Section 6: Measurements, Uncertainty and Spherical Symmetry Solutions
|
|
- Kelley Hoover
- 5 years ago
- Views:
Transcription
1 Physics 143a: Quantum Mechanics I Spring 015, Harvard Section 6: Measurements, Uncertainty and Spherical Symmetry Solutions Here is a summary of the most important points from the recent lectures, relevant for either solving homework problems, or for your general education. This material is covered in Chapter 3 of [?]. The nature of quantum measurement as best we know it is as follows: consider a Hermitian operator Q, and a wave function in the state Ψ = q n Ψ q n, 1 where q n are the orthonormal eigenvectors of Q. After a measurement, the wave function will collapse onto the state q n with probability q n Ψ. The uncertainty principle states that if, for a Hermitian operator A: with averages taken as A = Ψ A Ψ, then σ A A A σ A σ B 1 [A, B]. 3 An important application is Heisenberg s uncertainty principle, which says σ x σ px. 4 The Schrödinger equation for a particle of mass m in 3 spatial dimensions is i Ψ t = m Ψ + V rψ. 5 We solve this just as is 1 dimension, by solving the time-independent Schrödinger equation Ĥ Ψ = E Ψ. This time Ψ corresponds to a function of all spatial coordinates, Ψx, y, z. The normalization condition is 1 = d 3 r Ψ. 6 In the case where V is only a function of the distance to some origin: V = V x + y + z = V r, then we know that stationary states ψ are given by in spherical coordinates 1 ψ nlq = ur r Yq l θ, φ. 7 1 It is conventional to call the parameter q here m. But for now this can be confusing with the mass m, and so I will avoid mixing the two ms when possible. 1
2 Y q l are the spherical harmonics. The function ur obeys the equation d u m dr + V r + ll + 1 mr u = Eu. 8 u is defined for 0 r <. The boundary conditions on u are typically that ur = 0 = 0, so that ψ is continuous. Problem 1 BB84 Quantum Cryptography: Quantum mechanics gives us a way to send a string of bits 01001, etc., to a friend, with the absolute confidence that nobody else has read the message. The clever trick that we use is based on the disruptive nature of quantum measurement. Suppose that Alice wants to send her string of bits to her friend Bob, across a possibly insecure line of communication. However, suppose that she can send quantum bits. The basic idea is that Alice will send 4 bits, each with equal 1/4 probability, at times t = 0, 1,,...: ψ 1 = 0, ψ = 1, ψ 3 = , ψ 4 =. Now, Bob recieves this string of quantum bits. He has no idea what to expect, and so with equal probability he makes a measurement with one of the two Hermitian operators: σ z = , σ x = a What are the eigenvalues and eigenvectors of σ z and σ x? Try the quantum states above! Solution: Let s try σ z ψ 1 = = = 0 = ψ 1. So ψ 1 is an eigenvector of σ z with eigenvalue 1. Analogously, we can find ψ is an eigenvector of σ z with eigenvalue 1. Since σ z is a matrix, these are all eigenvalues. Now let s try, following the hint, ψ 3,4 with σ x : σ x ψ 3 = σ x ψ 4 = = = ψ = = ψ 4. So ψ 3 is the eigenvector of σ x with eigenvalue 1, and ψ 4 the eigenvector with eigenvalue 1. b There are 8 possibilities: Alice has sent ψ 1,,3,4 and Bob measures with σ z,x. Each is equally likely. Describe what quantum states Bob will have after the measurement, with what probability. Show that Bob does not alter the quantum state if Alice has sent ψ 1, and Bob measures with σ z, or if Alice has sent ψ 3,4 and Bob measures with σ x. Solution: Let s get four possibilities out of the way. If we make a measurement of a Hermitian operator on a quantum state which is an eigenstate, the quantum state is unchanged and we measure that eigenvalue with probability 1. So if we measure ψ 1 or ψ with σ z, we get +1 or 1 respectively; similarly, if we measure ψ 3 or ψ 4 with σ x, we get +1 or 1 respectively. But what if we measure ψ 1 with σ x? Well, we need to compute the overlap of ψ 1 with the eigenvectors of σ x, ψ 3,4 : ψ 3 ψ 1 = 0 0 = ψ 4 ψ 1 = 1.
