# Likewise, any operator, including the most generic Hamiltonian, can be written in this basis as H11 H

Save this PDF as:

Size: px
Start display at page:

Download "Likewise, any operator, including the most generic Hamiltonian, can be written in this basis as H11 H"

## Transcription

1 Finite Dimensional systems/ilbert space Finite dimensional systems form an important sub-class of degrees of freedom in the physical world To begin with, they describe angular momenta with fixed modulus and particle spins More generally, any set of closely spaced levels in the quantum system weakly connected to the rest of the spectrum can be dealt with approximately as a finite-dimensional system Any pair of levels in the spectrum addressed by near-resonant radiation also forms an effective two level system Neutrino flavors, isospins, tunneling states in the double-well potential, etc etc are all example of exact or well isolated finite-dimensional systems For simplicity, I will focus here on the two-level system (the larger the dimension of space the more complex is the general solution but most principles remain essentially the same I would rank it as number two on the list of most important QM problems after harmonic oscillator It does not really matter whether we are thinking of the spin-/, tunneling states in the double well potential, or isospin The formal description is identical for all of them, though the language used may depend on the context To make the point I will try to deliberately avoid the magneticfield/spin-/ projection which is the most common one in textbooks So, assume that we are dealing with the two-dimensional ilbert space characterized by two basis states e and e of whatever origin, eg they may be eigenstates of some specific amiltonian Then any state in this basis can be represented as ( α ψ = = α e β + β e Likewise, any operator, including the most generic amiltonian, can be written in this basis as ( Before we start solving for eigenstates and eigenvalues of this amiltonian, let us simplify it a little bit and introduce better notations Since Ĥ is ermitian, we have = I will call this non-diagonal (in a given basis matrix element a mixing matrix element and denote it as e iϕ If = 0 then our basis and amiltonian eigenstates coincide because ( 0 0 ( 0 0 ( 0 ( 0 ( = = E 0 e ( 0 = = E e E ( E E / * E It make perfect sense to count energy from the middle point (E + E /, so in the rest I will consider it as energy zero This results in the following generic amiltonian ( ξ e iϕ e iϕ ( ξ

2 where ξ = (E E / is often called an energy bias The dynamics of the system is determined from conventional i ( ( t α β ψ = Ĥ ψ i ξ = e iϕ e iϕ ξ ( α β and the best way to solve it to find the eigenvalue basis of Ĥ from ( ( ξ E e iϕ α e iϕ = 0 ( ξ E β The eigenvalue Equation is found from the zero determinant condition ξ E e iϕ e iϕ ξ E = 0 E ξ = 0 (3 and produces two eigenenergies Substituting this back to Eq ( we find the eigenvectors α = E ± = ± ξ + ±ɛ (4 ɛ + ξ eiϕ β, α + = ɛ ξ eiϕ β + (5 To complete the job we have to normalize them using α + β = This is best obtained through the parameterization (which satisfies the normalization condition by construction ( ( ( ( α sin θ e iϕ α+ cos θ e iϕ = =, + = =, (6 cos θ sin θ β with the mixing angle θ = tan, with cos θ = ξ + ɛ Now time evolution of an arbitrary state is a piece of cake: β + + ξ ɛ, sin θ = ξ ɛ (7 ψ(t = + ψ(t = 0 e iɛt + + ψ(t = 0 e +iɛt, all one has to do is to project an initial state on eigenvectors and write down the sum for the final answer Let us do it for two initial states: ψ(t = 0 = e and ψ(t = 0 = e In the first case we find ψ(t = α + e iɛt ( α+ β + + α e iɛt ( α β = e iɛt ( α+ α +β + + e iɛt ( α α β This defines the probability amplitude of finding the system in state at time t if it was in state at t = 0: U = e iɛt α + + e iɛt α = e iɛt cos θ + e iɛt sin θ = cos(ɛt i ξ ɛ sin(ɛt The probability itself is P = cos (ɛt + ξ ɛ sin (ɛt ɛ sin (ɛt

