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

Size: px
Start display at page:

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

Transcription

1 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 use the properties of the Pauli matrices that σ i = I and {σ i σ j } = 0 provided i j n σ) = n x σ x + n y σ y + n z σ z )n x σ x + n y σ y + n z σ z ) = n x σ x + n y σ y + n z σ z + n x n y σ x σ y + σ y σ x ) + n y n z σ y σ z + σ z σ y ) + n z n x σ z σ x + σ x σ z ) = n x + n y + n z)i = I is The unitary operator which generates a rotation by ϕ about the axis n in spinor space U = e iϕ n S/ h b) Show that U = cos ϕ/ + in σ sin ϕ/ First we note that n S/ h = n σ/ Second we expand the exponential in a Taylor series and note that since n σ) = I even terms in the series will involve the identity matrix and odd terms will involve n σ to the first power U = e iϕ n S/ h = e iϕ/)n σ) i n = I n! ϕ/)n + in σ) n=04 = I cos ϕ/ + in σ) sin ϕ/ n=35 i n n! ϕ/) n c) Show that UσU = nn σ) n n σ) cosϕ + n σ) sinϕ You might find it helpful to remember that a product of two Pauli matrices gives the identity matrix or plus or minus i times the third Pauli matrix according to the formula σ i σ j = δ ij + iǫ ijk σ k

2 Physics 505 Homework No 8 s S8- where the summation convention is assumed and where we have just used in place of I UσU = cos ϕ/ + in σ sin ϕ/)σcosϕ/ in σ sin ϕ/) = cos ϕ/ σ + i cos ϕ/ sinϕ/n σ σ σ n σ) + sin ϕ/ n σ σ n σ At this point we just grind out the i th component of each term The first term is just cos ϕ/ σ i For the second term we need n σ σ i σ i n σ = n j σ j σ i σ i n j σ j = n j [σ j σ i ] = in j ǫ jik σ k = iǫ ijk n j σ k = i n σ) i and the middle term is sin ϕ n σ) i For the third term we need n σ σ i n σ = n j σ j σ i n k σ k = n j n k σ j σ i σ k = n j n k δ ji + iǫ jil σ l )σ k = n i n σ + in j n k ǫ jil δ lk + iǫ lkm σ m ) = n i n σ n j ǫ jil ǫ lkm n k σ m = n i n σ n j ǫ jil n σ) l = n i n σ + ǫ ijl n j n σ) l = n i n σ + n n σ)) i The second term looks like what we want but we still have more work to do on the first and third terms We combine the two terms and use the identities cos ϕ/ = +cos ϕ)/ and sin ϕ/ = cos ϕ)/ to get σ + nn σ) + n n σ))/ + cos ϕσ nn σ) n n σ))/) We now use the vector identity A B C) = BA C) CA B) In the first term above σ + nn σ) + n n σ))/ = σ + nn σ) + nn σ) σn n))/ = nn σ)

3 Physics 505 Homework No 8 s S8-3 In the second term we have cos ϕσ nn σ) n n σ))/) = cos ϕσ nn σ) nn σ) + σn n))/ = cos ϕσ nn σ)) = cos ϕn n σ)) Putting everything together we get the desired result: UσU = nn σ) n n σ) cos ϕ + n σ) sinϕ d) Consider the special case n = e z and discuss infinitesimal rotations and In this case U + iσ z δϕ/ = + iδϕ/ 0 ) 0 iδϕ/ σ = UσU σ e x σ y δϕ + e y σ x δϕ The rotation about the z-axis carries a little bit of the x-component of the Pauli operator into the y-direction and a little bit of the y-component into the negative x-direction just as what happens with an ordinary vector under rotation e) For a spinor χ = α+ α ) calculate the transformed spinor χ for n = e z ) χ α+ = Uχ = cos ϕ/ + iσ z sin ϕ/) α ) 0 0 = cos ϕ/ + i sin ϕ/ = 0 α+ cos ϕ/ + iα + sin ϕ/ α cos ϕ/ iα sin ϕ/ )) ) α+ 0 α ) α = + e +iϕ/ ) α e iϕ/

4 Physics 505 Homework No 8 s S8-4 f) What happens to the spinor when ϕ = π? The spinor changes sign! A rotation of 4π is required to bring the spinor back to its original state! This is a weird property of fermions However expectation values involve χ so the sign cancels out! Spin precession a) The spinors = ) 0 = ) 0 are the eigenfunctions of σ z They are an orthonormal complete basis in which σ z is diagonal In this same basis what are the eigenfunctions spinors) of σ x and σ y? The eigenvalues are ± so the expression for the eigenspinors of σ x is 0 0 ) ) a = ± b ) a b which gives b = ±a In order that the spinors be normalized we take a = b = / Then the eigenspinors of σ x are x = ) x = ) For σ y 0 i i 0 ) ) a = ± b ) a b which gives b = ±ia Again we must take a = b = / and the eigenspinors of σ y are y = ) i y = ) i Now we consider a spin / particle in a uniform magnetic field For definiteness we suppose it s an electron so the Hamiltonian for the spin degree of freedom is H = ge mc S B e mc S B

