box by explicit integration that the wavefunction ψ x) = π x ' sin ) is normalized To verify that
More informationSuperposition of electromagnetic waves
Superposition of electromagnetic waves February 9, So far we have looked at properties of monochromatic plane waves. A more complete picture is found by looking at superpositions of many frequencies. Many
More informationUnderstand the basic principles of spectroscopy using selection rules and the energy levels. Derive Hund s Rule from the symmetrization postulate.
CHEM 5314: Advanced Physical Chemistry Overall Goals: Use quantum mechanics to understand that molecules have quantized translational, rotational, vibrational, and electronic energy levels. In a large
More information5.61 FIRST HOUR EXAM ANSWERS Fall, 2013
5.61 FIRST HOUR EXAM ANSWERS Fall, 013 I. A. Sketch ψ 5(x)ψ 5 (x) vs. x, where ψ 5 (x) is the n = 5 wavefunction of a particle in a box. Describe, in a few words, each of the essential qualitative features
More informationCHM320 EXAM #2 USEFUL INFORMATION
CHM30 EXAM # USEFUL INFORMATION Constants mass of electron: m e = 9.11 10 31 kg. Rydberg constant: R H = 109737.35 cm 1 =.1798 10 18 J. speed of light: c = 3.00 10 8 m/s Planck constant: 6.66 10 34 Js
More informationPHYSICS 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 informationbutthereisalsohyperfinesplittingwhichgivesanenergyshiftofe 1 n,hyperfine α 4.m e /m p ie a factor 10 smaller even than the Lamb shift.
13.7 lamb splitting in the above we said that 2p 1/2 and 2s 1/2 had the same energy - but actually there IS a (very small - of order α 5 ie factor 137 smaller than the fine structure) energy difference
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More informationUNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS. BSc and MPhys Undergraduate Programmes in Physics LEVEL HE2
Phys/Level /1/9/Semester, 009-10 (1 handout) UNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Undergraduate Programmes in Physics LEVEL HE PAPER 1 MATHEMATICAL,
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 informationPhysical Mathematics 2010: Problems 1 (week 2)
Physical Mathematics 2: Problems (week 2). Hellenic Calligraphy. ike it or not, Greek letters are very popular in physics. It will be easier to follow what is going on in the course if you know how they
More informationPeriodic functions: simple harmonic oscillator
Periodic functions: simple harmonic oscillator Recall the simple harmonic oscillator (e.g. mass-spring system) d 2 y dt 2 + ω2 0y = 0 Solution can be written in various ways: y(t) = Ae iω 0t y(t) = A cos
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 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 informationQM1 - Tutorial 2 Schrodinger Equation, Hamiltonian and Free Particle
QM - Tutorial Schrodinger Equation, Hamiltonian and Free Particle Yaakov Yuin November 07 Contents Hamiltonian. Denition...................................................... Example: Hamiltonian of a
More informationConnection Formula for Heine s Hypergeometric Function with q = 1
Connection Formula for Heine s Hypergeometric Function with q = 1 Ryu SASAKI Department of Physics, Shinshu University based on arxiv:1411.307[math-ph], J. Phys. A in 48 (015) 11504, with S. Odake nd Numazu
More informationPhysics 4617/5617: Quantum Physics Course Lecture Notes
Physics 4617/5617: Quantum Physics Course Lecture Notes Dr. Donald G. Luttermoser East Tennessee State University Edition 5.1 Abstract These class notes are designed for use of the instructor and students
More informationGriffiths Chapter 1. Dan Wysocki. February 12, = A exp( λu 2 ) d u. e cx2 d x = 1 = λ = A = π. λ exp( λ(x a)2 ) λ exp( λ(x a)2 ) = πa2.
Griffiths Chapter 1 Dan Wysocki February 12, 215 Problem 1. Consider the gaussian distribution ρ) = A ep λ a) 2 ), where A, a, and λ are positive real constants. a. Use Equation 1.16 to determine A. 1
More informationLecture 6. Four postulates of quantum mechanics. The eigenvalue equation. Momentum and energy operators. Dirac delta function. Expectation values
Lecture 6 Four postulates of quantum mechanics The eigenvalue equation Momentum and energy operators Dirac delta function Expectation values Objectives Learn about eigenvalue equations and operators. Learn
More informationBound and Scattering Solutions for a Delta Potential
Physics 342 Lecture 11 Bound and Scattering Solutions for a Delta Potential Lecture 11 Physics 342 Quantum Mechanics I Wednesday, February 20th, 2008 We understand that free particle solutions are meant
More informationQuantum Mechanics I - Session 9
Quantum Mechanics I - Session 9 May 5, 15 1 Infinite potential well In class, you discussed the infinite potential well, i.e. { if < x < V (x) = else (1) You found the permitted energies are discrete:
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationLecture #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box
561 Fall 017 Lecture #5 page 1 Last time: Lecture #5: Begin Quantum Mechanics: Free Particle and Particle in a 1D Box 1-D Wave equation u x = 1 u v t * u(x,t): displacements as function of x,t * nd -order:
More informationWave Mechanics Relevant sections in text: , 2.1
Wave Mechanics Relevant sections in text: 1.1 1.6, 2.1 The wave function We will now create a model for the system that we call a particle in one dimension. To do this we should define states and observables.