3 Likewise, for ψ(t = 0 = e we find and U = e iɛt β + + e iɛt β = e iɛt sin θ + e iɛt cos θ = cos(ɛt + i ξ ɛ sin(ɛt P == cos (ɛt + ξ ɛ sin (ɛt ɛ sin (ɛt The probabilities P and P are complementary to P and P, respectively, because j=, P ji = for i =, We see that oscillations have the largest amplitude of unity and the smallest frequency [sin (-squared oscillates twice as fast as sin(] for the symmetric two-level system when ξ = 0 As ξ is increased, the amplitude of oscillations decreases while the frequency increases as ɛ, ie the particle probes the other state at a much faster rate but only a tiny bit P ( t 0 ( P ( t ( / t / t It is time for homework and physical examples Problem 3 Two level system thermodynamics Use amplitudes U ii to compute the partition function of the two-level system from Z = Tr e β i e βĥ i, (recall the relation between the statistics and evolution in imaginary time discussed in the pathintegralpdf part of notes Check that your answer agrees with the one obtained from the sum of Gibbs exponentials based on known spectrum i=, Problem 33 Rabi oscillations Consider a two level system with the following time-dependent amiltonian: ( ( ĤT LS + ˆV ξ 0 0 δ e iωt rad = δ e iωt, 0 where the first term represent the two level system with the energy splitting ξ, and the second one describes coupling to the radiation field which induces transitions between the levels and is oscillating with frequency ω 3

4 a Consider first the case of exact resonance ω = ξ, and determine the probability of return P [int: use the so-called rotating frame basis which formally accounts for seeking the solution in the form ( α(te iωt ψ(t = β(t You will notice that for α(t and β(t the problem is reduced to the time-independent two-level system described in the notes] b Your solution in part a is often referred to as resonant Rabi oscillations More generally, Rabi oscillations are present in any two-level system selected by the near resonant radiation, so consider now the case ξ ω and assume that δ/ ξ ω is arbitrary Use the same approach to the solution and formulate an effective two-level amiltonian for the system in rotated frame, ie acting on α(t and β(t coefficients so that i d ( ( α α = dt β eff β c What is the frequency and amplitude of Rabi oscillations in the general near-resonant case discussed in part b? The general approach to an arbitrary n-dimensional system is along the same lines The amiltonian operator is represented by the n n matrix with ij = ji in some basis To solve it one has to diagonalize this matrix, ie to find its eigenvalues and eigenvectors, Ĥ ψ k = E k ψ k The dynamics immediately follows from χ(t = n k= ψ k χ(t = 0 e iekt ψ k Diagonalizing large matrix quickly becomes an impossible task analytically, except in special cases Numerically, nowadays one can diagonalize generic matrices for n as large as n For the lowest energy eigenstates more refined techniques are available which can handle ilbert spaces with up to several billion levels! Problem 34 Three level system TLS Consider the following amiltonian for the three level system (representing particle in the threewell potential with weak tunneling between the nearest-neighbor wells and the middle well having its energy shifted upwards 0 0 U, 0 0 a Diagonalize this amiltonian, ie find the spectrum of eigenenergies for this system Do not solve for eigenfunctions b Consider the lowest two states in the limit of U (keep terms proportional to /U only Given that the original model is symmetric with respect to left and right wells, determine an effective two-level amiltonian which has exactly the same lowest energy eigenvalues ( h h Ĥ eff = h h 4

5 It is representing an effective model for tunneling between left and right wells through a virtual intermediate state You may not be able to determine the sign of h Take it as negative for now The ammonia molecule Ammonia has the chemical form N 3 If we consider a simple picture with the molecule spinning about its symmetry axis, as shown in the figure (Note: N 3 has a tetrahedral structure with the hydrogen atoms forming a plane and the nitrogen being either above or below then there are two possible locations for nitrogen atom relative to the direction of spin, as shown (if the molecule was not spinning, then the two positions would be indistinguishable up to rotating the molecule as a rigid body upside down z N Center of mass N Of course from symmetry the energies of these two configurations should be the same ie taking these as basis states we have = = E 0 As usual we will consider E 0 as an energy zero In quantum mechanics particles have exponentially small but non-zero probability amplitudes to tunnel through the potential barrier and thus covert state e to state e We can represent this in terms of = = Thus our amiltonian becomes ( 0 Ĥ eff = 0 which has two eigenstates with energies and eigenfunctions ( E =, = E + = +, + = ( That it, the eigenstates are the simple even/odd mixtures of basis states and The tunneling (or mixing matrix element is generically negative when tunneling occurs between the lowest energy states in the potential The level splitting E + E = can be readily seen my the microwave 5

