# 16.1. PROBLEM SET I 197

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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, where ψx =Ae ikx + Be ikx, we have j left = k m A B ; similarly j right = k m C D. Current conservation demands that j left = j right, i.e. A + D = C + B all of the incoming beam is transferred to the outgoing beam, a statement of particle conservation. This conservation of current can be written as Ψ in Ψ in = Ψ out Ψ out. Therefore, using the expression for ψ out, we have Ψ in Ψ in = Ψ out Ψ out = Ψ in }{{} S S Ψ in, leading to the unitarity condition.! =I b From the unitarity condition, it follows that S t S = I = + r rt + r t rt + r t t + r Comparing the matrix elements, we obtained the required result. c For the δ-function scattering problem, the wavefunction is given by { e ψx = ikx + re ikx x<0 te ikx x>0 For the δ-function potential, the wavefunction must remain continuous as x = 0, and x ψ +ɛ x ψ ɛ = mav 0 ψ0. This translates to the conditions + r = t, and ikt +r = mav0 t. As a result, we obtain ik t =, ik + mav0 as required. Using the result from b, since the potential is symmetric, we have r = t t t = ik + γ k ik + γ k + γ = ik + γ γ ik + γ k + γ = γ ik + γ As a result, we obtain the required expression for r.. a We are told that Ĥ ψ = E ψ and Ĥ ψ = E ψ, where E E. Therefore ψ Ĥ ψ = ψ Ĥψ dx = ψe ψ dx = E ψ ψ. Since Ĥ is Hermitian, we can write ψ Ĥ ψ = Ĥψ ψ dx = E ψ ψ dx = E ψ ψ = E ψ ψ, since E and E are real. Therefore, E E ψ ψ = 0 and, if E E then ψ ψ = 0, i.e. ψ and ψ are orthogonal. b If Â ψ = ψ and Â ψ = ψ, then adding them, Â ψ + ψ = ψ + ψ and subtracting, Â ψ ψ = ψ ψ = ψ ψ. Hence we have an eigenvector of a = + corresponding to a normalized eigenvector ψ + ψ and an eigenvalue a = corresponding to eigenvector ψ ψ.

2 6.. PROBLEM SET I 98 c The time-dependent Schrödinger equation is Ĥψ = Eψ = i tψ, hence ψt =ψt = 0e iet/. Since ψ and ψ are eigenstates of the Hamiltonian Ĥ then we can write, ψt = [ ψ e iet/ ψ e iet/ ]. P = ψt = 0 ψt = [ ψ ψ ][ ψ e iet/ ψ e iet/ ] 4 = [ + cose E t/] = cos E E t/ 3. a Differentiating the left hand side of the given expression with respect to β, one obtains e βa [a, a ] e βa =. }{{} = Integrating, we therefore have that e βa ae βa = β+ integration constant. By setting β = 0 we can deduce that the constant must be a yielding the required result. Using this result, we have that e βa a β = e } βa {{ ae βa } 0 =β + a 0 = β 0. =β+a Lastly, to obtain the normalization, we have that β β = N 0 e β a β = N β a n 0 β n! i.e. N = e β / as required. n=0 n=0 = N β β n 0 β = N e β! =. n! b For the harmonic oscillator, the creation and annihilation operators are related to the phase space operators by ˆx = mω a + a, and ˆp = mω i a a. Therefore, we have mω ˆx = mω β + β, ˆp = i β β. Then, using the identity x = x x = x x, we have x = mω β a + aa + a a +a β = mω + β + β, ˆp = mω β a aa a a +a β = mω +β β. As a result, we find that x = mω mω and p = leading to the required expression. c The equation follows simply from the definion of the operator a and the solution may be checked by substitution. d Using the time-evolution of the stationary states, nt = e ient/ n0, where E n = ωn +/, it follows that βt = e iωt/ e β / βn n! e inωt n = e iωt/ e iωt β. Therefore, during the time-evolution, the coherent state form is preserved but the centre of mass and momentum follow that of the classical oscillator, x 0 t =A cosϕ + ωt, p 0 t =mωa sinϕ + ωt. The width of the wavepacket remains constant.