3 After the measurement, we get with equal probability the state ψ 3,4, and a measurement of the appropriate eigenvalue. An analogous argument works for ψ. And since the measurement of ψ 3,4 with σ z must return either ψ 1,, we can take the above results and immediately conclude that we re equally likely to get either ψ 1,, each with probability 1/, in this case. The next step in the protocol is that Alice publicly sends Bob a string of zs and xs. The n th letter in the string is a z if she sent ψ 1, at time n, and an x otherwise. Bob compares Alice s string with what he measured, and he returns to her a list of all times n at which he measured with the appropriate σ z,x operator. As we showed in part b, if Bob has directly received a quantum state from Alice, then it is these states which are unaltered. Bob now sends a list of the quantum states he has measured to Alice. Now, however, suppose that at each time step, Bob does not receive a bit directly from Alice. Instead, there s a quantum eavesdropper Eve who has been listening in. For our purposes, that means that at each time step, Eve makes a measurement on Alice s quantum state before Bob does. She uses the same two operators. As Eve also won t know which random states Alice is sending, she must, like Bob, choose her operators randomly. c What is the probability that Alice has sent the state ψ i, and Bob receives the state before his measurement ψ j? Make a table of these probabilities, Pi j. You should be able to exploit results from part b to do this quickly! Solution: Eve behaves just like Bob, and is equally likely to pick to measure with σ x or σ z. From part b it is straightforward to conclude: P1 1 = P = P3 3 = P4 4 = 1 P1 = P 1 = P3 4 = P4 3 = 0 P1 3, 4 = P 3, 4 = P3 1, = P4 1, = 1 4. d What is the probability ρ that Alice has sent ψ 1, and Bob measured with σ z, or that Alice has sent ψ 3,4 and Bob measured with σ x, and Bob s measurement of the quantum state disagrees with what Alice prepared? Solution: The probability that the quantum state passes through Eve unaltered is 1/, and altered is 1/. If the state is altered, it is altered because the measurement has been taken in a different basis. Bob is thus equally likely to measure the correct or incorrect result. So ρ = 1 1 = 1 4. If Alice and Bob disagree on at least a fraction ρ of the quantum states when they compare, then Alice knows that there is an eavesdropper in the channel. Otherwise she knows that nobody has been listening, and the channel is secure. She may now securely send her message to Bob across this channel. This scheme was developed in [?]. Problem The Energy-Time Uncertainty Principle, Redux: Let Ψ0 be the initial state of some quantum system with Hamiltonian H. Suppose that at some time τ, Ψτ Ψ0 = 0. 3
4 a Show that τ obeys the following inequality, where C is an O1 number: σ H τ C, where σ H is the uncertainty in the Hamiltonian of the state Ψ0. The simplest way to proceed is as follows: consider Xt e i H t/ Ψt Ψ0, and find an upper bound for ReXt. Solution: Let us explicitly write down the abstract evolution for the wave function. For simplicity we assume that there is a discrete basis of energy eigenstates, and we get Ψt e ient/ c n n. So we find, following the suggestion above: [ [ ] Xt = e i H t c n e n ] ient/ cm m = e ien H t/ c n. Using that cos y 1 y /, we find ReXt = c n cos E n H t c n 1 1 En H t = 1 σ H t. Now if Ψτ Ψ0 = 0, then ReXτ = 0. From our above inequality, this can t happen until which gives us C =. 0 1 σ H τ = σ H τ, b A more serious calculation [?] gives C = π/. Construct a quantum state Ψ0 and a Hamiltonian H such that the inequality above is realized with this value of C. Solution: Consider a harmonic oscillator of angular frequency ω in the following state: Ψ0 = The argument generalizes to many other choices this is just one case. We know that Ψt = 1 0 e iωt/ + 1 e 3iωt/ = e iωt e iωt/. At time τ = π/ω, we have Ψτ = i 0 1, Now ω σh = 1 3 ω + Ψτ Ψ0 = i = i = 0. And we find that the energy-time uncertainty principle is satisfied: [ 1 ω + 3 ω ] = 1 ω + 9 ω ω = ω σ H τ = ω π ω = π. 4
