Physics 215b: Problem Set 5
|
|
- Peter Simmons
- 6 years ago
- Views:
Transcription
1 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 ψx in the region x << 0 where V 0 as ψx < expikx + exp ikx and in the region x >> 0 again where V 0 as ψx > exp ikx + expikx. 2 The coefficients are related by the S-matrix S 2 S a Because we are looking at the steady-state, late-time behaviour, the probability density is constant in time and the continuity equation reduces to x J x Im ψ x ψ 0. 4 Since J is constant in space, we have that J < J >, which after some quick algebra translates to
2 We can rewrite the RHS using the defining S-matrix equation above to find A in Bin A in B in S S 6 Since this must be true for any choice of,, we must have S S I, 7 ie. S is unitary. A 2 2 complex matrix has 8 real degrees of freedom. We have four constraint equations from equating the matrix elements such that S S I, although only 3 of them are independent as the off-diagonal equations are conjugate to each other. The constraints are s 2 + S 2 2 s S 2 2 s s 2 + s 2 s The first two equations each give one real constraint because they only constrain the magnitude. The last equation constrains the real and imaginary parts and so gives two real constraints. We are left with 4 real degrees of freedom. b For a particle incident from the left, the transmission coefficient is given by T LR s 2 2 and for a particle incident from the right, the transmission coefficient is given by T RL s 2 2. The last constraint equation, when we take the magnitude, gives s 2 s 2 2 s 2 2 s which, substituting in the first two constraints, becomes s 2 2 s 2 2 s 2 2 s 2 2 s 2 2 s Thus T LR T RL. c If the system is time reversal invariant, then it leaves the S-matrix invariant. On the other hand, the action of time-reversal Θ on our wavefunctions is given by complex conjugation so that ψ < Θψx < A in exp ikx + A out expikx 2
3 and so that ψ > Θψx > B in expikx + B out exp ikx. 2 Ã in A out, Ã out A in, Bin B out, Bout B in. 3 Because the S-matrix is invariant, we have Ãout S 2 S 22 Ãin A in B in S 2 S 22 Conjugating this equation, and then using the S-matrix a second time, gives S S A out B out 4 5 so that S S I. Since the inverse of a matrix is unique, we conclude S S and hence that S is symmetric. Symmetry gives s 2 s 2. This is only one new constraint on the phase because unitarity already implied, as we showed above, that s 2 s 2. The most general symmetric unitary matrix then only has 3 real degrees of freedom. d If the Hamiltonian is parity invariant, then a parity transformation again leaves the S-matrix unchanged. The action of parity on the wavefunction exchanges the left and right asymptotic regions as well as leftmovers with rightmovers, so that ψ < P ψx > expikx + exp ikx 6 and ψ > P ψx < exp ikx + expikx. 7 Again we find Ãout S 2 S 22 Ãin S 2 S 22 8 We rewrite this as S 22 S 2 S 2 S 9 and hence conclude that s s 22. As the previous constraints already required these to have the same magnitude, this gives one additional constraint on the phase. The most general such unitary matrix has the form sin θe iφ i cos θe iφ 20 i cos θe iφ sin θe iφ 3
4 Problem 2 20 a An energy eigenstate must have a wavefunction of the form expikx + exp ikx, if x < 0 ψx expikx + exp ikx, if x > 0 2 as this wavefunction solves Schrodinger s equation on any open interval that does not contain the origin. It remains only to constrain the coefficients by the appropriate boundary conditions at the origin. These are. Continuity: This implies ψ0 ψ Equation of Motion: Integrating Schrodinger s equation over an arbitrarily small interval about the origin implies that [ dψx dx ] ɛ ɛ 2m 2 ɛ ɛ V xψx 2mV 0ψ The first constraint gives while the second gives iα + 24 where α mv0 k 2. While these constraints do not completely determine the coefficients, we will see that they do completely determine the S-matrix elements. We can determine the S-matrix elements most easily by an appropriate choice of in conditions. 0, then s / and s 2 /. Likewise, if 0, then s 2 / and s 22 /. So, first take 0. Then the constraints reduce to + 25 If and 2iα + 26 We rearrange to constraints to give and iα iα 27 + iα 28 This gives s iα/ + iα and s 2 / + iα. Next, take 0 and repeat to find s 22 iα/ + iα and s 2 / + iα. We conclude iα/ + iα / + iα S / + iα iα/ + iα k 2 + imv 0 imv 0 k 2 k 2 imv