5 Physics 505 Homework No 8 s S8-5 b) The state of the system is represented by the spinor χ = i h t χ+ χ ) What is the equation of motion for χ? After writing down the equation for the general case specialize to the case B = Be z Your results should depend on the frequency ω = eb/mc which will turn out to be the precession frequency It s just the Schroedinger equation also known as the Pauli equation in this case) χ+ χ ) = e ) ) mc S B χ+ χ+ = µ χ B σ B = µ χ B When B = Be z the equation of motion becomes i t χ+ χ ) = µ BB h B z B x + ib y ) ) 0 χ+ = ω ) ) 0 χ+ 0 χ 0 χ ) ) B x ib y χ+ B z where ω = µ B B/ h = eb/mc) is the frequency which will turn out to the precession frequency) c) In the case with B = Be z and in the Heisenberg representation what is the time dependence of the spin operator St)? In the Heisenberg representation χ ds z dt = ī h [H S z] = 0 so S z t) = S z 0) For the other two components it s more convenient to work with S x ±is y d dt S x ± is y ) = ī h [H S x ± is y ] = ieb mc h [S z S x ± is y ] = ieb mc h i hs y is x ) = ± ieb mc S x ± is y ) This means or S x t) ± is y t) = e ±iωt S x 0) ± is y 0)) S x t) = cos ωt S x 0) sin ωt S y 0) S y t) = sin ωt S x 0) + cos ωt S y 0)

6 Physics 505 Homework No 8 s S8-6 d) In the case with B = Be z and in the Schroedinger representation what is the time dependence of the spinor χt)? The argument of the exponential is χt) = e iht/ h χ0) iht h = iebt mc h S z = iωt h S z = iϕn S/ h provided we identify ϕ = ωt and n = e z This form is exactly the form for the unitary operator discussed in problem so the time dependence of χ is χt) = ) χ+ t) = cos ωt/) χ t) = e iωt/ 0 0 e +iωt/ e = iωt/ ) χ + 0) e +iωt/ χ 0) ) i sin ωt/) 0 0 ) ) χ+ 0) χ 0) )) ) χ+ 0) χ 0) This makes sense The up and down states are the stationary states of the Hamiltonian with energy hω/ for the up state The spin in the up state is aligned with B but the magnetic moment is anti-aligned so the energy is higher than in the down state which has energy hω/ e) At t = 0 the spin is aligned along the x axis What is the probability of getting h/ in a measurement of S z at time t The initial state must be to within a phase!) χ0) = x = ) From part e) we find the state at time t and project that onto z to find the amplitude for going to spin up along z and then square it to get the probability e P z ) = 0) iωt/ / ) e +iωt/ =

7 Physics 505 Homework No 8 s S8-7 which shouldn t be a surprise If the spin is pointed along the x axis it s equally likely to be up or down if a measurement is made along the z axis When the spin is in a magnetic field in the z-direction it precesses about the z-axis which does not change its z-component or the probability distribution of its z-component) f) At t = 0 the spin is aligned along the x axis What is the probability of getting h/ in a measurement of S x at time t We solve this part the same way except we project onto x P x ) = / / e ) iωt/ / ) e +iωt/ so the spin is in fact precessing with a frequency ω! = cos ωt/ = + cosωt))/ The solutions to parts e) and f) have been based on the Schroedinger representation In the Heisenberg representation we could ask about the expectation value of S x t) Recall that the state does not change We use S x t) from part c) S x t) = / / ) 0 he iωt / he iωt / 0 ) ) / / = h cos ωt which is consistent with the idea that the spin is precessing around the z-axis which means its projection on the x-axis oscillates between ± h/ 3 Magnetic Resonance This problem continues from where problem ended We have an electron in a uniform magnetic field along the z-axis In addition we have a small time variable field along the x-axis: B x = B p cos ωt where ω is determined by the field in the z-direction ω = eb/mc We suppose that at t = 0 the electron has spin down along the z-axis The experimental picture is that the magnetic moment of the electron is aligned with the field so the spin is anti-aligned) and the probe field is turned on at t = 0 What happens? In particular what is S z t)? Use the interaction representation to solve this problem Hint: The probe field is oscillating along the x-axis However you can think of it as the sum of two fields each rotating in the xy-plane with angular velocity ω One rotates in the positive direction and one rotates in the negative direction When you transform the Hamiltonian due to the probe field to the interaction representation one of these will wind up stationary and the other will wind up rotating at ω Make an argument for why the ω term can be ignored and then proceed with the stationary term