6 spectroscopy by radiating the molecule (see the Rabi oscillations problem Next, one can apply an electric field to couple E ( E to the molecules dipole moment µ oriented along the molecule s ẑ-axis having opposite directions in states and If the field is applied along the ẑ-axis then it adds a bias to our two-level system ξ = µe The resulting spectrum as a function of electric field strength is shown in the figure In E the absence of tunneling one would have an exact crossing of levels in zero field Mixing between the levels leads to the so-called avoided crossing and ( E opens a gap between the ground and first excited states One can easily see that in the strong-field limit µe the eigenstates are very well represented by the original and states Indeed, the mixing angle in this limit is small θ / µe At the avoided crossing point the eigenstates are equal superpositions of the original states, and as the electric field is varied, the nature of the ground state continuously changes from to An interesting problem (not discussed in this course is to look at transition probabilities between the states when the amiltonian parameter ξ is changed at a finite rate from large negative to large positive values the so-called Majorana, or Landau-Zener, transition (the attachment of names depends on who set the problem first, or solved it semiclassically, or exactly K 0 K0 and neutrino oscillations There are many other examples which are accurately described by the two-level, or n-level, system One is the K 0 K0 system ere the relevant degrees of freedom are particle/antiparticle pairs, one having positive strangeness quantum number S = +, and the other having negative strangeness S = (we are not really concerned here what this word means : Because we are talking about particle/antiparticle pair the two have identical masses Due to weak interaction these prime particle states are actually mixed, ie they can convert one into another This leads to the description of the K 0 K0 system in the form ( m + M δ δ m + M In the real world then a state which starts as a K 0 becomes a K 0 later on then back K 0, etc One slight complication is that these states are unstable on a longer timescale and decay into pions (the decay is dominated by the two-pion decay of the symmetric superposition state This results in damped oscillations but they reamain clearly visible, as in the figure below, which shows the 6

7 probability for the particle which starts out at time t = 0 as K 0 to be detected at time t as K 0 P 0 0 K K P e t 0 (0 sec 3 t 0 (0 sec Neutrino oscillations are similar One of the mysteries was that the flux of electron-neutrinos coming from the sun is about a factor of three smaller then expected from very robust calculations, according to both the omestake gold mine experiment in South Dakota and to the Kamiokande experiment in Japan An explanation for this phenomenon is that neutrinos have a (small mass and that there are mixing terms between neutrinos of different kind Taking the basis states as electron-, ν e, and muon-neutrinos, ν µ, the amiltonian then reads It leads to the superposition eigenstates ( m δm δm m ν + = cos θ ν e + sin θ ν µ, ν = sin θ ν e + cos θ ν µ Now the low energy solar neutrinos can be detected in the state ν e since they can create an electron via the weak interaction, and detectors were designed for this purpose owever, the state ν µ is sterile since the energy is not large enough to allow a muon to be created [Note: m µ 0 m e ] Thus the explanation for the solar neutrino deficit is that most of them convert to ν µ state by the time they arrive to Earth In the figure you can see a model plot for conversion from ν e to ν µ state There many more examples from atomic physics, cavity QED, amorphous solids, magnetism, quantum information (qubits you name it The origin of two states is, of course, important in defining the matrix elements in the effective amiltonian, but the rest of the description is absolutely universal I will do spins-/ in magnetic field later in the course, but you probably will be bored 7

### Time 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 information

### Physics 505 Homework No. 8 Solutions S Spinor rotations. Somewhat based on a problem in Schwabl.

Physics 505 Homework No 8 s S8- Spinor rotations Somewhat based on a problem in Schwabl a) Suppose n is a unit vector We are interested in n σ Show that n σ) = I where I is the identity matrix We will

More information

### Physics 342 Lecture 27. Spin. Lecture 27. Physics 342 Quantum Mechanics I

Physics 342 Lecture 27 Spin Lecture 27 Physics 342 Quantum Mechanics I Monday, April 5th, 2010 There is an intrinsic characteristic of point particles that has an analogue in but no direct derivation from