5 and note that it takes precisely the form we described in Problem. b The Landauer formular gives the conductance as G e2 s 2 2 e2 + α 2 e 2 k 2 3 k m 2 V c Rather than solve for the coefficients in all three regions, it is more useful to solve for a transfer matrix T such that T so that we can determine both ingoing and outgoing data in one region from the corresponding data in another. Using the S-matrix, one finds s 2 s s 2 s 2 s s 22 s 22 In part a we calculated the s-matrix for a delta function potential at x 0. A potential at a general location x a is found simply by shifting the coordinate x y a. Under the shift we find the coefficients shift by A a,in e ika, A a,out e ika, B a,in e ika, and B a,out e ika. We absorb the shifts into the S-matrix to find S a k 2 + imv 0 imv 0 e i2ka k 2 k 2 imv 0 e 2ika Now, for two potential barriers, since the T -matrix relates one region to another, they compose by multiplication so that T tot T a T a 34 where we have shown that iα iαe 2ika T a iαe 2ika + iα 35 This gives e 4iak α + i 2 2iα sin2ak + cos2ak T tot 2iα sin2ak + cos2ak e 4iak + + iα 2 36 We now simply invert the previous process to find the S-matrix from the T-matrix: S t 2 t 22 t
6 so that S tot 2iα sin2ak + cos2ak e 4iak + + iα 2 2iα sin2ak + cos2ak 38 where α mv 0 /k 2. d The Landauer formula gives the conductance as G e2 s 2 2 e2 α 4 + 2α α 2 cos4ak + 4α sin4ak 39 where α mv 0 /k 2. The resistance do not add because of resonant behaviour when the inter-potential distance is commensurate with the wavelength of the electron. Rather than plot a few values here, check out the dynamic plot at Exciting! Problem 3 20 We have a particle of mass m, whose position is given by the operator ˆX, and who is confined to a harmonic potential of natural frequency ω 0, which interacts with a free particle of mass M whose position is given by the operator ˆR, via a potential V ˆX ˆR V 0 exp 2a 2 ˆX ˆR 2 We wish to calculate the cross section to find an outgoing free particle of energy E 2 k 2 /2M from an ingoing free particle with energy E 2 k 2 /2M 40 d 2 σ dωde Γ2π3 ρe k /M 4 where the density of states is ρe Mk/ 2 and the amplitude Γ is given by Γ f 2π f V i 2 δe f E i 42 where we sum over all possible final states of the harmonic oscillator. We denote the total energy of the final state of the particle and the oscillator to be E f E + N + 3/2ω and the total energy of the initial state E i E + 3/2ω. If you re confused where this comes from, consult the first few pages of Sakurai Chapter 6. Relative to the notation in the book, I have simply chosen the normalization L 2π. 6
7 a Let us begin to compute. Our first step is to find the matrix elements f V i 2 : Let a state with momentum k and oscillator level n have the wavefunction Then we have x, r n, k k, n V 0, k 2π 3 Doing the fourier transform and setting q k k gives k, n V 0, k V 0 2π 3 2π3/2 a 3 e a 2 2π 3/2 eikx ψ n r 43 d 3 x d 3 r e ixk k e 2a 2 x r2 ψ nrψ 0 r 44 2 q2 We then need to compute the harmonic oscillator matrix element d 3 r e irq ψ nrψ 0 r 45 n exp iq ˆR 0 46 Using the hint given to you, we rewrite the exponential as exp iq 2mω a + a exp q 2 8mω exp i 2mω qa exp i 2mω qa 47 where a a x, a y, a z. Only the first term of the annihilation operator exponential survives. The creation operator exponential gives n exp iq 2mω a 0 i where n tot n x + n y + n z. So we have that 2mω ntot qx nx nx! q ny y q nz z ny! nz! 48 k, n V 0, k 2 V 2 0 a 6 2 2π 3 e a q 2 e 4mω q2 2mω ntot q 2nx x n x! q 2ny y n y! q 2nz z n z! 49 and hence that Γ V 0 2 a 6 2 2π 2 e a q 2 e 4mω q2 n ntot qx 2nx 2mω n x! q 2ny y n y! q 2nz z n z! δ E f E i. 50 Now E i E + 3/2ω and E f E + ωn tot + 3/2 so that the delta function enforces that n tot E E ω N 5 7