8 Physics 505 Homework No 8 s S8-8 The additional term in the Hamiltonian is V t) = eb p mc S x cos ωt = ω p S x cos ωt In the interaction picture the operators evolve just as they did in the Heisenberg picture so we can use problem part c) for the St) In particular S z t) = S z 0) so that was easy! Transforming the extra term in the Hamiltonian requires a bit more work First as mentioned in the hint we can write the extra magnetic field as the sum of B + + B with Then the extra term is V + + V with B ± = B p cosωt e x ± sin ωt e y ) V ± = eb p mc S x cos ωt ± S y sin ωt) Now we must transform this extra term to the Heisenberg representation based on the Hamiltonian without V Again we use problem c) and replace S x t) by its expression in terms of S x 0) and S y 0) and similarly for S y t) So V I± t) = ω p cos ωts x 0) cos ωt sinωts y 0) ± sin ωts x 0) ± cos ωt sinωts y 0) ) V I+ t) = ω p S x 0) V I t) = ω p cos ωts x 0) sin ωts y 0)) The integral of the Hamiltonian with time gives the phase of the wave function V I is oscillating about 0) twice as fast as anything else so it essentially never has time to build up an appreciable phase Thus we ignore it and continue with V I+ t) which is just a constant The state evolves as states in the Schroedinger representation but with the interaction Hamiltonian rather than the full Hamiltonian i h t χ = V I+t)χ χt) = e iv I+t)/ h χ0) = e iω p e x S0)t/ h χ0) Again we make use of problem to write this as χt) = cos ω p t/) + iσ x 0) sin ω p t/))χ0)

9 Physics 505 Homework No 8 s S8-9 The state at t = 0 is spin down along the z-axis so ) ) ) χ+ cos ωp t/ i sin ω = p t/ 0 = i sin ω p t/ cos ω p t/ χ ) i sin ωp t/ cos ω p t/ So we have found how the state evolves due to the probe field and we know how S z evolves due to the main field and we can calculate S z ) ) h/ 0 i sin ωp t/ S z t) = +i sin ω p t/ cos ω p t/ ) = h 0 h/ cos ω p t/ cos ω pt After all that we have a very simple answer! The spin flips up and down with frequency ω p Physically the Heisenberg representation in this case corresponds to a frame rotating with the precessing spin Since we made the probe field oscillate with the same frequency as the rotation it split into two pieces one that s stationary in the rotating frame and one that s rotating like mad and has no net effect Since the field is stationary in the rotating frame it causes the spin to precess about this field in the rotating frame and it oscillates between aligned and anti-aligned You have just discovered the principle of magnetic resonance imaging which generally uses nuclear spins but the idea is the same) A sample is placed in a strong field this causes the spins to precess Because the environment of the spins depends not only on the applied field but the field due to neighbors the actual precession frequency will vary with chemical environment The probe field is scanned in frequency slowly compared to ω and ω p ) and there will be an absorption and emission of energy by the sample as the resonant frequency is passed The strength of the signal depends on the density of aligned spins A lot can be learned about the sample by studying how the spins precess and flip! 4 Combining angular momentum Two electron spins S and S can be summed to produce a total angular momentum J = S + S The states j ms s can be expanded in the states s m s m Determine the expansion Addition of two spin / angular momenta produces a spin system with j = and m = 0 + the triplet) and a spin system with j = 0 and m = 0 So Recall that + / / = / / / / L ± l m = h ll + ) mm ± ) l m ± We operate with J = S + S to get + ) ) 0 / / = /)/ + ) /)/ ) / / / / + /)/ + ) /)/ ) / / / /

10 Physics 505 Homework No 8 s S8-0 or 0 / / = / / / / + / / / / ) If we operate with J again we find Finally / / = / / / / 0 0 / / = / / / / / / / / ) We find this by noting that it must be orthogonal to 0 / / and it must contain the same m and m states that add up to 0 Comment: we will learn later that the wave functions for Fermions must be antisymmetric that is change sign) when any two Fermions are exchanged The singlet state above is anti-symmetric so this must be combined with a symmetric state for the spatial wave function to have a suitable overall state Similarly the triplet state must be combined with an anti-symmetric spatial wave function

Sommerfeld (1920) noted energy levels of Li deduced from spectroscopy looked like H, with slight adjustment of principal quantum number:

Sommerfeld (1920) noted energy levels of Li deduced from spectroscopy looked like H, with slight adjustment of principal quantum number: Spin. Historical Spectroscopy of Alkali atoms First expt. to suggest need for electron spin: observation of splitting of expected spectral lines for alkali atoms: i.e. expect one line based on analogy

More information

Lecture3 (and part of lecture 4).

Lecture3 (and part of lecture 4). Lecture3 (and part of lecture 4). Angular momentum and spin. Stern-Gerlach experiment Spin Hamiltonian Evolution of spin with time Evolution of spin in precessing magnetic field. In classical mechanics

More information

PHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.

PHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m. PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms

More information

QM and Angular Momentum

QM and Angular Momentum Chapter 5 QM and Angular Momentum 5. Angular Momentum Operators In your Introductory Quantum Mechanics (QM) course you learned about the basic properties of low spin systems. Here we want to review that

More information

conventions and notation

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

Physics 4022 Notes on Density Matrices

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

Dirac Equation. Chapter 1

Dirac Equation. Chapter 1 Chapter Dirac Equation This course will be devoted principally to an exposition of the dynamics of Abelian and non-abelian gauge theories. These form the basis of the Standard Model, that is, the theory

More information

Notes on Spin Operators and the Heisenberg Model. Physics : Winter, David G. Stroud

Notes on Spin Operators and the Heisenberg Model. Physics : Winter, David G. Stroud Notes on Spin Operators and the Heisenberg Model Physics 880.06: Winter, 003-4 David G. Stroud In these notes I give a brief discussion of spin-1/ operators and their use in the Heisenberg model. 1. Spin

More information

4. Two-level systems. 4.1 Generalities

4. Two-level systems. 4.1 Generalities 4. Two-level systems 4.1 Generalities 4. Rotations and angular momentum 4..1 Classical rotations 4.. QM angular momentum as generator of rotations 4..3 Example of Two-Level System: Neutron Interferometry

More information

Introduction to Group Theory

Introduction to Group Theory Chapter 10 Introduction to Group Theory Since symmetries described by groups play such an important role in modern physics, we will take a little time to introduce the basic structure (as seen by a physicist)

More information

Basic Physical Chemistry Lecture 2. Keisuke Goda Summer Semester 2015

Basic Physical Chemistry Lecture 2. Keisuke Goda Summer Semester 2015 Basic Physical Chemistry Lecture 2 Keisuke Goda Summer Semester 2015 Lecture schedule Since we only have three lectures, let s focus on a few important topics of quantum chemistry and structural chemistry

More information

Coherent states, beam splitters and photons

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

arxiv: v1 [quant-ph] 4 Sep 2008

arxiv: v1 [quant-ph] 4 Sep 2008 The Schroedinger equation as the time evolution of a deterministic and reversible process Gabriele Carcassi Brookhaven National Laboratory, Upton, NY 11973 (Dated: September 4, 008) In this paper we derive

More information

Brief review of Quantum Mechanics (QM)

Brief review of Quantum Mechanics (QM) Brief review of Quantum Mechanics (QM) Note: This is a collection of several formulae and facts that we will use throughout the course. It is by no means a complete discussion of QM, nor will I attempt

More information

Physics 70007, Fall 2009 Answers to Final Exam

Physics 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 information

Chapter 16 The relativistic Dirac equation. from my book: Understanding Relativistic Quantum Field Theory. Hans de Vries

Chapter 16 The relativistic Dirac equation. from my book: Understanding Relativistic Quantum Field Theory. Hans de Vries Chapter 6 The relativistic Dirac equation from my book: Understanding Relativistic Quantum Field Theory Hans de Vries November, 28 2 Chapter Contents 6 The relativistic Dirac equation 6. The linearized

More information

Rotational motion of a rigid body spinning around a rotational axis ˆn;

Rotational motion of a rigid body spinning around a rotational axis ˆn; Physics 106a, Caltech 15 November, 2018 Lecture 14: Rotations The motion of solid bodies So far, we have been studying the motion of point particles, which are essentially just translational. Bodies with

More information

221B Lecture Notes Many-Body Problems I (Quantum Statistics)

221B Lecture Notes Many-Body Problems I (Quantum Statistics) 221B Lecture Notes Many-Body Problems I (Quantum Statistics) 1 Quantum Statistics of Identical Particles If two particles are identical, their exchange must not change physical quantities. Therefore, a

More information

5.61 Physical Chemistry Lecture #36 Page

5.61 Physical Chemistry Lecture #36 Page 5.61 Physical Chemistry Lecture #36 Page 1 NUCLEAR MAGNETIC RESONANCE Just as IR spectroscopy is the simplest example of transitions being induced by light s oscillating electric field, so NMR is the simplest

More information

1.3 Translational Invariance

1.3 Translational Invariance 1.3. TRANSLATIONAL INVARIANCE 7 Version of January 28, 2005 Thus the required rotational invariance statement is verified: [J, H] = [L + 1 Σ, H] = iα p iα p = 0. (1.49) 2 1.3 Translational Invariance One

More information

Quantum Dynamics. March 10, 2017

Quantum Dynamics. March 10, 2017 Quantum Dynamics March 0, 07 As in classical mechanics, time is a parameter in quantum mechanics. It is distinct from space in the sense that, while we have Hermitian operators, X, for position and therefore

More information

B2.III Revision notes: quantum physics

B2.III Revision notes: quantum physics B.III Revision notes: quantum physics Dr D.M.Lucas, TT 0 These notes give a summary of most of the Quantum part of this course, to complement Prof. Ewart s notes on Atomic Structure, and Prof. Hooker s

More information

a = ( a σ )( b σ ) = a b + iσ ( a b) mω 2! x + i 1 2! x i 1 2m!ω p, a = mω 2m!ω p Physics 624, Quantum II -- Final Exam

a = ( a σ )( b σ ) = a b + iσ ( a b) mω 2! x + i 1 2! x i 1 2m!ω p, a = mω 2m!ω p Physics 624, Quantum II -- Final Exam Physics 624, Quantum II -- Final Exam Please show all your work on the separate sheets provided (and be sure to include your name). You are graded on your work on those pages, with partial credit where

More information

Spin resonance. Basic idea. PSC 3151, (301)