5 Problem 3 Quantum Wires: Consider a very thin strip of a semiconductor, with a square cross section of side length a. Electrons of mass m kg are approximately constrained to move in the three dimensional potential { 0 z, y a/ V x, y, z. otherwise There will be electrons in energy eigenstates for every E < µ in the system; µ > 0. semiconductor we can estimate µ 1 ev J. a What are the eigenvectors and eigenvalues of the Hamiltonian? Solution: The eigenvalue equation is If we make the ansatz then we find the equation m x + y + z ψ = Eψ. ψx, y, z = XxY yzz, [ X m X + Y ] Y + Z XY Z = EXY Z. Z For a typical X /X is a function only of X; likewise for Y and Z. It is only possible for this equation to be satisfied if X X = E Y x, Y = E Z y, Z = E z, with E x, E y and E z constants, and E = E x + E y + E z. Now we have three 1d problems to analyze. In the x direction, there are no constraints at all, and so we know from 1d that Xx = e ikxx and E x = kx/m. In the y direction, we are constrained to the region y < a/, and this is just the particle in a box. The wave functions are Y y = a sin ny π y a, E y = n yπ a ma. An identical result holds for Zz: Zz = a sin nz π z a, E z = n zπ a ma. Thus the eigenvectors/eigenvalues are ψ kx,ny,n z = ny π a eikxx sin y a nz π sin z a, E = a a m k x + ny π a nz π +. a b For what value of a will the dynamics of all electrons be describable by the Hamiltonian of a one dimensional free particle in the potential V x = constant? Compare your answer to the typical interatomic spacing, 0. nm. Solution: We need for all E < µ eigenstates to have n z = n y = 1. We want for the lowest energy in the say n y =, n z = 1 state to be positive. This requires that π µ < π + 4 = 5 π m a a ma. 5
6 a < 5 π mµ. Plugging in for the numbers above, we estimate a > 1 nm. direction! This is about 5 atoms thick in each Wires of width smaller than this, made of typical materials, will behave as effectively one dimensional quantum systems. This width is too small to be fabricated cleanly in most labs. But recent materials such as carbon nanotubes have allowed for realistic, effectively one dimensional quantum systems to be created and used for engineering purposes. Problem 4 Binding to an Impurity in a Metal: Consider a particle of mass m in three dimensions in the potential V r = αδr R, where α, R > 0. This is a crude model for attractive interactions between a free electron and an impurity in a metal. a Find all bound states of this potential with l = 0 angular momentum, including both the stationary state and the energy. Does a bound state always exist? Solution: We write our bound state as ψr = ur/r, where d u αδr Ru = Eu. m dr Look for states with E < 0. Then defining me κ, we find that demanding normalizability: { A sinhκr r < R ur Be κr r > R. The boundary condition that u0 = 0 follows from demanding that the wave function ψ is smooth as r 0. Continuity at r = R gives us that We also require that A sinhκr = Be κr. αur = αbe κr = u R + u R = κbe κr Aκ coshκr m m mα = κ 1 + cothκr. Does this always have a solution? Well, if κr is small, we have mα = κ 1 κr + Oκ0 = κr +. R This only has a solution if α > mr. You can check by numerically plotting the right hand side that at larger κ our conclusions are unaltered. 6