8 the delta function is only satisfied when this is an integer. So we can truncate the sum to n x,n y,n z n totn q 2nx x n x! q 2ny y n y! q 2nz z n z! 52 N! q2 N 53 So at last we write the amplitude as where N is an integer. b Γ V 0 2 a 6 N 2 2π 2 N! e a q 2 e 4mω q2 q2 δ E + Nω E 54 2mω At last we put this together to find the cross section d 2 σ dωde 2πM 2 V 2 N! 4 0 a 6 N E 2 E e a q 2 e 4mω q2 q2 δ E + Nω E 55 2mω The oscillator is in its highest possible state when there is minimal energy carried by the scattered particle. Because the energy of the oscillator is quantized, this does not necessarily happen when E 0, but when the integer part. E N ω 56 8
Quantum 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 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 information8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Wednesday April Exam 2
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Wednesday April 18 2012 Exam 2 Last Name: First Name: Check Recitation Instructor Time R01 Barton Zwiebach 10:00 R02
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler March 8, 011 1 Lecture 19 1.1 Second Quantization Recall our results from simple harmonic oscillator. We know the Hamiltonian very well so no need to repeat here.
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler April, 20 Lecture 2. Scattering Theory Reviewed Remember what we have covered so far for scattering theory. We separated the Hamiltonian similar to the way in
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 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 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. Problem Set 7. Due Tuesday April 9 at 11.00AM
8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Thursday April 4 Problem Set 7 Due Tuesday April 9 at 11.00AM Assigned Reading: E&R 6 all Li. 7 1 9, 8 1 Ga. 4 all, 6
More informationSimple one-dimensional potentials
Simple one-dimensional potentials Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 Ninth lecture Outline 1 Outline 2 Energy bands in periodic potentials 3 The harmonic oscillator 4 A charged particle
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 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 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 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 informationPhysics 443, Solutions to PS 1 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
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 informationψ( ) k (r) which take the asymtotic form far away from the scattering center: k (r) = E kψ (±) φ k (r) = e ikr
Scattering Theory Consider scattering of two particles in the center of mass frame, or equivalently scattering of a single particle from a potential V (r), which becomes zero suciently fast as r. The initial
More informationPhys 622 Problems Chapter 6
1 Problem 1 Elastic scattering Phys 622 Problems Chapter 6 A heavy scatterer interacts with a fast electron with a potential V (r) = V e r/r. (a) Find the differential cross section dσ dω = f(θ) 2 in the
More informationPHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep
Problem set 1 Due Sep 15 2005 1. Let V be the set of all complex valued functions of a real variable θ, that are periodic with period 2π. That is u(θ + 2π) = u(θ), for all u V. (1) (i) Show that this V
More informationQuantum Mechanics: Particles in Potentials
Quantum Mechanics: Particles in Potentials 3 april 2010 I. Applications of the Postulates of Quantum Mechanics Now that some of the machinery of quantum mechanics has been assembled, one can begin to apply
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 informationQMI PRELIM Problem 1. All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work.
QMI PRELIM 013 All problems have the same point value. If a problem is divided in parts, each part has equal value. Show all your work. Problem 1 L = r p, p = i h ( ) (a) Show that L z = i h y x ; (cyclic
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 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 informationPhysics 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 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 informationEigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator.