Spin resonance. Basic idea. PSC 3151, (301) Spin Resonance Phys623 Spring 2018 Prof. Ted Jacobson PSC 3151, (301)405-6020 jacobson@physics.umd.edu Spin resonance Spin resonance refers to the enhancement of a spin flipping probability in a magnetic

More information

Problem 1: Spin 1 2. particles (10 points)

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

Representations of Lorentz Group

Representations of Lorentz Group Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: How do we find the smallest (irreducible) representations of the

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 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

and S is in the state ( )

and S is in the state ( ) Physics 517 Homework Set #8 Autumn 2016 Due in class 12/9/16 1. Consider a spin 1/2 particle S that interacts with a measuring apparatus that is another spin 1/2 particle, A. The state vector of the system

More information

1 Mathematical preliminaries

1 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 information

Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 22, March 20, 2006

Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 22, March 20, 2006 Chem 350/450 Physical Chemistry II Quantum Mechanics 3 Credits Spring Semester 006 Christopher J. Cramer Lecture, March 0, 006 Some material in this lecture has been adapted from Cramer, C. J. Essentials

More information

Plan for the rest of the semester. ψ a

Plan for the rest of the semester. ψ a Plan for the rest of the semester ϕ ψ a ϕ(x) e iα(x) ϕ(x) 167 Representations of Lorentz Group based on S-33 We defined a unitary operator that implemented a Lorentz transformation on a scalar field: and

More information

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

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

More information

Physics 221A Fall 1996 Notes 13 Spins in Magnetic Fields

Physics 221A Fall 1996 Notes 13 Spins in Magnetic Fields Physics 221A Fall 1996 Notes 13 Spins in Magnetic Fields A nice illustration of rotation operator methods which is also important physically is the problem of spins in magnetic fields. The earliest experiments

More information

Lecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition)

Lecture 10: A (Brief) Introduction to Group Theory (See Chapter 3.13 in Boas, 3rd Edition) Lecture 0: A (Brief) Introduction to Group heory (See Chapter 3.3 in Boas, 3rd Edition) Having gained some new experience with matrices, which provide us with representations of groups, and because symmetries

More information

Lecture #6 NMR in Hilbert Space

Lecture #6 NMR in Hilbert Space Lecture #6 NMR in Hilbert Space Topics Review of spin operators Single spin in a magnetic field: longitudinal and transverse magnetiation Ensemble of spins in a magnetic field RF excitation Handouts and

More information

Lecture 19 (Nov. 15, 2017)

Lecture 19 (Nov. 15, 2017) Lecture 19 8.31 Quantum Theory I, Fall 017 8 Lecture 19 Nov. 15, 017) 19.1 Rotations Recall that rotations are transformations of the form x i R ij x j using Einstein summation notation), where R is an

More information

Angular momentum and spin

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

Math 320, spring 2011 before the first midterm

Math 320, spring 2011 before the first midterm Math 320, spring 2011 before the first midterm Typical Exam Problems 1 Consider the linear system of equations 2x 1 + 3x 2 2x 3 + x 4 = y 1 x 1 + 3x 2 2x 3 + 2x 4 = y 2 x 1 + 2x 3 x 4 = y 3 where x 1,,

More information

NANOSCALE SCIENCE & TECHNOLOGY

NANOSCALE 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 information

r R A 1 r R B + 1 ψ(r) = αψ A (r)+βψ B (r) (5) where we assume that ψ A and ψ B are ground states: ψ A (r) = π 1/2 e r R A ψ B (r) = π 1/2 e r R B.

r R A 1 r R B + 1 ψ(r) = αψ A (r)+βψ B (r) (5) where we assume that ψ A and ψ B are ground states: ψ A (r) = π 1/2 e r R A ψ B (r) = π 1/2 e r R B. Molecules Initial questions: What are the new aspects of molecules compared to atoms? What part of the electromagnetic spectrum can we probe? What can we learn from molecular spectra? How large a molecule

More information

Solutions to exam : 1FA352 Quantum Mechanics 10 hp 1

Solutions 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 information

Physics Capstone. Karsten Gimre May 19, 2009

Physics Capstone. Karsten Gimre May 19, 2009 Physics Capstone Karsten Gimre May 19, 009 Abstract Quantum mechanics is, without a doubt, one of the major advancements of the 0th century. It can be incredibly difficult to understand the fundamental

More information

G : Quantum Mechanics II

G : Quantum Mechanics II G5.666: Quantum Mechanics II Notes for Lecture 5 I. REPRESENTING STATES IN THE FULL HILBERT SPACE Given a representation of the states that span the spin Hilbert space, we now need to consider the problem

More information

Quantum Mechanics Solutions. λ i λ j v j v j v i v i.