7 b Compare to what happens for the δ well in one dimension. Solution: Unlike in one dimension, here there s not always a bound state. If it does exist, then the energy is given by solving the above equation for κ which can t be done exactly, and the wave function is as found above. But if the impurity is weak enough, it can t trap any electrons in three dimensions. This is a crude model that suggests the following result, which turns out to be rigorous. In one dimension, any amount of impurities at all can trap electrons in a metal and localize them, which means that it is very hard to maintain an electrical current namely, a one dimensional metal becomes an insulator for any impurity density. However, we must have a finite strength of impurities modeled by α in a three dimensional metal before we get an insulator. [1] D. J. Grififths. Introduction to Quantum Mechanics Prentice Hall, nd ed., 004 [] C. H. Bennett and G. Brassard. Quantum cryptography: Public key distribution and coin tossing, Proceedings of IEEE International Conference on Computers, Systems and Signal Processing [3] L. Vaidman. Minimum time for the evolution to an orthogonal quantum state, American Journal of Physics
Ph 219/CS 219. Exercises Due: Friday 3 November 2006
Ph 9/CS 9 Exercises Due: Friday 3 November 006. Fidelity We saw in Exercise. that the trace norm ρ ρ tr provides a useful measure of the distinguishability of the states ρ and ρ. Another useful measure
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 informationQuantum Error Correcting Codes and Quantum Cryptography. Peter Shor M.I.T. Cambridge, MA 02139
Quantum Error Correcting Codes and Quantum Cryptography Peter Shor M.I.T. Cambridge, MA 02139 1 We start out with two processes which are fundamentally quantum: superdense coding and teleportation. Superdense
More information1 1D Schrödinger equation: Particle in an infinite box
1 OF 5 NOTE: This problem set is to be handed in to my mail slot (SMITH) located in the Clarendon Laboratory by 5:00 PM (noon) Tuesday, 24 May. 1 1D Schrödinger equation: Particle in an infinite box Consider
More informationThe Schrödinger Equation
Chapter 13 The Schrödinger Equation 13.1 Where we are so far We have focused primarily on electron spin so far because it s a simple quantum system (there are only two basis states!), and yet it still
More informationQuantum Physics II (8.05) Fall 2002 Assignment 11
Quantum Physics II (8.05) Fall 00 Assignment 11 Readings Most of the reading needed for this problem set was already given on Problem Set 9. The new readings are: Phase shifts are discussed in Cohen-Tannoudji
More informationSection 11: Review. µ1 x < 0
Physics 14a: Quantum Mechanics I Section 11: Review Spring 015, Harvard Below are some sample problems to help study for the final. The practice final handed out is a better estimate for the actual length
More information1 1D Schrödinger equation: Particle in an infinite box
1 OF 5 1 1D Schrödinger equation: Particle in an infinite box Consider a particle of mass m confined to an infinite one-dimensional well of width L. The potential is given by V (x) = V 0 x L/2, V (x) =
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 informationLecture: Quantum Information
Lecture: Quantum Information Transcribed by: Crystal Noel and Da An (Chi Chi) November 10, 016 1 Final Proect Information Find an issue related to class you are interested in and either: read some papers
More informationQuantum Mechanics Solutions
Quantum Mechanics Solutions (a (i f A and B are Hermitian, since (AB = B A = BA, operator AB is Hermitian if and only if A and B commute So, we know that [A,B] = 0, which means that the Hilbert space H
More informationLecture 4: Postulates of quantum mechanics
Lecture 4: Postulates of quantum mechanics Rajat Mittal IIT Kanpur The postulates of quantum mechanics provide us the mathematical formalism over which the physical theory is developed. For people studying
More informationEntanglement and information