PHYS208 spring 2008 Eigenmodes for coupled harmonic vibrations. Algebraic Method for Harmonic Oscillator. 07.02.2008 Adapted from the text Light - Atom Interaction PHYS261 autumn 2007 Go to list of topics
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 informationin terms of the classical frequency, ω = , puts the classical Hamiltonian in the form H = p2 2m + mω2 x 2
One of the most important problems in quantum mechanics is the simple harmonic oscillator, in part because its properties are directly applicable to field theory. The treatment in Dirac notation is particularly
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 Mechanics: Vibration and Rotation of Molecules
Quantum Mechanics: Vibration and Rotation of Molecules 8th April 2008 I. 1-Dimensional Classical Harmonic Oscillator The classical picture for motion under a harmonic potential (mass attached to spring
More informationUniversity of Illinois at Chicago Department of Physics. Quantum Mechanics Qualifying Examination. January 7, 2013 (Tuesday) 9:00 am - 12:00 noon
University of Illinois at Chicago Department of Physics Quantum Mechanics Qualifying Examination January 7, 13 Tuesday 9: am - 1: noon Full credit can be achieved from completely correct answers to 4 questions
More informationChemistry 432 Problem Set 4 Spring 2018 Solutions
Chemistry 4 Problem Set 4 Spring 18 Solutions 1. V I II III a b c A one-dimensional particle of mass m is confined to move under the influence of the potential x a V V (x) = a < x b b x c elsewhere and
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 informationQuantum Mechanics C (130C) Winter 2014 Final exam
University of California at San Diego Department of Physics Prof. John McGreevy Quantum Mechanics C (130C Winter 014 Final exam Please remember to put your name on your exam booklet. This is a closed-book
More informationP3317 HW from Lecture 7+8 and Recitation 4
P3317 HW from Lecture 7+8 and Recitation 4 Due Friday Tuesday September 25 Problem 1. In class we argued that an ammonia atom in an electric field can be modeled by a two-level system, described by a Schrodinger
More informationColumbia University Department of Physics QUALIFYING EXAMINATION
Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 10, 2018 10:00AM to 12:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of
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 informationQuiz 6: Modern Physics Solution
Quiz 6: Modern Physics Solution Name: Attempt all questions. Some universal constants: Roll no: h = Planck s constant = 6.63 10 34 Js = Reduced Planck s constant = 1.06 10 34 Js 1eV = 1.6 10 19 J d 2 TDSE
More informationAdvanced Quantum Mechanics
Advanced Quantum Mechanics Rajdeep Sensarma sensarma@theory.tifr.res.in Scattering Theory Ref : Sakurai, Modern Quantum Mechanics Taylor, Quantum Theory of Non-Relativistic Collisions Landau and Lifshitz,
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 informationSummary of Last Time Barrier Potential/Tunneling Case I: E<V 0 Describes alpha-decay (Details are in the lecture note; go over it yourself!!) Case II:
Quantum Mechanics and Atomic Physics Lecture 8: Scattering & Operators and Expectation Values http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Summary of Last Time Barrier Potential/Tunneling Case
More informationTopics Covered: Motion in a central potential, spherical harmonic oscillator, hydrogen atom, orbital electric and magnetic dipole moments
PHYS85 Quantum Mechanics I, Fall 9 HOMEWORK ASSIGNMENT Topics Covered: Motion in a central potential, spherical harmonic oscillator, hydrogen atom, orbital electric and magnetic dipole moments. [ pts]
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 informationConventions for fields and scattering amplitudes
Conventions for fields and scattering amplitudes Thomas DeGrand 1 1 Department of Physics, University of Colorado, Boulder, CO 80309 USA (Dated: September 21, 2017) Abstract This is a discussion of conventions
More informationPHYS 508 (2015-1) Final Exam January 27, Wednesday.
PHYS 508 (2015-1) Final Exam January 27, Wednesday. Q1. Scattering with identical particles The quantum statistics have some interesting consequences for the scattering of identical particles. This is
More information5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:
5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ 1 0 0 0 ξ 1 0 0 0 ξ 3, ξ
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More information1 Planck-Einstein Relation E = hν
C/CS/Phys C191 Representations and Wavefunctions 09/30/08 Fall 2008 Lecture 8 1 Planck-Einstein Relation E = hν This is the equation relating energy to frequency. It was the earliest equation of quantum
More informationLecture 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7
Lecture 4: Equations of motion and canonical quantization Read Sakurai Chapter 1.6 and 1.7 In Lecture 1 and 2, we have discussed how to represent the state of a quantum mechanical system based the superposition
More informationTime dependent perturbation theory 1 D. E. Soper 2 University of Oregon 11 May 2012
Time dependent perturbation theory D. E. Soper University of Oregon May 0 offer here some background for Chapter 5 of J. J. Sakurai, Modern Quantum Mechanics. The problem Let the hamiltonian for a system
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 informationLecture 7. More dimensions
Lecture 7 More dimensions 67 68 LECTURE 7. MORE DIMENSIONS 7.1 Introduction In this lecture we generalize the concepts introduced so far to systems that evolve in more than one spatial dimension. While
More 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 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 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 informationReview of Quantum Mechanics, cont.