Quantum 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 information

C/CS/Phys C191 Particle-in-a-box, Spin 10/02/08 Fall 2008 Lecture 11

C/CS/Phys C191 Particle-in-a-box, Spin 10/02/08 Fall 2008 Lecture 11 C/CS/Phys C191 Particle-in-a-box, Spin 10/0/08 Fall 008 Lecture 11 Last time we saw that the time dependent Schr. eqn. can be decomposed into two equations, one in time (t) and one in space (x): space

More information

Physics 215 Quantum Mechanics I Assignment 8

Physics 215 Quantum Mechanics I Assignment 8 Physics 15 Quantum Mechanics I Assignment 8 Logan A. Morrison March, 016 Problem 1 Let J be an angular momentum operator. Part (a) Using the usual angular momentum commutation relations, prove that J =

More information

LSZ reduction for spin-1/2 particles

LSZ reduction for spin-1/2 particles LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free

More information

REVIEW REVIEW. A guess for a suitable initial state: Similarly, let s consider a final state: Summary of free theory:

REVIEW REVIEW. A guess for a suitable initial state: Similarly, let s consider a final state: Summary of free theory: LSZ reduction for spin-1/2 particles based on S-41 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free

More information

Physics 115C Homework 3

Physics 115C Homework 3 Physics 115C Homework 3 Problem 1 In this problem, it will be convenient to introduce the Einstein summation convention. Note that we can write S = i S i i where the sum is over i = x,y,z. In the Einstein

More information

Lecture 4. 1 de Broglie wavelength and Galilean transformations 1. 2 Phase and Group Velocities 4. 3 Choosing the wavefunction for a free particle 6

Lecture 4. 1 de Broglie wavelength and Galilean transformations 1. 2 Phase and Group Velocities 4. 3 Choosing the wavefunction for a free particle 6 Lecture 4 B. Zwiebach February 18, 2016 Contents 1 de Broglie wavelength and Galilean transformations 1 2 Phase and Group Velocities 4 3 Choosing the wavefunction for a free particle 6 1 de Broglie wavelength

More information

But what happens when free (i.e. unbound) charged particles experience a magnetic field which influences orbital motion? e.g. electrons in a metal.

But what happens when free (i.e. unbound) charged particles experience a magnetic field which influences orbital motion? e.g. electrons in a metal. Lecture 5: continued But what happens when free (i.e. unbound charged particles experience a magnetic field which influences orbital motion? e.g. electrons in a metal. Ĥ = 1 2m (ˆp qa(x, t2 + qϕ(x, t,

More information

Particle Physics. Michaelmas Term 2009 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims

Particle Physics. Michaelmas Term 2009 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims Particle Physics Michaelmas Term 2009 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2009 205 Introduction/Aims Symmetries play a central role in particle physics;

More information

Addition of Angular Momenta

Addition of Angular Momenta Addition of Angular Momenta What we have so far considered to be an exact solution for the many electron problem, should really be called exact non-relativistic solution. A relativistic treatment is needed

More information

2 Canonical quantization

2 Canonical quantization Phys540.nb 7 Canonical quantization.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system?.1.1.lagrangian Lagrangian mechanics is a reformulation of classical mechanics.

More information

Quantum Mechanics C (130C) Winter 2014 Assignment 7

Quantum Mechanics C (130C) Winter 2014 Assignment 7 University of California at San Diego Department of Physics Prof. John McGreevy Quantum Mechanics C (130C) Winter 014 Assignment 7 Posted March 3, 014 Due 11am Thursday March 13, 014 This is the last problem

More information

Damped harmonic motion

Damped harmonic motion Damped harmonic motion March 3, 016 Harmonic motion is studied in the presence of a damping force proportional to the velocity. The complex method is introduced, and the different cases of under-damping,

More information

Assignment 8. [η j, η k ] = J jk

Assignment 8. [η j, η k ] = J jk Assignment 8 Goldstein 9.8 Prove directly that the transformation is canonical and find a generating function. Q 1 = q 1, P 1 = p 1 p Q = p, P = q 1 q We can establish that the transformation is canonical

More information

Atomic Structure. Chapter 8

Atomic Structure. Chapter 8 Atomic Structure Chapter 8 Overview To understand atomic structure requires understanding a special aspect of the electron - spin and its related magnetism - and properties of a collection of identical

More information

LS coupling. 2 2 n + H s o + H h f + H B. (1) 2m

LS coupling. 2 2 n + H s o + H h f + H B. (1) 2m LS coupling 1 The big picture We start from the Hamiltonian of an atomic system: H = [ ] 2 2 n Ze2 1 + 1 e 2 1 + H s o + H h f + H B. (1) 2m n e 4πɛ 0 r n 2 4πɛ 0 r nm n,m Here n runs pver the electrons,

More information

Physics 557 Lecture 5

Physics 557 Lecture 5 Physics 557 Lecture 5 Group heory: Since symmetries and the use of group theory is so much a part of recent progress in particle physics we will take a small detour to introduce the basic structure (as

More information

Rotations in Quantum Mechanics

Rotations in Quantum Mechanics Rotations in Quantum Mechanics We have seen that physical transformations are represented in quantum mechanics by unitary operators acting on the Hilbert space. In this section, we ll think about the specific

More information

Lecture 10. The Dirac equation. WS2010/11: Introduction to Nuclear and Particle Physics

Lecture 10. The Dirac equation. WS2010/11: Introduction to Nuclear and Particle Physics Lecture 10 The Dirac equation WS2010/11: Introduction to Nuclear and Particle Physics The Dirac equation The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist

More information

Chapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1)

Chapter 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 information

Lecture 19: Building Atoms and Molecules

Lecture 19: Building Atoms and Molecules Lecture 19: Building Atoms and Molecules +e r n = 3 n = 2 n = 1 +e +e r y even Lecture 19, p 1 Today Nuclear Magnetic Resonance Using RF photons to drive transitions between nuclear spin orientations in

More information

Lecture 11: Long-wavelength expansion in the Neel state Energetic terms

Lecture 11: Long-wavelength expansion in the Neel state Energetic terms Lecture 11: Long-wavelength expansion in the Neel state Energetic terms In the last class we derived the low energy effective Hamiltonian for a Mott insulator. This derivation is an example of the kind

More information

16.1. PROBLEM SET I 197

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

2.1 Green Functions in Quantum Mechanics

2.1 Green Functions in Quantum Mechanics Chapter 2 Green Functions and Observables 2.1 Green Functions in Quantum Mechanics We will be interested in studying the properties of the ground state of a quantum mechanical many particle system. We

More information

1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12

1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to

More information

Lecture 21 Relevant sections in text: 3.1

Lecture 21 Relevant sections in text: 3.1 Lecture 21 Relevant sections in text: 3.1 Angular momentum - introductory remarks The theory of angular momentum in quantum mechanics is important in many ways. The myriad of results of this theory, which

More information

Total Angular Momentum for Hydrogen

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

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

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

129 Lecture Notes More on Dirac Equation

129 Lecture Notes More on Dirac Equation 19 Lecture Notes More on Dirac Equation 1 Ultra-relativistic Limit We have solved the Diraction in the Lecture Notes on Relativistic Quantum Mechanics, and saw that the upper lower two components are large

More information

9 Angular Momentum I. Classical analogy, take. 9.1 Orbital Angular Momentum

9 Angular Momentum I. Classical analogy, take. 9.1 Orbital Angular Momentum 9 Angular Momentum I So far we haven t examined QM s biggest success atomic structure and the explanation of atomic spectra in detail. To do this need better understanding of angular momentum. In brief:

More information

Coupling of Angular Momenta Isospin Nucleon-Nucleon Interaction

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

Chapter 4. Q. A hydrogen atom starts out in the following linear combination of the stationary. (ψ ψ 21 1 ). (1)

Chapter 4. Q. A hydrogen atom starts out in the following linear combination of the stationary. (ψ ψ 21 1 ). (1) Tor Kjellsson Stockholm University Chapter 4 4.5 Q. A hydrogen atom starts out in the following linear combination of the stationary states n, l, m =,, and n, l, m =,, : Ψr, 0 = ψ + ψ. a Q. Construct Ψr,

More information

QUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer

QUANTUM MECHANICS. Franz Schwabl. Translated by Ronald Kates. ff Springer Franz Schwabl QUANTUM MECHANICS Translated by Ronald Kates Second Revised Edition With 122Figures, 16Tables, Numerous Worked Examples, and 126 Problems ff Springer Contents 1. Historical and Experimental

More information

Shimming of a Magnet for Calibration of NMR Probes UW PHYSICS REU 2013

Shimming of a Magnet for Calibration of NMR Probes UW PHYSICS REU 2013 Shimming of a Magnet for Calibration of NMR Probes RACHEL BIELAJEW UW PHYSICS REU 2013 Outline Background The muon anomaly The g-2 Experiment NMR Design Helmholtz coils producing a gradient Results Future

More information

Appendix: SU(2) spin angular momentum and single spin dynamics

Appendix: SU(2) spin angular momentum and single spin dynamics Phys 7 Topics in Particles & Fields Spring 03 Lecture v0 Appendix: SU spin angular momentum and single spin dynamics Jeffrey Yepez Department of Physics and Astronomy University of Hawai i at Manoa Watanabe

More information

E = φ 1 A The dynamics of a particle with mass m and charge q is determined by the Hamiltonian

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

Galilean Transformations

Galilean Transformations Chapter 7 Galilean Transformations Something needs to be understood a bit better. Go back to the (dimensionless) q, p variables, which satisfy [q, p] =. (7.) i The finite version of the infinestimal translation

More information

Problem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension

Problem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension 105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1

More information

Tutorial 5 Clifford Algebra and so(n)

Tutorial 5 Clifford Algebra and so(n) Tutorial 5 Clifford Algebra and so(n) 1 Definition of Clifford Algebra A set of N Hermitian matrices γ 1, γ,..., γ N obeying the anti-commutator γ i, γ j } = δ ij I (1) is the basis for an algebra called

More information

Quantum Physics II (8.05) Fall 2002 Outline

Quantum Physics II (8.05) Fall 2002 Outline Quantum Physics II (8.05) Fall 2002 Outline 1. General structure of quantum mechanics. 8.04 was based primarily on wave mechanics. We review that foundation with the intent to build a more formal basis

More information

The Dirac Field. Physics , Quantum Field Theory. October Michael Dine Department of Physics University of California, Santa Cruz