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 8, February 3, 2006 & L "
Chem 352/452 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 26 Christopher J. Cramer Lecture 8, February 3, 26 Solved Homework (Homework for grading is also due today) Evaluate
More informationQuantum Physics III (8.06) Spring 2006 Solution Set 4
Quantum Physics III 8.6 Spring 6 Solution Set 4 March 3, 6 1. Landau Levels: numerics 4 points When B = 1 tesla, then the energy spacing ω L is given by ω L = ceb mc = 197 1 7 evcm3 1 5 ev/cm 511 kev =
More information8.04 Quantum Physics Lecture IV. ψ(x) = dkφ (k)e ikx 2π
Last time Heisenberg uncertainty ΔxΔp x h as diffraction phenomenon Fourier decomposition ψ(x) = dkφ (k)e ikx π ipx/ h = dpφ(p)e (4-) πh φ(p) = φ (k) (4-) h Today how to calculate φ(k) interpretation of
More information1 Orders of Magnitude
Quantum Mechanics M.T. 00 J.F. Wheater Problems These problems cover all the material we will be studying in the lectures this term. The Synopsis tells you which problems are associated with which lectures.
More informationAppendix B: The Transfer Matrix Method
Y D Chong (218) PH441: Quantum Mechanics III Appendix B: The Transfer Matrix Method The transfer matrix method is a numerical method for solving the 1D Schrödinger equation, and other similar equations
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 informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationSample Quantum Chemistry Exam 1 Solutions
Chemistry 46 Fall 217 Dr Jean M Standard September 27, 217 Name SAMPE EXAM Sample Quantum Chemistry Exam 1 Solutions 1 (24 points Answer the following questions by selecting the correct answer from the
More informationSection 4: Harmonic Oscillator and Free Particles Solutions
Physics 143a: Quantum Mechanics I Section 4: Harmonic Oscillator and Free Particles Solutions Spring 015, Harvard Here is a summary of the most important points from the recent lectures, relevant for either
More informationSemi-Classical Theory of Radiative Transitions
Semi-Classical Theory of Radiative Transitions Massimo Ricotti ricotti@astro.umd.edu University of Maryland Semi-Classical Theory of Radiative Transitions p.1/13 Atomic Structure (recap) Time-dependent
More informationStatistical Interpretation
Physics 342 Lecture 15 Statistical Interpretation Lecture 15 Physics 342 Quantum Mechanics I Friday, February 29th, 2008 Quantum mechanics is a theory of probability densities given that we now have an
More informationPhysics 505 Homework No. 1 Solutions S1-1
Physics 505 Homework No s S- Some Preliminaries Assume A and B are Hermitian operators (a) Show that (AB) B A dx φ ABψ dx (A φ) Bψ dx (B (A φ)) ψ dx (B A φ) ψ End (b) Show that AB [A, B]/2+{A, B}/2 where
More informationLecture 2: simple QM problems
Reminder: http://www.star.le.ac.uk/nrt3/qm/ Lecture : simple QM problems Quantum mechanics describes physical particles as waves of probability. We shall see how this works in some simple applications,
More 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 informationFourier transforms. c n e inπx. f (x) = Write same thing in an equivalent form, using n = 1, f (x) = l π
Fourier transforms We can imagine our periodic function having periodicity taken to the limits ± In this case, the function f (x) is not necessarily periodic, but we can still use Fourier transforms (related
More informationQM1 - Tutorial 5 Scattering
QM1 - Tutorial 5 Scattering Yaakov Yudkin 3 November 017 Contents 1 Potential Barrier 1 1.1 Set Up of the Problem and Solution...................................... 1 1. How to Solve: Split Up Space..........................................
More informationWave Phenomena Physics 15c
Wave Phenomena Physics 5c Lecture Fourier Analysis (H&L Sections 3. 4) (Georgi Chapter ) What We Did Last ime Studied reflection of mechanical waves Similar to reflection of electromagnetic waves Mechanical
More informationLesson 3: Linear differential equations of the first order Solve each of the following differential equations by two methods.