More information

### Physics 4022 Notes on Density Matrices

Physics 40 Notes on Density Matrices Definition: For a system in a definite normalized state ψ > the density matrix ρ is ρ = ψ >< ψ 1) From Eq 1 it is obvious that in the basis defined by ψ > and other

More information

### 1 Recall what is Spin

C/CS/Phys C191 Spin measurement, initialization, manipulation by precession10/07/08 Fall 2008 Lecture 10 1 Recall what is Spin Elementary particles and composite particles carry an intrinsic angular momentum

More information

### Radiating Dipoles in Quantum Mechanics

Radiating Dipoles in Quantum Mechanics Chapter 14 P. J. Grandinetti Chem. 4300 Oct 27, 2017 P. J. Grandinetti (Chem. 4300) Radiating Dipoles in Quantum Mechanics Oct 27, 2017 1 / 26 P. J. Grandinetti (Chem.

More information

### conventions and notation

Ph95a lecture notes, //0 The Bloch Equations A quick review of spin- conventions and notation The quantum state of a spin- particle is represented by a vector in a two-dimensional complex Hilbert space

More information

### Principles of Molecular Spectroscopy

Principles of Molecular Spectroscopy What variables do we need to characterize a molecule? Nuclear and electronic configurations: What is the structure of the molecule? What are the bond lengths? How strong

More information

### 16.1. 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 information

### Mesoscopic field state superpositions in Cavity QED: present status and perspectives

Mesoscopic field state superpositions in Cavity QED: present status and perspectives Serge Haroche, Ein Bokek, February 21 st 2005 Entangling single atoms with larger and larger fields: an exploration

More information

### ECE 487 Lecture 5 : Foundations of Quantum Mechanics IV Class Outline:

ECE 487 Lecture 5 : Foundations of Quantum Mechanics IV Class Outline: Linearly Varying Potential Triangular Potential Well Time-Dependent Schrödinger Equation Things you should know when you leave Key

More information

### Lecture 4 Quantum mechanics in more than one-dimension

Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts

More information

### Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Wednesday, January 14, 015 1:00PM to 3:00PM Modern Physics Section 3. Quantum Mechanics Two hours are permitted for the completion of this

More information

### Two-level systems coupled to oscillators

Two-level systems coupled to oscillators RLE Group Energy Production and Conversion Group Project Staff Peter L. Hagelstein and Irfan Chaudhary Introduction Basic physical mechanisms that are complicated

More information

### Rotations and vibrations of polyatomic molecules

Rotations and vibrations of polyatomic molecules When the potential energy surface V( R 1, R 2,..., R N ) is known we can compute the energy levels of the molecule. These levels can be an effect of: Rotation

More information

### Problem 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 information

### Quantum 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 information

### Physics 129, Fall 2010; Prof. D. Budker

Physics 129, Fall 2010; Prof. D. Budker Some introductory thoughts Reductionists science Identical particles are truly so (bosons, fermions) We will be using (relativistic) QM where initial conditions

More information

### Lecture 3 (Part 1) Physics 4213/5213

September 8, 2000 1 FUNDAMENTAL QED FEYNMAN DIAGRAM Lecture 3 (Part 1) Physics 4213/5213 1 Fundamental QED Feynman Diagram The most fundamental process in QED, is give by the definition of how the field

More information

### Parity violation. no left-handed ν\$ are produced

Parity violation Wu experiment: b decay of polarized nuclei of Cobalt: Co (spin 5) decays to Ni (spin 4), electron and anti-neutrino (spin ½) Parity changes the helicity (H). Ø P-conservation assumes a

More information

### Assignment 2 Solutions. 1. The general state of a spin half particle with spin component S n = S ˆn = 1 2 can be shown to be given by

PHYSICS 301 QUANTUM PHYSICS I (007) Assignment Solutions 1. The general state of a spin half particle with spin component S n = S ˆn = 1 can be shown to be given by S n = 1 = cos( 1 θ) S z = 1 + eiφ sin(

More information

### Angular momentum and spin

Luleå tekniska universitet Avdelningen för Fysik, 007 Hans Weber Angular momentum and spin Angular momentum is a measure of how much rotation there is in particle or in a rigid body. In quantum mechanics

More information

### Damped Harmonic Oscillator

Damped Harmonic Oscillator Wednesday, 23 October 213 A simple harmonic oscillator subject to linear damping may oscillate with exponential decay, or it may decay biexponentially without oscillating, or