The Dirac Field. Physics , Quantum Field Theory. October Michael Dine Department of Physics University of California, Santa Cruz Michael Dine Department of Physics University of California, Santa Cruz October 2013 Lorentz Transformation Properties of the Dirac Field First, rotations. In ordinary quantum mechanics, ψ σ i ψ (1) is

More information

Physics 505 Homework No. 1 Solutions S1-1

Physics 505 Homework No. 1 Solutions S1-1 Physics 505 Homework No s S- Some Preliminaries Assume A and B are Hermitian operators (a) Show that (AB) B A dx φ ABψ dx (A φ) Bψ dx (B (A φ)) ψ dx (B A φ) ψ End (b) Show that AB [A, B]/2+{A, B}/2 where

More information

1 Simple Harmonic Oscillator

1 Simple Harmonic Oscillator Physics 1a Waves Lecture 3 Caltech, 10/09/18 1 Simple Harmonic Oscillator 1.4 General properties of Simple Harmonic Oscillator 1.4.4 Superposition of two independent SHO Suppose we have two SHOs described

More information

Second quantization: where quantization and particles come from?

Second 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 information

charges q r p = q 2mc 2mc L (1.4) ptles µ e = g e

charges q r p = q 2mc 2mc L (1.4) ptles µ e = g e APAS 5110. Atomic and Molecular Processes. Fall 2013. 1. Magnetic Moment Classically, the magnetic moment µ of a system of charges q at positions r moving with velocities v is µ = 1 qr v. (1.1) 2c charges

More information

Intermission: Let s review the essentials of the Helium Atom

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

Physics Sep Example A Spin System

Physics Sep Example A Spin System Physics 30 7-Sep-004 4- Example A Spin System In the last lecture, we discussed the binomial distribution. Now, I would like to add a little physical content by considering a spin system. Actually this

More information

Physics 139B Solutions to Homework Set 4 Fall 2009

Physics 139B Solutions to Homework Set 4 Fall 2009 Physics 139B Solutions to Homework Set 4 Fall 9 1. Liboff, problem 1.16 on page 594 595. Consider an atom whose electrons are L S coupled so that the good quantum numbers are j l s m j and eigenstates

More information

Time Independent Perturbation Theory Contd.

Time Independent Perturbation Theory Contd. Time Independent Perturbation Theory Contd. A summary of the machinery for the Perturbation theory: H = H o + H p ; H 0 n >= E n n >; H Ψ n >= E n Ψ n > E n = E n + E n ; E n = < n H p n > + < m H p n

More information

Lecture #13 1. Incorporating a vector potential into the Hamiltonian 2. Spin postulates 3. Description of spin states 4. Identical particles in

Lecture #13 1. Incorporating a vector potential into the Hamiltonian 2. Spin postulates 3. Description of spin states 4. Identical particles in Lecture #3. Incorporating a vector potential into the Hamiltonian. Spin postulates 3. Description of spin states 4. Identical particles in classical and QM 5. Exchange degeneracy - the fundamental problem

More information

Rotation Matrices. Chapter 21

Rotation Matrices. Chapter 21 Chapter 1 Rotation Matrices We have talked at great length last semester and this about unitary transformations. The putting together of successive unitary transformations is of fundamental importance.

More information

d 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5)

d 3 r d 3 vf( r, v) = N (2) = CV C = n where n N/V is the total number of molecules per unit volume. Hence e βmv2 /2 d 3 rd 3 v (5) LECTURE 12 Maxwell Velocity Distribution Suppose we have a dilute gas of molecules, each with mass m. If the gas is dilute enough, we can ignore the interactions between the molecules and the energy will

More information

GROUP THEORY PRIMER. New terms: so(2n), so(2n+1), symplectic algebra sp(2n)

GROUP THEORY PRIMER. New terms: so(2n), so(2n+1), symplectic algebra sp(2n) GROUP THEORY PRIMER New terms: so(2n), so(2n+1), symplectic algebra sp(2n) 1. Some examples of semi-simple Lie algebras In the previous chapter, we developed the idea of understanding semi-simple Lie algebras

More information

Advanced Statistical Mechanics

Advanced Statistical Mechanics Advanced Statistical Mechanics Victor Gurarie Fall 0 Week 3: D and D transverse field Ising models, exact solution D Ising Model: Peierls Argument D Ising model is defined on a D lattice. Each spin of

More information

C/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 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 information

The Einstein A and B Coefficients

The Einstein A and B Coefficients The Einstein A and B Coefficients Austen Groener Department of Physics - Drexel University, Philadelphia, Pennsylvania 19104, USA Quantum Mechanics III December 10, 010 Abstract In this paper, the Einstein

More information

(1.1) In particular, ψ( q 1, m 1 ; ; q N, m N ) 2 is the probability to find the first particle

(1.1) In particular, ψ( q 1, m 1 ; ; q N, m N ) 2 is the probability to find the first particle Chapter 1 Identical particles 1.1 Distinguishable particles The Hilbert space of N has to be a subspace H = N n=1h n. Observables Ân of the n-th particle are self-adjoint operators of the form 1 1 1 1

More information