Lesson 3: Linear differential equations of the first der Solve each of the following differential equations by two methods. Exercise 3.1. Solution. Method 1. It is clear that y + y = 3 e dx = e x is an
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 informationTime dependent Schrodinger equation
Lesson: Time dependent Schrodinger equation Lesson Developer: Dr. Monika Goyal, College/Department: Shyam Lal College (Day), University of Delhi Table of contents 1.1 Introduction 1. Dynamical evolution
More informationQuantum Mechanics (Draft 2010 Nov.)
Quantum Mechanics (Draft 00 Nov) For a -dimensional simple harmonic quantum oscillator, V (x) = mω x, it is more convenient to describe the dynamics by dimensionless position parameter ρ = x/a (a = h )
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 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 informationLecture 10: Solving the Time-Independent Schrödinger Equation. 1 Stationary States 1. 2 Solving for Energy Eigenstates 3
Contents Lecture 1: Solving the Time-Independent Schrödinger Equation B. Zwiebach March 14, 16 1 Stationary States 1 Solving for Energy Eigenstates 3 3 Free particle on a circle. 6 1 Stationary States
More informationMandl and Shaw reading assignments
Mandl and Shaw reading assignments Chapter 2 Lagrangian Field Theory 2.1 Relativistic notation 2.2 Classical Lagrangian field theory 2.3 Quantized Lagrangian field theory 2.4 Symmetries and conservation
More informationThe Position Representation in Quantum Mechanics
The Position Representation in Quantum Mechanics Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 13, 2007 The x-coordinate of a particle is associated with
More informationElectron in a Box. A wave packet in a square well (an electron in a box) changing with time.
Electron in a Box A wave packet in a square well (an electron in a box) changing with time. Last Time: Light Wave model: Interference pattern is in terms of wave intensity Photon model: Interference in
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
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 informationPHYS Handout 6
PHYS 060 Handout 6 Handout Contents Golden Equations for Lectures 8 to Answers to examples on Handout 5 Tricks of the Quantum trade (revision hints) Golden Equations (Lectures 8 to ) ψ Â φ ψ (x)âφ(x)dxn
More informationPhysics 220. Exam #2. May 23 May 30, 2014
Physics 0 Exam # May 3 May 30, 014 Name Please read and follow these instructions carefully: Read all problems carefully before attempting to solve them. Your work must be legible, with clear organization,
More information= k, (2) p = h λ. x o = f1/2 o a. +vt (4)
Traveling Functions, Traveling Waves, and the Uncertainty Principle R.M. Suter Department of Physics, Carnegie Mellon University Experimental observations have indicated that all quanta have a wave-like
More informationApplied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation
22.101 Applied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation References -- R. M. Eisberg, Fundamentals of Modern Physics (Wiley & Sons, New York, 1961). R. L. Liboff, Introductory
More informationNotes on wavefunctions IV: the Schrödinger equation in a potential and energy eigenstates.
Notes on wavefunctions IV: the Schrödinger equation in a potential and energy eigenstates. We have now seen that the wavefunction for a free electron changes with time according to the Schrödinger Equation
More informationPY 351 Modern Physics - Lecture notes, 3
PY 351 Modern Physics - Lecture notes, 3 Copyright by Claudio Rebbi, Boston University, October 2016. These notes cannot be duplicated and distributed without explicit permission of the author. Time dependence
More informationPHY293 Lecture #18. h2 2 [c 1 Ψ 1 + c 2 Ψ 2 ] = 0. + c 2
PHY93 Lecture #18 December 5, 017 1. Linear Superposition of Solutions to the Schrodinger Equation Yesterday saw solutions to the TDSE, that have definite energy and momentum These are specific, or special
More informationQ U A N T U M M E C H A N I C S : L E C T U R E 5
Q U A N T U M M E C H A N I C S : L E C T U R E 5 salwa al saleh Abstract This lecture discusses the formal solution of Schrödinger s equation for a free particle. Including the separation of variables
More informationChemistry 3502/4502. Exam III. March 28, ) Circle the correct answer on multiple-choice problems.
A Chemistry 352/452 Exam III March 28, 25 1) Circle the correct answer on multiple-choice problems. 2) There is one correct answer to every multiple-choice problem. There is no partial credit. On the short-answer
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 informationWave Phenomena Physics 15c. Lecture 11 Dispersion
Wave Phenomena Physics 15c Lecture 11 Dispersion What We Did Last Time Defined Fourier transform f (t) = F(ω)e iωt dω F(ω) = 1 2π f(t) and F(w) represent a function in time and frequency domains Analyzed
More informationSample Quantum Chemistry Exam 2 Solutions
Chemistry 46 Fall 7 Dr. Jean M. Standard Name SAMPE EXAM Sample Quantum Chemistry Exam Solutions.) ( points) Answer the following questions by selecting the correct answer from the choices provided. a.)