More information

### i = cos 2 0i + ei sin 2 1i

Chapter 10 Spin 10.1 Spin 1 as a Qubit In this chapter we will explore quantum spin, which exhibits behavior that is intrinsically quantum mechanical. For our purposes the most important particles are

More information

### Intermission: Let s review the essentials of the Helium Atom

PHYS3022 Applied Quantum Mechanics Problem Set 4 Due Date: 6 March 2018 (Tuesday) T+2 = 8 March 2018 All problem sets should be handed in not later than 5pm on the due date. Drop your assignments in the

More information

### Optical Lattices. Chapter Polarization

Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose- Hubbard model used to describe ultracold atoms in optical lattices. We show how the AC-Stark

More information

### Electronic structure theory: Fundamentals to frontiers. 1. Hartree-Fock theory

Electronic structure theory: Fundamentals to frontiers. 1. Hartree-Fock theory MARTIN HEAD-GORDON, Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley National

More information

### E = 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 information

### The Stern-Gerlach experiment and spin

The Stern-Gerlach experiment and spin Experiments in the early 1920s discovered a new aspect of nature, and at the same time found the simplest quantum system in existence. In the Stern-Gerlach experiment,

More information

### Atomic cross sections

Chapter 12 Atomic cross sections The probability that an absorber (atom of a given species in a given excitation state and ionziation level) will interact with an incident photon of wavelength λ is quantified

More information

### The 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 information

### Landau s Fermi Liquid Theory

Thors Hans Hansson Stockholm University Outline 1 Fermi Liquids Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids How? 2 Landau s Phenomenological Approach The free Fermi gas

More information

### The Spin (continued). February 8, 2012

The Spin continued. Magnetic moment of an electron Particle wave functions including spin Stern-Gerlach experiment February 8, 2012 1 Magnetic moment of an electron. The coordinates of a particle include

More information

### Complex Numbers. The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition,

Complex Numbers Complex Algebra The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition, and (ii) complex multiplication, (x 1, y 1 )

More information

### Chapter 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 information

### Complex Numbers. The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition,

Complex Numbers Complex Algebra The set of complex numbers can be defined as the set of pairs of real numbers, {(x, y)}, with two operations: (i) addition, and (ii) complex multiplication, (x 1, y 1 )

More information

### Coherent states, beam splitters and photons

Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.

More information

### 26 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 information

### 22.02 Intro to Applied Nuclear Physics

22.02 Intro to Applied Nuclear Physics Mid-Term Exam Solution Problem 1: Short Questions 24 points These short questions require only short answers (but even for yes/no questions give a brief explanation)

More information

### Adiabatic quantum computation a tutorial for computer scientists

Adiabatic quantum computation a tutorial for computer scientists Itay Hen Dept. of Physics, UCSC Advanced Machine Learning class UCSC June 6 th 2012 Outline introduction I: what is a quantum computer?

More information

### Jacopo Ferretti Sapienza Università di Roma

Jacopo Ferretti Sapienza Università di Roma NUCLEAR RESONANCES: FROM PHOTOPRODUCTION TO HIGH PHOTON VIRTUALITIES ECT*, TRENTO (ITALY), -6 OCTOBER 05 Three quark QM vs qd Model A relativistic Interacting

More information

### Total Angular Momentum for Hydrogen

Physics 4 Lecture 7 Total Angular Momentum for Hydrogen Lecture 7 Physics 4 Quantum Mechanics I Friday, April th, 008 We have the Hydrogen Hamiltonian for central potential φ(r), we can write: H r = p

More information

### PHYS Quantum Mechanics I - Fall 2011 Problem Set 7 Solutions

PHYS 657 - Fall PHYS 657 - Quantum Mechanics I - Fall Problem Set 7 Solutions Joe P Chen / joepchen@gmailcom For our reference, here are some useful identities invoked frequentl on this problem set: J

More information

### PHYSICAL SCIENCES PART A

PHYSICAL SCIENCES PART A 1. The calculation of the probability of excitation of an atom originally in the ground state to an excited state, involves the contour integral iωt τ e dt ( t τ ) + Evaluate the

More information

### There is light at the end of the tunnel. -- proverb. The light at the end of the tunnel is just the light of an oncoming train. --R.