Ph95a lecture notes for 0/29/0 Entanglement and information Lately we ve spent a lot of time examining properties of entangled states such as ab è 2 0 a b è Ý a 0 b è. We have learned that they exhibit
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 informationProblem Set: TT Quantum Information
Problem Set: TT Quantum Information Basics of Information Theory 1. Alice can send four messages A, B, C, and D over a classical channel. She chooses A with probability 1/, B with probability 1/4 and C
More informationSecurity Implications of Quantum Technologies
Security Implications of Quantum Technologies Jim Alves-Foss Center for Secure and Dependable Software Department of Computer Science University of Idaho Moscow, ID 83844-1010 email: jimaf@cs.uidaho.edu
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 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 informationHarmonic Oscillator I
Physics 34 Lecture 7 Harmonic Oscillator I Lecture 7 Physics 34 Quantum Mechanics I Monday, February th, 008 We can manipulate operators, to a certain extent, as we would algebraic expressions. By considering
More informationIntroduction to Quantum Cryptography
Università degli Studi di Perugia September, 12th, 2011 BunnyTN 2011, Trento, Italy This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Quantum Mechanics
More informationLecture #1. Review. Postulates of quantum mechanics (1-3) Postulate 1
L1.P1 Lecture #1 Review Postulates of quantum mechanics (1-3) Postulate 1 The state of a system at any instant of time may be represented by a wave function which is continuous and differentiable. Specifically,
More 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 informationAn Introduction to Quantum Information. By Aditya Jain. Under the Guidance of Dr. Guruprasad Kar PAMU, ISI Kolkata
An Introduction to Quantum Information By Aditya Jain Under the Guidance of Dr. Guruprasad Kar PAMU, ISI Kolkata 1. Introduction Quantum information is physical information that is held in the state of
More informationOpinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6. Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- 6.6 Simple Harmonic
More informationQuantum Cryptography
Quantum Cryptography Umesh V. Vazirani CS 161/194-1 November 28, 2005 Why Quantum Cryptography? Unconditional security - Quantum computers can solve certain tasks exponentially faster; including quantum
More informationCS120, Quantum Cryptography, Fall 2016
CS10, Quantum Cryptography, Fall 016 Homework # due: 10:9AM, October 18th, 016 Ground rules: Your homework should be submitted to the marked bins that will be by Annenberg 41. Please format your solutions
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 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 informationBasics on quantum information
Basics on quantum information Mika Hirvensalo Department of Mathematics and Statistics University of Turku mikhirve@utu.fi Thessaloniki, May 2016 Mika Hirvensalo Basics on quantum information 1 of 52 Brief
More informationA Genetic Algorithm to Analyze the Security of Quantum Cryptographic Protocols
A Genetic Algorithm to Analyze the Security of Quantum Cryptographic Protocols Walter O. Krawec walter.krawec@gmail.com Iona College Computer Science Department New Rochelle, NY USA IEEE WCCI July, 2016
More informationEPR paradox, Bell inequality, etc.
EPR paradox, Bell inequality, etc. Compatible and incompatible observables AA, BB = 0, then compatible, can measure simultaneously, can diagonalize in one basis commutator, AA, BB AAAA BBBB If we project
More informationQuantum Mechanics Exercises and solutions
Quantum Mechanics Exercises and solutions P.J. Mulders Department of Physics and Astronomy, Faculty of Sciences, Vrije Universiteit Amsterdam De Boelelaan 181, 181 HV Amsterdam, the Netherlands email:
More informationQuantum Mechanics Solutions. λ i λ j v j v j v i v i.
Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =
More information1 Measurement and expectation values
C/CS/Phys 191 Measurement and expectation values, Intro to Spin 2/15/05 Spring 2005 Lecture 9 1 Measurement and expectation values Last time we discussed how useful it is to work in the basis of energy
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 informationPage 404. Lecture 22: Simple Harmonic Oscillator: Energy Basis Date Given: 2008/11/19 Date Revised: 2008/11/19
Page 404 Lecture : Simple Harmonic Oscillator: Energy Basis Date Given: 008/11/19 Date Revised: 008/11/19 Coordinate Basis Section 6. The One-Dimensional Simple Harmonic Oscillator: Coordinate Basis Page
More informationChapter 13: Photons for quantum information. Quantum only tasks. Teleportation. Superdense coding. Quantum key distribution
Chapter 13: Photons for quantum information Quantum only tasks Teleportation Superdense coding Quantum key distribution Quantum teleportation (Theory: Bennett et al. 1993; Experiments: many, by now) Teleportation
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 information+ = OTP + QKD = QC. ψ = a. OTP One-Time Pad QKD Quantum Key Distribution QC Quantum Cryptography. θ = 135 o state 1
Quantum Cryptography Quantum Cryptography Presented by: Shubhra Mittal Instructor: Dr. Stefan Robila Intranet & Internet Security (CMPT-585-) Fall 28 Montclair State University, New Jersey Introduction
More informationUncertainty Principle
Uncertainty Principle n n A Fourier transform of f is a function of frequency v Let be Δv the frequency range n It can be proved that ΔtΔv 1/(4 π) n n If Δt is small, f corresponds to a small interval
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 informationQuantum mechanics in one hour
Chapter 2 Quantum mechanics in one hour 2.1 Introduction The purpose of this chapter is to refresh your knowledge of quantum mechanics and to establish notation. Depending on your background you might
More informationBasics on quantum information
Basics on quantum information Mika Hirvensalo Department of Mathematics and Statistics University of Turku mikhirve@utu.fi Thessaloniki, May 2014 Mika Hirvensalo Basics on quantum information 1 of 49 Brief
More informationIf electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.
CHEM 2060 Lecture 18: Particle in a Box L18-1 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk
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 informationNANOSCALE SCIENCE & TECHNOLOGY
. NANOSCALE SCIENCE & TECHNOLOGY V Two-Level Quantum Systems (Qubits) Lecture notes 5 5. Qubit description Quantum bit (qubit) is an elementary unit of a quantum computer. Similar to classical computers,
More informationA New Wireless Quantum Key Distribution Protocol based on Authentication And Bases Center (AABC)
A New Wireless Quantum Key Distribution Protocol based on Authentication And Bases Center (AABC) Majid Alshammari and Khaled Elleithy Department of Computer Science and Engineering University of Bridgeport
More informationPhysible: Interactive Physics Collection MA198 Proposal Rough Draft
Physible: Interactive Physics Collection MA98 Proposal Rough Draft Brian Campbell-Deem Professor George Francis November 6 th 205 Abstract Physible conglomerates four smaller, physics-related programs
More informationLecture 10: The Schrödinger Equation. Lecture 10, p 2
Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that
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 informationPhysics 741 Graduate Quantum Mechanics 1 Solutions to Midterm Exam, Fall x i x dx i x i x x i x dx
Physics 74 Graduate Quantum Mechanics Solutions to Midterm Exam, Fall 4. [ points] Consider the wave function x Nexp x ix (a) [6] What is the correct normaliation N? The normaliation condition is. exp,
More informationLecture 10: The Schrödinger Equation. Lecture 10, p 2
Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that
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 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 informationPhysics 202 Laboratory 5. Linear Algebra 1. Laboratory 5. Physics 202 Laboratory
Physics 202 Laboratory 5 Linear Algebra Laboratory 5 Physics 202 Laboratory We close our whirlwind tour of numerical methods by advertising some elements of (numerical) linear algebra. There are three
More informationCS/Ph120 Homework 1 Solutions
CS/Ph0 Homework Solutions October, 06 Problem : State discrimination Suppose you are given two distinct states of a single qubit, ψ and ψ. a) Argue that if there is a ϕ such that ψ = e iϕ ψ then no measurement
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More informationAn Introduction to Quantum Information and Applications
An Introduction to Quantum Information and Applications Iordanis Kerenidis CNRS LIAFA-Univ Paris-Diderot Quantum information and computation Quantum information and computation How is information encoded
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 informationQuantum Communication. Serge Massar Université Libre de Bruxelles
Quantum Communication Serge Massar Université Libre de Bruxelles Plan Why Quantum Communication? Prepare and Measure schemes QKD Using Entanglement Teleportation Communication Complexity And now what?