3 6.. PROBLEM SET I a Starting with the electron Hamiltonian Ĥ = m ˆp qa, substitution of the expression for the vector potential leads to the required result. b Setting ω = m = =, we immediately obtained the required dimensionless form of the Hamiltonian. c Straightforward substitution of the differential operators leads to the required identities. As a result, we can confirm that [a, a ]= 4 [ z, z] [z, z ] =, and similarly [b, b ] =. In the complex coordinate representation, Ĥ = i z + z i 4 z z + z z + 4 z + z = z z + 8 zz + z z z z = z + z z + z = a a + d The angular momentum operator is given by = ˆL z =x ˆp z = ix y y x = i z + z z z z z z + z = z + z z + z + z + z z + z = a a b b e In the coordinate representation, the condition a 0, 0 = 0 translates to the equation z + z r 0, 0 =0 4 We thus obtain the Gaussian expression for the wavefunction. Then, using the relation 0,m = b m m! 0, 0, we have r 0,m = m/ z + z m e zz/4 = m! 4 π πm m! zm e zz/4, as required. f If we populate the states of the lowest Landau with electrons, starting from states of the lowest angular momentum m, the wavefunction is a Slater determinant involving entries φ m r j =zj m /4 e zj. Taking the determinant, and making use of the Vandemonde determinant identity, one obtains the required many-electron wavefunction. 5. The spin operator in the θ, φ direction, Ŝθφ, can be found by forming the scalar product of the spin operator Ŝ with a unit vector in the θ, φ direction, sin θ cos φ, sin θ sin φ, cos θ. Therefore Ŝ θφ = cos θ e iφ sin θ e iφ. sin θ cos θ We need the eigenvalues of the matrix, i.e. cos θ sin θe iφ u sin θe iφ cos θ v u = λ v. Eliminating u and v, we find λ = and hence the eigenvalues of Ŝθφ are ±/, as expected. Substituting the values λ = ± back into the equations relating u and v, we can infer the ratios, u v = e iφ cotθ/ and e iφ tanθ/. So, in matrix notation, the eigenstates are cosθ/ e iφ sinθ/ and sinθ/ e iφ cosθ/,

4 6.. PROBLEM SET I 00 for eigenvalues +/ and / respectively. The spin states in the x-direction are obtained by setting θ = π/, φ = 0, and the spin states in the y-direction are obtained by setting φ = π/4 in these general formulae. 6. The four states are given by φ = χ + χ +, φ = χ χ, φ 3 = [χ + χ +χ χ + ], and φ 4 = [χ + χ χ χ + ], where φ, φ and φ 3 are symmetric under particle interchange and φ 4 is antisymmetric. Since Ŝ+Ŝ = Ŝ x + Ŝ y + iŝyŝx ŜxŜy =Ŝ Ŝ z + Ŝz, Ŝ = Ŝ+Ŝ + Ŝ z Ŝz. 6. Noting that Ŝz = Ŝ z + Ŝ z, and likewise for Ŝ±, and using the basic results for the operation of the spin operators on the single particle states, Ŝ z χ ± = ± χ ±, Ŝ χ ± = χ, Ŝ ± χ ± = 0, we can now calculate the effect of applying Ŝ to each state. For example Ŝ φ = [χ + χ + + χ + χ + χ + χ + ] = φ, where the three terms in square brackets correspond to the three operators on the right hand side of Eq. 6.. Likewise φ. Clearly Ŝzφ 3 =0=Ŝzφ 4, so only the Ŝ+Ŝ term need be considered. We find Ŝ + Ŝ χ + χ = Ŝ+χ χ = χ + χ + χ χ + φ 3. Similarly Ŝ+Ŝ χ χ + = φ 3 so that Ŝ φ 3 = φ 3, Ŝ φ 4 =0. Thus, φ, φ, φ 3 all have eigenvalue and hence S =, while φ 4 has S = 0. The state ψ is clearly an eigenstate of Ŝz with eigenvalue 0, and must therefore be a linear combination of φ 3 and φ 4. The S = component is thus the φ 3 term in the wavefunction with amplitude c 3 = φ 3 ψ = 3 χ +χ + = /3+ /6 3 χ χ + The probability of S = is therefore c 3 = 3+ 6 =0.97. χ +χ + χ χ + 7. Taking as a basis the states of the Ŝz operator, we can deduce the form of the Ŝ x operator most easily from the action of the spin raising and lower operators. Noting that, for spin S =, Ŝ± S =,m = [ mm + ] /,m±, we can construct the matrix elements of Ŝ±. The latter are given by Ŝ + = , Ŝ = Ŝ +. Then, using the relation, Ŝx = Ŝ+ Ŝ, we obtained the matrix elements of the operator and corresponding eigenstates, Ŝ x = , m x = m x =0 m x = 0