A vast time bubble has been projected into the future to the precise moment of the end of the universe. This is, of course, impossible. --D. Adams, The Hitchhiker s Guide to the Galaxy There is light at

More information

### Neutrino Oscillations and the Matter Effect

Master of Science Examination Neutrino Oscillations and the Matter Effect RAJARSHI DAS Committee Walter Toki, Robert Wilson, Carmen Menoni Overview Introduction to Neutrinos Two Generation Mixing and Oscillation

More information

### Paradigms in Physics: Quantum Mechanics

Paradigms in Physics: Quantum Mechanics David H. McIntyre Corinne A. Manogue Janet Tate Oregon State University 23 November 2010 Copyright 2010 by David H. McIntyre, Corinne A. Manogue, Janet Tate CONTENTS

More information

### Electron spins in nonmagnetic semiconductors

Electron spins in nonmagnetic semiconductors Yuichiro K. Kato Institute of Engineering Innovation, The University of Tokyo Physics of non-interacting spins Optical spin injection and detection Spin manipulation

More information

### Neutrinos From The Sky and Through the Earth

Neutrinos From The Sky and Through the Earth Kate Scholberg, Duke University DNP Meeting, October 2016 Neutrino Oscillation Nobel Prize! The fourth Nobel for neutrinos: 1988: neutrino flavor 1995: discovery

More information

### Physics 161 Homework 2 - Solutions Wednesday August 31, 2011

Physics 161 Homework 2 - s Wednesday August 31, 2011 Make sure your name is on every page, and please box your final answer. Because we will be giving partial credit, be sure to attempt all the problems,

More information

### The 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 information

### Electric properties of molecules

Electric properties of molecules For a molecule in a uniform electric fielde the Hamiltonian has the form: Ĥ(E) = Ĥ + E ˆµ x where we assume that the field is directed along the x axis and ˆµ x is the

More information

### 9 Atomic Coherence in Three-Level Atoms

9 Atomic Coherence in Three-Level Atoms 9.1 Coherent trapping - dark states In multi-level systems coherent superpositions between different states (atomic coherence) may lead to dramatic changes of light

More information

### 2 Feynman rules, decay widths and cross sections

2 Feynman rules, decay widths and cross sections 2.1 Feynman rules Normalization In non-relativistic quantum mechanics, wave functions of free particles are normalized so that there is one particle in

More information

### Columbia 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 information

### Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction

Lecture 5 Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction WS0/3: Introduction to Nuclear and Particle Physics,, Part I I. Angular Momentum Operator Rotation R(θ): in polar coordinates the

More information

### PAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES)

Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 5 and Transition probabilities and transition dipole moment, Overview of selection rules CHE_P8_M5 TABLE

More information

### Lecture 11 Spin, orbital, and total angular momentum Mechanics. 1 Very brief background. 2 General properties of angular momentum operators

Lecture Spin, orbital, and total angular momentum 70.00 Mechanics Very brief background MATH-GA In 9, a famous experiment conducted by Otto Stern and Walther Gerlach, involving particles subject to a nonuniform

More information

### Photon Coupling with Matter, u R

1 / 16 Photon Coupling with Matter, u R Consider the up quark. We know that the u R has electric charge 2 3 e (where e is the proton charge), and that the photon A is a linear combination of the B and

More information

### eigenvalues eigenfunctions

Born-Oppenheimer Approximation Atoms and molecules consist of heavy nuclei and light electrons. Consider (for simplicity) a diatomic molecule (e.g. HCl). Clamp/freeze the nuclei in space, a distance r

More information

### Electric Dipole Paradox: Question, Answer, and Interpretation

Electric Dipole Paradox: Question, Answer, and Interpretation Frank Wilczek January 16, 2014 Abstract Non-vanishing electric dipole moments for the electron, neutron, or other entities are classic signals

More information

### 7 Three-level systems

7 Three-level systems In this section, we will extend our treatment of atom-light interactions to situations with more than one atomic energy level, and more than one independent coherent driving field.

More information

### 4 Matrix product states

Physics 3b Lecture 5 Caltech, 05//7 4 Matrix product states Matrix product state (MPS) is a highly useful tool in the study of interacting quantum systems in one dimension, both analytically and numerically.