More informationProblem 1: A 3-D Spherical Well(10 Points)
Problem : A 3-D Spherical Well( Points) For this problem, consider a particle of mass m in a three-dimensional spherical potential well, V (r), given as, V = r a/2 V = W r > a/2. with W >. All of the following
More informationCryptography and Security Final Exam
Cryptography and Security Final Exam Serge Vaudenay 29.1.2018 duration: 3h no documents allowed, except one 2-sided sheet of handwritten notes a pocket calculator is allowed communication devices are not
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 informationGenerators for Continuous Coordinate Transformations
Page 636 Lecture 37: Coordinate Transformations: Continuous Passive Coordinate Transformations Active Coordinate Transformations Date Revised: 2009/01/28 Date Given: 2009/01/26 Generators for Continuous
More informationLecture 45: The Eigenvalue Problem of L z and L 2 in Three Dimensions, ct d: Operator Method Date Revised: 2009/02/17 Date Given: 2009/02/11
Page 757 Lecture 45: The Eigenvalue Problem of L z and L 2 in Three Dimensions, ct d: Operator Method Date Revised: 2009/02/17 Date Given: 2009/02/11 The Eigenvector-Eigenvalue Problem of L z and L 2 Section
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 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 informationPhysics is becoming too difficult for physicists. David Hilbert (mathematician)
Physics is becoming too difficult for physicists. David Hilbert (mathematician) Simple Harmonic Oscillator Credit: R. Nave (HyperPhysics) Particle 2 X 2-Particle wave functions 2 Particles, each moving
More information26 Group Theory Basics
26 Group Theory Basics 1. Reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian.
More informationLecture 11 September 30, 2015
PHYS 7895: Quantum Information Theory Fall 015 Lecture 11 September 30, 015 Prof. Mark M. Wilde Scribe: Mark M. Wilde This document is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike
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 informationIntroduction to Quantum Mechanics
Introduction to Quantum Mechanics R. J. Renka Department of Computer Science & Engineering University of North Texas 03/19/2018 Postulates of Quantum Mechanics The postulates (axioms) of quantum mechanics
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Petros Wallden Lecture 3: Basic Quantum Mechanics 26th September 2016 School of Informatics, University of Edinburgh Resources 1. Quantum Computation and Quantum Information
More informationSection 10: Many Particle Quantum Mechanics Solutions
Physics 143a: Quantum Mechanics I Section 10: Many Particle Quantum Mechanics Solutions Spring 015, Harvard Here is a summary of the most important points from this week (with a few of my own tidbits),
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More informationPage 712. Lecture 42: Rotations and Orbital Angular Momentum in Two Dimensions Date Revised: 2009/02/04 Date Given: 2009/02/04
Page 71 Lecture 4: Rotations and Orbital Angular Momentum in Two Dimensions Date Revised: 009/0/04 Date Given: 009/0/04 Plan of Attack Section 14.1 Rotations and Orbital Angular Momentum: Plan of Attack
More informationCreation and Destruction Operators and Coherent States
Creation and Destruction Operators and Coherent States WKB Method for Ground State Wave Function state harmonic oscillator wave function, We first rewrite the ground < x 0 >= ( π h )1/4 exp( x2 a 2 h )
More informationE = φ 1 A The dynamics of a particle with mass m and charge q is determined by the Hamiltonian
Lecture 9 Relevant sections in text: 2.6 Charged particle in an electromagnetic field We now turn to another extremely important example of quantum dynamics. Let us describe a non-relativistic particle
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 21 Quantum Mechanics in Three Dimensions Lecture 21 Physics 342 Quantum Mechanics I Monday, March 22nd, 21 We are used to the temporal separation that gives, for example, the timeindependent
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 informationQuantum Cryptography
Quantum Cryptography (Notes for Course on Quantum Computation and Information Theory. Sec. 13) Robert B. Griffiths Version of 26 March 2003 References: Gisin = N. Gisin et al., Rev. Mod. Phys. 74, 145