5 6.. PROBLEM SET I 0 When placed in a magnetic field, B, the molecules will acquire an energy µbm z, where µ is the magnetic moment of the molecule, which in this case equals twice the magnetic moment of the proton, and m z is the eigenvalue of Ŝ z. At t = 0, the molecules enter the magnetic field in the m x = state, after which their wavefunction evolves with time in the usual way, i.e. ψt =e iµbŝzt/ ψ0 = e iµbt e iµbt = e iµbt e iµbt Thus, if µbt = n + π, with n an integer, the molecules will be in a pure m x = state, and none will pass the second filter. The time is given by t = L/v = L m/e, where L = 0 mm and m and E are the mass and energy of the molecules respectively. We thus have / n + π E µ = = JT BL m and hence the proton magnetic moment is JT. Note that the result can also be obtained by treating the problem as one of classical precession. The couple = µb = LΩ, where L = is the angular momentum and Ω the angular frequency of precession. If Ωt = n + π, the molecules have precessed into the m x = state, and the result readily follows.. 8. Substituting for the definition of the spin raising and lowering operators using the Holstein-Primakoff transformation, the commutator is obtained as S [Ŝ+, Ŝ ]= = a a S + a a a a S a a S a a + / {}}{ aa a a S a a + a a aa = a a S S. / a a a a S With Ŝz = S a a, we obtain the required commutation relation [Ŝ+, Ŝ ]= Ŝz. 9. For the case l =, l =, a table of possible ways of forming each value of M = m + m is shown right. The largest value of M is 3, so the largest value of L must be 3. There must also be a state with M = corresponding to L = 3, but we have two states with M =. Therefore there must be a state with L = as well. We need two states with M =, one for each of the L =3, multiplets, but we actually have three states with M =, so there must be an L = state as well. All of the M states are now accounted for. For the case l = 3, l =, we can again for a table see right. Following the same logic as before, we see that states with L =4, 3, just account for all the states. To construct the states explicitly, we start by writing the L =3M = 3 state, since there is only one way of forming M = 3, viz. 3, 3 =,,. We then operate with the lowering operator ˆL, which is simply the sum of the lowering operators for the two separate particles. Recalling that: ˆL l, m = ll + mm l, m, we obtain 6 3, =, 0, + 4,,, where the first term on the right hand side comes from lowering the l = state and the second from lowering the l = state. Hence 3, = /3, 0, + /3,,. The state, must be the orthogonal linear combination, i.e., = /3, 0, /3,,. Further states could be computed in the same way if required. M m,m 3,, 0,, 0 0,, 0, 0, 0,, 0,, 0 0,, 3, M m,m 4 3, 3 3, 0, 3,, 0,,, 0 0, 0, 0, 0, 0,, 0,,, 0 3, 3, 3, 0 4 3,