More information

### 6. Molecular structure and spectroscopy I

6. Molecular structure and spectroscopy I 1 6. Molecular structure and spectroscopy I 1 molecular spectroscopy introduction 2 light-matter interaction 6.1 molecular spectroscopy introduction 2 Molecular

More information

### The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field:

SPIN 1/2 PARTICLE Stern-Gerlach experiment The experiment consists of studying the deflection of a beam of neutral ground state paramagnetic atoms (silver) in inhomogeneous magnetic field: A silver atom

More information

### ( ) /, so that we can ignore all

Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The ac-stark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)

More information

### Towards Landau-Zener-Stückelberg Interferometry on Single-molecule Magnets

Towards Landau-Zener-Stückelberg Interferometry on Single-molecule Magnets Changyun Yoo Advisor: Professor Jonathan Friedman May 5, 2015 Submitted to the Department of Physics and Astronomy in partial

More information

### Angular momentum. Quantum mechanics. Orbital angular momentum

Angular momentum 1 Orbital angular momentum Consider a particle described by the Cartesian coordinates (x, y, z r and their conjugate momenta (p x, p y, p z p. The classical definition of the orbital angular

More information

### Chapter 1. Harmonic Oscillator. 1.1 Energy Analysis

Chapter 1 Harmonic Oscillator Figure 1.1 illustrates the prototypical harmonic oscillator, the mass-spring system. A mass is attached to one end of a spring. The other end of the spring is attached to

More information

### INTERMOLECULAR INTERACTIONS

5.61 Physical Chemistry Lecture #3 1 INTERMOLECULAR INTERACTIONS Consider the interaction between two stable molecules (e.g. water and ethanol) or equivalently between two noble atoms (e.g. helium and

More information

### Lecture 4 Quantum mechanics in more than one-dimension

Lecture 4 Quantum mechanics in more than one-dimension Background Previously, we have addressed quantum mechanics of 1d systems and explored bound and unbound (scattering) states. Although general concepts

More information

### CHAPTER 13 LECTURE NOTES

CHAPTER 13 LECTURE NOTES Spectroscopy is concerned with the measurement of (a) the wavelengths (or frequencies) at which molecules absorb/emit energy, and (b) the amount of radiation absorbed at these

More information

### Introduction to Instantons. T. Daniel Brennan. Quantum Mechanics. Quantum Field Theory. Effects of Instanton- Matter Interactions.

February 18, 2015 1 2 3 Instantons in Path Integral Formulation of mechanics is based around the propagator: x f e iht / x i In path integral formulation of quantum mechanics we relate the propagator to

More information

### Simple Explanation of Fermi Arcs in Cuprate Pseudogaps: A Motional Narrowing Phenomenon

Simple Explanation of Fermi Arcs in Cuprate Pseudogaps: A Motional Narrowing Phenomenon ABSTRACT: ARPES measurements on underdoped cuprates above the superconducting transition temperature exhibit the

More information

### Approximation Methods in QM

Chapter 3 Approximation Methods in QM Contents 3.1 Time independent PT (nondegenerate)............... 5 3. Degenerate perturbation theory (PT)................. 59 3.3 Time dependent PT and Fermi s golden

More information

### Correlations between spin accumulation and degree of time-inverse breaking for electron gas in solid

Correlations between spin accumulation and degree of time-inverse breaking for electron gas in solid V.Zayets * Spintronic Research Center, National Institute of Advanced Industrial Science and Technology

More information

### Physics 504, Lecture 9 Feb. 21, 2011

Last Latexed: February 17, 011 at 15:8 1 1 Ionosphere, Redux Physics 504, Lecture 9 Feb. 1, 011 Let us return to the resonant cavity formed by the surface of the Earth a spherical shell of radius r = R

More information

### Group representation theory and quantum physics

Group representation theory and quantum physics Olivier Pfister April 29, 2003 Abstract This is a basic tutorial on the use of group representation theory in quantum physics, in particular for such systems

More information

### The KTY formalism and the neutrino oscillation probability including nonadiabatic contributions. Tokyo Metropolitan University.