Review of Quantum Mechanics, cont. 1 Probabilities In analogy to the development of a wave intensity from a wave amplitude, the probability of a particle with a wave packet amplitude, ψ, between x and
More information20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R
20 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 2. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian
More informationPage 684. Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02
Page 684 Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02 Time Transformations Section 12.5 Symmetries: Time Transformations Page 685 Time Translation
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
More informationAn Algebraic Approach to Reflectionless Potentials in One Dimension. Abstract
An Algebraic Approach to Reflectionless Potentials in One Dimension R.L. Jaffe Center for Theoretical Physics, 77 Massachusetts Ave., Cambridge, MA 02139-4307 (Dated: January 31, 2009) Abstract We develop
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 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 informationQuantum Mechanics II Lecture 11 (www.sp.phy.cam.ac.uk/~dar11/pdf) David Ritchie
Quantum Mechanics II Lecture (www.sp.phy.cam.ac.u/~dar/pdf) David Ritchie Michaelmas. So far we have found solutions to Section 4:Transitions Ĥ ψ Eψ Solutions stationary states time dependence with time
More informationPhysics 215 Quantum Mechanics 1 Assignment 5
Physics 15 Quantum Mechanics 1 Assignment 5 Logan A. Morrison February 10, 016 Problem 1 A particle of mass m is confined to a one-dimensional region 0 x a. At t 0 its normalized wave function is 8 πx
More informationThe Sommerfeld Polynomial Method: Harmonic Oscillator Example
Chemistry 460 Fall 2017 Dr. Jean M. Standard October 2, 2017 The Sommerfeld Polynomial Method: Harmonic Oscillator Example Scaling the Harmonic Oscillator Equation Recall the basic definitions of the harmonic
More informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
More informationSummary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization:
The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory:
More information1.1 A Scattering Experiment
1 Transfer Matrix In this chapter we introduce and discuss a mathematical method for the analysis of the wave propagation in one-dimensional systems. The method uses the transfer matrix and is commonly
More informationPhysics 70007, Fall 2009 Answers to Final Exam
Physics 70007, Fall 009 Answers to Final Exam December 17, 009 1. Quantum mechanical pictures a Demonstrate that if the commutation relation [A, B] ic is valid in any of the three Schrodinger, Heisenberg,
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationChapter 29. Quantum Chaos
Chapter 29 Quantum Chaos What happens to a Hamiltonian system that for classical mechanics is chaotic when we include a nonzero h? There is no problem in principle to answering this question: given a classical
More informationQualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!
Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics
More informationScattering at a quantum barrier
Scattering at a quantum barrier John Robinson Department of Physics UMIST October 5, 004 Abstract Electron scattering is widely employed to determine the structures of quantum systems, with the electron
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 information3 Quantization of the Dirac equation
3 Quantization of the Dirac equation 3.1 Identical particles As is well known, quantum mechanics implies that no measurement can be performed to distinguish particles in the same quantum state. Elementary
More informationP3317 HW from Lecture and Recitation 7
P3317 HW from Lecture 1+13 and Recitation 7 Due Oct 16, 018 Problem 1. Separation of variables Suppose we have two masses that can move in 1D. They are attached by a spring, yielding a Hamiltonian where
More informationkg meter ii) Note the dimensions of ρ τ are kg 2 velocity 2 meter = 1 sec 2 We will interpret this velocity in upcoming slides.