6 6.. PROBLEM SET I 0 0. The commutation relations are given by [J i,j j ]=iɛ ijk J k. a First define the eigenstates ψ m : J z ψ m = mψ m. To see if J ± ψ m is an eigenstate of J z, we need to look at J z J ± ψ m, which is equal to J ± J z ψ m [J ±,J z ]ψ m. The required commutator is [J ±,J z ] = [J x,j z ] ± i[j y,j z ], from the definition of J ±. From the basic commutators given at the start, this is [J ±,J z ]= ij y ± J x = ± J ± if treating ± like a number is confusing, do this separately for J + and J. Going back to J z J ± ψ m, we can now write this as J ± J z ψ m + ±J ± ψ m. The first term is just J ± mψ m, so this is m ± J ± ψ m. Thus, J ± ψ m is an eigenstate of J z, with eigenvalue m ±. This establishes the raising and lowering property of J ±. b Two electrons would have a total spin of S = or 0. Adding a third spin / particle creates total spin S =3/ or / from the S = two-particle state. The S = 0 two-particle state becomes S =/ only on adding the third particle, so total S =3/ or / are the only possibilities. c The states with well-defined values of m, m, and m 3 for the z spin components of all particles are the uncoupled basis. Where all particles are spin up, this state may be written as. This state is also the m S =3/ state of total S =3/ there is no other way to get m + m + m 3 =3/in the uncoupled basis. We can therefore write S =3/,m S = 3/ =. To get from here to S =3/,m S =/, we need to apply J = S + S + S 3. In other words, the total lowering operator is the sum of the lowering operator for each separate spin reasonably enough this follows from the definition of J and J x = S x + S x + S x 3, etc. Now, we need to use the given normalization result. This says that J S =3/,m S =3/ = 5/4 3/4 S =3/,m S =/ = 3 S =3/,m S =/. Notice that the total quantum number, J, is the same as the overall spin quantum number, S in this case. Therefore S = 3/,m S = / = / 3J S = 3/,m S = 3/. Using the given normalization result again for a single state, S /, / = 3/4+/4 /, /. This establishes the required result.

### Lecture 12. The harmonic oscillator

Lecture 12 The harmonic oscillator 107 108 LECTURE 12. THE HARMONIC OSCILLATOR 12.1 Introduction In this chapter, we are going to find explicitly the eigenfunctions and eigenvalues for the time-independent

### 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

### Solutions to chapter 4 problems

Chapter 9 Solutions to chapter 4 problems Solution to Exercise 47 For example, the x component of the angular momentum is defined as ˆL x ŷˆp z ẑ ˆp y The position and momentum observables are Hermitian;

### Angular Momentum - set 1

Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x = 0,

### Angular Momentum - set 1

Angular Momentum - set PH0 - QM II August 6, 07 First of all, let us practise evaluating commutators. Consider these as warm up problems. Problem : Show the following commutation relations ˆx, ˆL x ] =

### 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

### 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

### PHY413 Quantum Mechanics B Duration: 2 hours 30 minutes

BSc/MSci Examination by Course Unit Thursday nd May 4 : - :3 PHY43 Quantum Mechanics B Duration: hours 3 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO

### 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

### 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

### Sample Quantum Chemistry Exam 2 Solutions

Chemistry 46 Fall 7 Dr. Jean M. Standard Name SAMPE EXAM Sample Quantum Chemistry Exam Solutions.) ( points) Answer the following questions by selecting the correct answer from the choices provided. a.)

### 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

### 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

### 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

### 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

### Summary: angular momentum derivation

Summary: angular momentum derivation L = r p L x = yp z zp y, etc. [x, p y ] = 0, etc. (-) (-) (-3) Angular momentum commutation relations [L x, L y ] = i hl z (-4) [L i, L j ] = i hɛ ijk L k (-5) Levi-Civita

### 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

### 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

### Waves and the Schroedinger Equation

Waves and the Schroedinger Equation 5 april 010 1 The Wave Equation We have seen from previous discussions that the wave-particle duality of matter requires we describe entities through some wave-form

### Quantum mechanics II

jaroslav.hamrle@vsb.cz February 3, 2014 Outline 1 Introduction 2 Duality 3 Wavefunction 4 Free particle 5 Schödinger equation 6 Formalism of quantum mechanics 7 Angular momentum Non-relativistic description

### 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

### Quantum Mechanics Final Exam Solutions Fall 2015

171.303 Quantum Mechanics Final Exam Solutions Fall 015 Problem 1 (a) For convenience, let θ be a real number between 0 and π so that a sinθ and b cosθ, which is possible since a +b 1. Then the operator