More informationWAVE PACKETS & SUPERPOSITION
1 WAVE PACKETS & SUPERPOSITION Reading: Main 9. PH41 Fourier notes 1 x k Non dispersive wave equation x ψ (x,t) = 1 v t ψ (x,t) So far, we know that this equation results from application of Newton s law
More informationCh 125a Problem Set 1
Ch 5a Problem Set Due Monday, Oct 5, 05, am Problem : Bra-ket notation (Dirac notation) Bra-ket notation is a standard and convenient way to describe quantum state vectors For example, φ is an abstract
More informationElectromagnetism HW 1 math review
Electromagnetism HW math review Problems -5 due Mon 7th Sep, 6- due Mon 4th Sep Exercise. The Levi-Civita symbol, ɛ ijk, also known as the completely antisymmetric rank-3 tensor, has the following properties:
More informationx(t) = R cos (ω 0 t + θ) + x s (t)
Formula Sheet Final Exam Springs and masses: dt x(t + b d x(t + kx(t = F (t dt More general differential equation with harmonic driving force: m d Steady state solutions: where d dt x(t + Γ d dt x(t +
More informationBME I5000: Biomedical Imaging
1 BME I5000: Biomedical Imaging Lecture FT Fourier Transform Review Lucas C. Parra, parra@ccny.cuny.edu Blackboard: http://cityonline.ccny.cuny.edu/ Fourier Transform (1D) The Fourier Transform (FT) is
More informationAbout solving time dependent Schrodinger equation
About solving time dependent Schrodinger equation (Griffiths Chapter 2 Time Independent Schrodinger Equation) Given the time dependent Schrodinger Equation: Ψ Ψ Ψ 2 1. Observe that Schrodinger time dependent
More informationHilbert Space Problems
Hilbert Space Problems Prescribed books for problems. ) Hilbert Spaces, Wavelets, Generalized Functions and Modern Quantum Mechanics by Willi-Hans Steeb Kluwer Academic Publishers, 998 ISBN -7923-523-9
More informationFourier Series and Integrals
Fourier Series and Integrals Fourier Series et f(x) beapiece-wiselinearfunctionon[, ] (Thismeansthatf(x) maypossessa finite number of finite discontinuities on the interval). Then f(x) canbeexpandedina
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 informationPh2b Quiz - 2. Instructions
Ph2b Quiz - 2 Instructions 1. Your solutions are due by Monday, February 26th, 2018 at 4pm in the quiz box outside 201 E. Bridge. 2. Late quizzes will not be accepted, except in very special circumstances.
More informationThe Dirac δ-function
The Dirac δ-function Elias Kiritsis Contents 1 Definition 2 2 δ as a limit of functions 3 3 Relation to plane waves 5 4 Fourier integrals 8 5 Fourier series on the half-line 9 6 Gaussian integrals 11 Bibliography
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 13, 2016 3:10PM to 5:10PM Modern Physics Section 4. Relativity and Applied Quantum Mechanics Two hours are permitted
More informationThe One-Dimensional Finite-Difference Time-Domain (FDTD) Algorithm Applied to the Schrödinger Equation
The One-Dimensional Finite-Difference Time-Domain (FDTD) Algorithm Applied to the Schrödinger Equation James. Nagel University of Utah james.nagel@utah.edu 1. NTODUCTON For anyone who has ever studied
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 informationmultiply both sides of eq. by a and projection overlap
Fourier Series n x n x f xa ancos bncos n n periodic with period x consider n, sin x x x March. 3, 7 Any function with period can be represented with a Fourier series Examples (sawtooth) (square wave)
More informationPHY 396 K. Solutions for problems 1 and 2 of set #5.
PHY 396 K. Solutions for problems 1 and of set #5. Problem 1a: The conjugacy relations  k,  k,, Ê k, Ê k, follow from hermiticity of the Âx and Êx quantum fields and from the third eq. 6 for the polarization
More informationTP computing lab - Integrating the 1D stationary Schrödinger eq
TP computing lab - Integrating the 1D stationary Schrödinger equation September 21, 2010 The stationary 1D Schrödinger equation The time-independent (stationary) Schrödinger equation is given by Eψ(x)
More informationReview of Fundamental Equations Supplementary notes on Section 1.2 and 1.3
Review of Fundamental Equations Supplementary notes on Section. and.3 Introduction of the velocity potential: irrotational motion: ω = u = identity in the vector analysis: ϕ u = ϕ Basic conservation principles:
More informationLECTURE 4 WAVE PACKETS
LECTURE 4 WAVE PACKETS. Comparison between QM and Classical Electrons Classical physics (particle) Quantum mechanics (wave) electron is a point particle electron is wavelike motion described by * * F ma
More information