The KTY formalism and the neutrino oscillation probability including nonadiabatic contributions Tokyo Metropolitan niversity Osamu Yasuda Based on Phys.Rev. D89 (2014) 093023 21 December 2014@Miami2014

More information

### UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT. PART I Qualifying Examination. August 20, 2013, 5:00 p.m. to 8:00 p.m.

UNIVERSITY OF MISSOURI-COLUMBIA PHYSICS DEPARTMENT PART I Qualifying Examination August 20, 2013, 5:00 p.m. to 8:00 p.m. Instructions: The only material you are allowed in the examination room is a writing

More information

### Harmonic 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 information

### Chapter 6. Quantum Theory of the Hydrogen Atom

Chapter 6 Quantum Theory of the Hydrogen Atom 1 6.1 Schrodinger s Equation for the Hydrogen Atom Symmetry suggests spherical polar coordinates Fig. 6.1 (a) Spherical polar coordinates. (b) A line of constant

More information

### 2. Electric Dipole Start from the classical formula for electric dipole radiation. de dt = 2. 3c 3 d 2 (2.1) qr (2.2) charges q

APAS 50. Internal Processes in Gases. Fall 999. Transition Probabilities and Selection Rules. Correspondence between Classical and Quantum Mechanical Transition Rates According to the correspondence principle

More information

### Relativistic corrections of energy terms

Lectures 2-3 Hydrogen atom. Relativistic corrections of energy terms: relativistic mass correction, Darwin term, and spin-orbit term. Fine structure. Lamb shift. Hyperfine structure. Energy levels of the

More information

### Quantum computation and quantum information

Quantum computation and quantum information Chapter 7 - Physical Realizations - Part 2 First: sign up for the lab! do hand-ins and project! Ch. 7 Physical Realizations Deviate from the book 2 lectures,

More information

### ODEs. September 7, Consider the following system of two coupled first-order ordinary differential equations (ODEs): A =

ODEs September 7, 2017 In [1]: using Interact, PyPlot 1 Exponential growth and decay Consider the following system of two coupled first-order ordinary differential equations (ODEs): d x/dt = A x for the

More information

### 8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 19. Problem Set 6. Due Wednesday April 3 at 10.

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 19 Problem Set 6 Due Wednesday April 3 at 10.00AM Assigned Reading: E&R 6 all, G, H Li. 7 1 9, 8 1 Ga.

More information

### Electron 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 information

### 129 Lecture Notes Relativistic Quantum Mechanics

19 Lecture Notes Relativistic Quantum Mechanics 1 Need for Relativistic Quantum Mechanics The interaction of matter and radiation field based on the Hamitonian H = p e c A m Ze r + d x 1 8π E + B. 1 Coulomb

More information

### Chapter 8 Magnetic Resonance

Chapter 8 Magnetic Resonance 9.1 Electron paramagnetic resonance 9.2 Ferromagnetic resonance 9.3 Nuclear magnetic resonance 9.4 Other resonance methods TCD March 2007 1 A resonance experiment involves

More information

### The Λ(1405) is an anti-kaon nucleon molecule. Jonathan Hall, Waseem Kamleh, Derek Leinweber, Ben Menadue, Ben Owen, Tony Thomas, Ross Young

The Λ(1405) is an anti-kaon nucleon molecule Jonathan Hall, Waseem Kamleh, Derek Leinweber, Ben Menadue, Ben Owen, Tony Thomas, Ross Young The Λ(1405) The Λ(1405) is the lowest-lying odd-parity state of

More information

### Lecture 9 Superconducting qubits Ref: Clarke and Wilhelm, Nature 453, 1031 (2008).

Lecture 9 Superconducting qubits Ref: Clarke and Wilhelm, Nature 453, 1031 (2008). Newcomer in the quantum computation area ( 2000, following experimental demonstration of coherence in charge + flux qubits).

More information

### Quantum 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 information

### C. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.

Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is

More information

### Quantum 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 information

### Exact diagonalization methods

Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Exact diagonalization methods Anders W. Sandvik, Boston University Representation of states in the computer bit

More information

### Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,

Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you

More information

### 2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements

1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical

More information

### Lecture 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 information

### The Quantum Heisenberg Ferromagnet

The Quantum Heisenberg Ferromagnet Soon after Schrödinger discovered the wave equation of quantum mechanics, Heisenberg and Dirac developed the first successful quantum theory of ferromagnetism W. Heisenberg,

More information