### 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

### 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,

### Quantum Mechanics I Physics 5701

Quantum Mechanics I Physics 5701 Z. E. Meziani 02/23//2017 Physics 5701 Lecture Outline 1 General Formulation of Quantum Mechanics 2 Measurement of physical quantities and observables 3 Representations

### 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,

### Statistical 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

### CHM 532 Notes on Creation and Annihilation Operators

CHM 53 Notes on Creation an Annihilation Operators These notes provie the etails concerning the solution to the quantum harmonic oscillator problem using the algebraic metho iscusse in class. The operators

### The Postulates of Quantum Mechanics

p. 1/23 The Postulates of Quantum Mechanics We have reviewed the mathematics (complex linear algebra) necessary to understand quantum mechanics. We will now see how the physics of quantum mechanics fits

### 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

### 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

### Exam in TFY4205 Quantum Mechanics Saturday June 10, :00 13:00

NTNU Page 1 of 9 Institutt for fysikk Contact during the exam: Professor Arne Brataas Telephone: 7359367 Exam in TFY5 Quantum Mechanics Saturday June 1, 6 9: 13: Allowed help: Alternativ C Approved Calculator.

### Collection of formulae Quantum mechanics. Basic Formulas Division of Material Science Hans Weber. Operators

Basic Formulas 17-1-1 Division of Material Science Hans Weer The de Broglie wave length λ = h p The Schrödinger equation Hψr,t = i h t ψr,t Stationary states Hψr,t = Eψr,t Collection of formulae Quantum

### 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

### 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.

### Problem Set # 8 Solutions

Id: hw.tex,v 1.4 009/0/09 04:31:40 ike Exp 1 MIT.111/8.411/6.898/18.435 Quantum Information Science I Fall, 010 Sam Ocko November 15, 010 Problem Set # 8 Solutions 1. (a) The eigenvectors of S 1 S are

### Spin Dynamics Basic Theory Operators. Richard Green SBD Research Group Department of Chemistry

Spin Dynamics Basic Theory Operators Richard Green SBD Research Group Department of Chemistry Objective of this session Introduce you to operators used in quantum mechanics Achieve this by looking at:

### David J. Starling Penn State Hazleton PHYS 214

All the fifty years of conscious brooding have brought me no closer to answer the question, What are light quanta? Of course today every rascal thinks he knows the answer, but he is deluding himself. -Albert

### Fermionic Algebra and Fock Space

Fermionic Algebra and Fock Space Earlier in class we saw how harmonic-oscillator-like bosonic commutation relations [â α,â β ] = 0, [ ] â α,â β = 0, [ ] â α,â β = δ α,β (1) give rise to the bosonic Fock

### E = hν light = hc λ = ( J s)( m/s) m = ev J = ev

Problem The ionization potential tells us how much energy we need to use to remove an electron, so we know that any energy left afterwards will be the kinetic energy of the ejected electron. So first we

### 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

### 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

### 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

### Quantum Mechanics. L. Del Debbio, University of Edinburgh Version 0.9

Quantum Mechanics L. Del Debbio, University of Edinburgh Version 0.9 November 26, 2010 Contents 1 Quantum states 1 1.1 Introduction..................................... 2 1.1.1 Experiment with classical

### PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10: Solutions

PHYS851 Quantum Mechanics I, Fall 009 HOMEWORK ASSIGNMENT 10: Solutions Topics Covered: Tensor product spaces, change of coordinate system, general theory of angular momentum Some Key Concepts: Angular

### Qualification Exam: Quantum Mechanics

Qualification Exam: Quantum Mechanics Name:, QEID#76977605: October, 2017 Qualification Exam QEID#76977605 2 1 Undergraduate level Problem 1. 1983-Fall-QM-U-1 ID:QM-U-2 Consider two spin 1/2 particles

### Physics 443, Solutions to PS 2

. Griffiths.. Physics 443, Solutions to PS The raising and lowering operators are a ± mω ( iˆp + mωˆx) where ˆp and ˆx are momentum and position operators. Then ˆx mω (a + + a ) mω ˆp i (a + a ) The expectation

### 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

### Quantum Many Body Theory

Quantum Many Body Theory Victor Gurarie Week 4 4 Applications of the creation and annihilation operators 4.1 A tight binding model Consider bosonic particles which hops around on the lattice, for simplicity,