More information04. Five Principles of Quantum Mechanics
04. Five Principles of Quantum Mechanics () States are represented by vectors of length. A physical system is represented by a linear vector space (the space of all its possible states). () Properties
More informationQuantum Theory of Angular Momentum and Atomic Structure
Quantum Theory of Angular Momentum and Atomic Structure VBS/MRC Angular Momentum 0 Motivation...the questions Whence the periodic table? Concepts in Materials Science I VBS/MRC Angular Momentum 1 Motivation...the
More informationA Review of Perturbation Theory
A Review of Perturbation Theory April 17, 2002 Most quantum mechanics problems are not solvable in closed form with analytical techniques. To extend our repetoire beyond just particle-in-a-box, a number
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 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 informationChapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1)
Tor Kjellsson Stockholm University Chapter 6 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: where α is a constant. H = αδ(x a/ ( a Find the first-order correction
More informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationLecture 3,4: Multiparty Computation
CS 276 Cryptography January 26/28, 2016 Lecture 3,4: Multiparty Computation Instructor: Sanjam Garg Scribe: Joseph Hui 1 Constant-Round Multiparty Computation Last time we considered the GMW protocol,
More informationQuantum Cryptography. Areas for Discussion. Quantum Cryptography. Photons. Photons. Photons. MSc Distributed Systems and Security
Areas for Discussion Joseph Spring Department of Computer Science MSc Distributed Systems and Security Introduction Photons Quantum Key Distribution Protocols BB84 A 4 state QKD Protocol B9 A state QKD
More informationTechnical Report Communicating Secret Information Without Secret Messages
Technical Report 013-605 Communicating Secret Information Without Secret Messages Naya Nagy 1, Marius Nagy 1, and Selim G. Akl 1 College of Computer Engineering and Science Prince Mohammad Bin Fahd University,
More informationQuantum Entanglement, Quantum Cryptography, Beyond Quantum Mechanics, and Why Quantum Mechanics Brad Christensen Advisor: Paul G.
Quantum Entanglement, Quantum Cryptography, Beyond Quantum Mechanics, and Why Quantum Mechanics Brad Christensen Advisor: Paul G. Kwiat Physics 403 talk: December 2, 2014 Entanglement is a feature of compound
More informationC/CS/Phys C191 Quantum Mechanics in a Nutshell 10/06/07 Fall 2009 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell 10/06/07 Fall 2009 Lecture 12 In this lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to this course. Topics
More informationUnitary evolution: this axiom governs how the state of the quantum system evolves in time.
CS 94- Introduction Axioms Bell Inequalities /7/7 Spring 7 Lecture Why Quantum Computation? Quantum computers are the only model of computation that escape the limitations on computation imposed by the
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 informationLecture 10. Central potential
Lecture 10 Central potential 89 90 LECTURE 10. CENTRAL POTENTIAL 10.1 Introduction We are now ready to study a generic class of three-dimensional physical systems. They are the systems that have a central
More informationPhysics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I
Physics 342 Lecture 26 Angular Momentum Lecture 26 Physics 342 Quantum Mechanics I Friday, April 2nd, 2010 We know how to obtain the energy of Hydrogen using the Hamiltonian operator but given a particular
More informationAQI: Advanced Quantum Information Lecture 6 (Module 2): Distinguishing Quantum States January 28, 2013
AQI: Advanced Quantum Information Lecture 6 (Module 2): Distinguishing Quantum States January 28, 2013 Lecturer: Dr. Mark Tame Introduction With the emergence of new types of information, in this case
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 informationA Refinement of Quantum Mechanics by Algorithmic Randomness and Its Application to Quantum Cryptography
Copyright c 017 The Institute of Electronics, Information and Communication Engineers SCIS 017 017 Symposium on Cryptography and Information Security Naha, Japan, Jan. 4-7, 017 The Institute of Electronics,
More information