II. Generalizing the 1-dimensional wave equation First generalize the notation. i) "q" has meant transverse deflection of the string. Replace q Ψ, where Ψ may indicate other properties of the medium that
More informationSection 9 Variational Method. Page 492
Section 9 Variational Method Page 492 Page 493 Lecture 27: The Variational Method Date Given: 2008/12/03 Date Revised: 2008/12/03 Derivation Section 9.1 Variational Method: Derivation Page 494 Motivation
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 informationFrom Particles to Fields
From Particles to Fields Tien-Tsan Shieh Institute of Mathematics Academic Sinica July 25, 2011 Tien-Tsan Shieh (Institute of MathematicsAcademic Sinica) From Particles to Fields July 25, 2011 1 / 24 Hamiltonian
More informationPHY413 Quantum Mechanics B Duration: 2 hours 30 minutes
BSc/MSci Examination by Course Unit Thursday nd May 4 : - :3 PHY43 Quantum Mechanics B Duration: hours 3 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO
More informationElectronic Structure of Crystalline Solids
Electronic Structure of Crystalline Solids Computing the electronic structure of electrons in solid materials (insulators, conductors, semiconductors, superconductors) is in general a very difficult problem
More informationQuantum Mechanics Final Exam Solutions Fall 2015
171.303 Quantum Mechanics Final Exam Solutions Fall 015 Problem 1 (a) For convenience, let θ be a real number between 0 and π so that a sinθ and b cosθ, which is possible since a +b 1. Then the operator
More 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 informationTopological Insulator Surface States and Electrical Transport. Alexander Pearce Intro to Topological Insulators: Week 11 February 2, / 21
Topological Insulator Surface States and Electrical Transport Alexander Pearce Intro to Topological Insulators: Week 11 February 2, 2017 1 / 21 This notes are predominately based on: J.K. Asbóth, L. Oroszlány
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 informationTime-Reversal Symmetric Two-Dimensional Topological Insulators: The Bernevig-Hughes-Zhang Model
Time-Reversal Symmetric Two-Dimensional Topological Insulators: The Bernevig-Hughes-Zhang Model Alexander Pearce Intro to Topological Insulators: Week 5 November 26, 2015 1 / 22 This notes are based on
More informationTransmission across potential wells and barriers
3 Transmission across potential wells and barriers The physics of transmission and tunneling of waves and particles across different media has wide applications. In geometrical optics, certain phenomenon
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 information1 Equal-time and Time-ordered Green Functions
1 Equal-time and Time-ordered Green Functions Predictions for observables in quantum field theories are made by computing expectation values of products of field operators, which are called Green functions
More informationSolutions to exam : 1FA352 Quantum Mechanics 10 hp 1
Solutions to exam 6--6: FA35 Quantum Mechanics hp Problem (4 p): (a) Define the concept of unitary operator and show that the operator e ipa/ is unitary (p is the momentum operator in one dimension) (b)
More informationNon-stationary States and Electric Dipole Transitions
Pre-Lab Lecture II Non-stationary States and Electric Dipole Transitions You will recall that the wavefunction for any system is calculated in general from the time-dependent Schrödinger equation ĤΨ(x,t)=i
More information4.3 Lecture 18: Quantum Mechanics
CHAPTER 4. QUANTUM SYSTEMS 73 4.3 Lecture 18: Quantum Mechanics 4.3.1 Basics Now that we have mathematical tools of linear algebra we are ready to develop a framework of quantum mechanics. The framework
More informationGeneral Exam Part II, Fall 1998 Quantum Mechanics Solutions
General Exam Part II, Fall 1998 Quantum Mechanics Solutions Leo C. Stein Problem 1 Consider a particle of charge q and mass m confined to the x-y plane and subject to a harmonic oscillator potential V
More informationChapter 18: Scattering in one dimension. 1 Scattering in One Dimension Time Delay An Example... 5
Chapter 18: Scattering in one dimension B. Zwiebach April 26, 2016 Contents 1 Scattering in One Dimension 1 1.1 Time Delay.......................................... 4 1.2 An Example..........................................
More informationTopological insulator part I: Phenomena
Phys60.nb 5 Topological insulator part I: Phenomena (Part II and Part III discusses how to understand a topological insluator based band-structure theory and gauge theory) (Part IV discusses more complicated
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 informationOne-dimensional potentials: potential step
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 where a current of particles
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 information