### 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

### 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

### 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

### 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

### LSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (07/2017) =!2 π 2 a cos π x

LSU Dept. of Physics and Astronomy Qualifying Exam Quantum Mechanics Question Bank (7/17) 1. For a particle trapped in the potential V(x) = for a x a and V(x) = otherwise, the ground state energy and eigenfunction

### Joint Entrance Examination for Postgraduate Courses in Physics EUF

Joint Entrance Examination for Postgraduate Courses in Physics EUF Second Semester 013 Part 1 3 April 013 Instructions: DO NOT WRITE YOUR NAME ON THE TEST. It should be identified only by your candidate

### 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

### 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

### 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)

### 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

### Physics 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,

### 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

### 4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam

Lecture Notes for Quantum Physics II & III 8.05 & 8.059 Academic Year 1996/1997 4. Supplementary Notes on Time and Space Evolution of a Neutrino Beam c D. Stelitano 1996 As an example of a two-state system

### I. CSFs Are Used to Express the Full N-Electron Wavefunction

Chapter 11 One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N- Electron Configuration Functions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon

### Physics 401: Quantum Mechanics I Chapter 4

Physics 401: Quantum Mechanics I Chapter 4 Are you here today? A. Yes B. No C. After than midterm? 3-D Schroedinger Equation The ground state energy of the particle in a 3D box is ( 1 2 +1 2 +1 2 ) π2

### Review of the Formalism of Quantum Mechanics

Review of the Formalism of Quantum Mechanics The postulates of quantum mechanics are often stated in textbooks. There are two main properties of physics upon which these postulates are based: 1)the probability

### 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

### Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition

### 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

### Further Quantum Mechanics Problem Set

CWPP 212 Further Quantum Mechanics Problem Set 1 Further Quantum Mechanics Christopher Palmer 212 Problem Set There are three problem sets, suitable for use at the end of Hilary Term, beginning of Trinity

### The Position Representation in Quantum Mechanics

The Position Representation in Quantum Mechanics Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 13, 2007 The x-coordinate of a particle is associated with

### 1 Position Representation of Quantum State Function

C/CS/Phys C191 Quantum Mechanics in a Nutshell II 10/09/07 Fall 2007 Lecture 13 1 Position Representation of Quantum State Function We will motivate this using the framework of measurements. Consider first

### 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.

### Quantum Mechanics in One Dimension. Solutions of Selected Problems

Chapter 6 Quantum Mechanics in One Dimension. Solutions of Selected Problems 6.1 Problem 6.13 (In the text book) A proton is confined to moving in a one-dimensional box of width.2 nm. (a) Find the lowest

### Lecture 20. Parity and Time Reversal

Lecture 20 Parity and Time Reversal November 15, 2009 Lecture 20 Time Translation ( ψ(t + ɛ) = Û[T (ɛ)] ψ(t) = I iɛ ) h Ĥ ψ(t) Ĥ is the generator of time translations. Û[T (ɛ)] = I iɛ h Ĥ [Ĥ, Ĥ] = 0 time

### PHYS-454 The position and momentum representations

PHYS-454 The position and momentum representations 1 Τhe continuous spectrum-a n So far we have seen problems where the involved operators have a discrete spectrum of eigenfunctions and eigenvalues.! n

### What is a Photon? Foundations of Quantum Field Theory. C. G. Torre

What is a Photon? Foundations of Quantum Field Theory C. G. Torre May 1, 2018 2 What is a Photon? Foundations of Quantum Field Theory Version 1.0 Copyright c 2018. Charles Torre, Utah State University.

### Lecture 5. Hartree-Fock Theory. WS2010/11: Introduction to Nuclear and Particle Physics

Lecture 5 Hartree-Fock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particle-number representation: General formalism The simplest starting point for a many-body state is a system of

### Lecture #9 Redfield theory of NMR relaxation

Lecture 9 Redfield theory of NMR relaxation Topics Redfield theory recap Relaxation supermatrix Dipolar coupling revisited Scalar relaxation of the st kind Handouts and Reading assignments van de Ven,

### Quantum Mechanics II

Quantum Mechanics II Prof. Boris Altshuler April, 20 Lecture 2. Scattering Theory Reviewed Remember what we have covered so far for scattering theory. We separated the Hamiltonian similar to the way in

### Harmonic oscillator - Vibration energy of molecules

Harmonic oscillator - Vibration energy of molecules The energy of a molecule is approximately the sum of the energies of translation of the electrons (kinetic energy), of inter-atomic vibration, of rotation

### Quantum Physics Lecture 9

Quantum Physics Lecture 9 Potential barriers and tunnelling Examples E < U o Scanning Tunelling Microscope E > U o Ramsauer-Townsend Effect Angular Momentum - Orbital - Spin Pauli exclusion principle potential

### Two and Three-Dimensional Systems

0 Two and Three-Dimensional Systems Separation of variables; degeneracy theorem; group of invariance of the two-dimensional isotropic oscillator. 0. Consider the Hamiltonian of a two-dimensional anisotropic

### Symmetries and Supersymmetries in Trapped Ion Hamiltonian Models

Proceedings of Institute of Mathematics of NAS of Ukraine 004, Vol. 50, Part, 569 57 Symmetries and Supersymmetries in Trapped Ion Hamiltonian Models Benedetto MILITELLO, Anatoly NIKITIN and Antonino MESSINA

### Non-relativistic scattering

Non-relativistic scattering Contents Scattering theory 2. Scattering amplitudes......................... 3.2 The Born approximation........................ 5 2 Virtual Particles 5 3 The Yukawa Potential

### If 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

### Physics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics

Physics 221A Fall 1996 Notes 12 Orbital Angular Momentum and Spherical Harmonics We now consider the spatial degrees of freedom of a particle moving in 3-dimensional space, which of course is an important

### Coherent states of the harmonic oscillator

Coherent states of the harmonic oscillator In these notes I will assume knowlege about the operator metho for the harmonic oscillator corresponing to sect..3 i Moern Quantum Mechanics by J.J. Sakurai.

### Chapter 12. Linear Molecules

Chapter 1. Linear Molecules Notes: Most of the material presented in this chapter is taken from Bunker and Jensen (1998), Chap. 17. 1.1 Rotational Degrees of Freedom For a linear molecule, it is customary

### Accelerator Physics Homework #7 P470 (Problems: 1-4)

Accelerator Physics Homework #7 P470 (Problems: -4) This exercise derives the linear transfer matrix for a skew quadrupole, where the magnetic field is B z = B 0 a z, B x = B 0 a x, B s = 0; with B 0 a

### 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

### Numerical Simulations of High-Dimensional Mode-Coupling Models in Molecular Dynamics

Dickinson College Dickinson Scholar Student Honors Theses By Year Student Honors Theses 5-22-2016 Numerical Simulations of High-Dimensional Mode-Coupling Models in Molecular Dynamics Kyle Lewis Liss Dickinson

### Multipole Expansion for Radiation;Vector Spherical Harmonics

Multipole Expansion for Radiation;Vector Spherical Harmonics Michael Dine Department of Physics University of California, Santa Cruz February 2013 We seek a more systematic treatment of the multipole expansion

### 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

### Lectures 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

### G : Quantum Mechanics II

G5.666: Quantum Mechanics II Notes for Lecture 7 I. A SIMPLE EXAMPLE OF ANGULAR MOMENTUM ADDITION Given two spin-/ angular momenta, S and S, we define S S S The problem is to find the eigenstates of the

### df(x) = h(x) dx Chemistry 4531 Mathematical Preliminaries Spring 2009 I. A Primer on Differential Equations Order of differential equation

Chemistry 4531 Mathematical Preliminaries Spring 009 I. A Primer on Differential Equations Order of differential equation Linearity of differential equation Partial vs. Ordinary Differential Equations

### Lecture 4: Resonant Scattering

Lecture 4: Resonant Scattering Sep 16, 2008 Fall 2008 8.513 Quantum Transport Analyticity properties of S-matrix Poles and zeros in a complex plane Isolated resonances; Breit-Wigner theory Quasi-stationary

### 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