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PHYSICS 851  Quantum Mechanics  Michigan State University Study Resources
 Michigan State University
 M. Moore
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Evolution In Time Notes
School: Michigan State University
Course: Quantum Mechanics
' p tvd v $ p s V T V Q RT ` w e V G G f Q V QT R Q P cfw_!W&YSW5!o5HYFi!$&SS&a y s $ # p d p d !iSVWqYYF3` t!UY!ya y t&qW!&WYSoAt5&5` " T GT T " V f Q s G "RR V QT T e e ` V ` Q FR&V!eG!kGSRYV&WTYFih`3`!Q5!YG y x!3 HWYYF3 y HhHYWST x x !x7HFv9id Q ` "

Lect28_TwoBodyProb
School: Michigan State University
Course: Quantum Mechanics
Lecture 28: The Quantum Twobody Problem Phy851 Fall 2009 Two interacting particles Consider a system of two particles with no external fields By symmetry, the interaction energy can only depend on the separation distance: rr P12 P22 H= + + V R1 R2 2 m1

Lect27_OrbAngMom
School: Michigan State University
Course: Quantum Mechanics
Lecture 27: Orbital Angular Momentum Phy851 Fall 2009 The General Theory of Angular Momentum Starting point: Assume you have three operators that satisfy the commutation relations: [ J x , J y ] = ihJ z [ J y , J z ] = ihJ x [ J z , J x ] = ihJ y Let: 2

Lect26_AngMom2
School: Michigan State University
Course: Quantum Mechanics
Lecture 26: Angular Momentum II Phy851 Fall 2009 The Angular Momentum Operator The angular momentum operator is defined as: rrr L = R P It is a vector operator: r r r r L = Lx e x + L y e y + Lz e z According to the definition of the crossproduct, the

Lect25_AngMom1
School: Michigan State University
Course: Quantum Mechanics
Lecture 25: Introduction to the Quantum Theory of Angular Momentum Phy851 Fall 2009 Goals 1. Understand how to use different coordinate systems in QM (cartesian, spherical,) 2. Derive the quantum mechanical properties of Angular Momentum Use an algebraic

Lect24_TensorProduct
School: Michigan State University
Course: Quantum Mechanics
Lecture 24: Tensor Product States Phy851 Fall 2009 Basis sets for a particle in 3D Clearly the Hilbert space of a particle in three dimensions is not the same as the Hilbert space for a particle in onedimension In one dimension, X and P are incompatible

Lect23_HeisUncPrinc
School: Michigan State University
Course: Quantum Mechanics
Lecture 23: Heisenberg Uncertainty Principle Phy851 Fall 2009 Heisenberg Uncertainty Relation Most of us are familiar with the Heisenberg Uncertainty relation between position and momentum: h x p 2 How do we know this is true? Are the similar relations

Lect22_CohStates
School: Michigan State University
Course: Quantum Mechanics
Lecture 22: Coherent States Phy851 Fall 2009 Summary Properties of the QM SHO: memorize P2 1 H= + m 2 X 2 2m 2 A= = h m 1 X 1 X i P + i P A = h h 2 2 (A + A ) P = i h A A X= 2 2 ( ) 1 H = h A A + 2 H n = h (n + 1 / 2) n A n = n n 1 A n = n + 1 n + 1 n=

Lect4_BasisSet
School: Michigan State University
Course: Quantum Mechanics
Hermitian Operators Definition: an operator is said to be Hermitian if it satisfies: A=A Alternatively called self adjoint In QM we will see that all observable properties must be represented by Hermitian operators Theorem: all eigenvalues of a Hermitia

Lect3_Operators
School: Michigan State University
Course: Quantum Mechanics
Operators In QM, an operator is an object that acts on a ket, transforming it into another ket Let A represent a generic operator An operator is a linear map A:HH A=  Operators are linear: A(a 1+b 2) = aA 1+bA2 a and b are arbitrary cnumbers No

Lect2_DiracNot
School: Michigan State University
Course: Quantum Mechanics
Lecture I: Dirac Notation To describe a physical system, QM assigns a complex number (`amplitude) to each distinct available physical state. (Or alternately: two real numbers) What is a `distinct physical state? Consider a system with M distinct availab

Lect29_CentralPot
School: Michigan State University
Course: Quantum Mechanics
Lecture 29: Motion in a Central Potential Phy851 Fall 2009 Side Remarks Counting quantum numbers: 3N quantum numbers to specify a basis state for N particles in 3dimensions It will go up to 5N when we include spin When does it work: All of the stand

Lect30_Hydrogen
School: Michigan State University
Course: Quantum Mechanics
Lecture 30: The Hydrogen Atom Phy851 Fall 2009 Example 2: Hydrogen Atom The Hamiltonian for a system consisting of an electron and a proton is: Pp2 Pe2 e2 H= + rr 2me 2m p 4 0 Re R p In COM and relative coordinates, the Hamiltonian is separable: H = H C

Electromagnetic Field Notes
School: Michigan State University
Course: Quantum Mechanics
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Adding Angular Momentum Notes
School: Michigan State University
Course: Quantum Mechanics
k 9g i bf n w h " m p pf " p e e b w up g#cuXtl" f xq Ggtgr0! p w n `f ` ` h d p w ! b p $ !f b h df h p " `f b " b e `f u i 0gtsguic gttcfw_g(r'4q (#yugitsrsqu#rgt" ! h " f ` f e f b " b b f p $h ` w bf b e w `f ` e m w u ` " n w b ` wf `f ! p w n w !

TimeDependent Interactions Notes
School: Michigan State University
Course: Quantum Mechanics
#%~ ) xl%~ U x x d gt t r ! r dt m " q $ g 90hhpU#s&x! k ~ ~ x~ x~ 95~ ' ~ ~ k x ' ~ U ~ k x 0#" drxdd#gIfQq9gh9YqhhRtfAd9pthR#"hhUtAp&90990phs d g g d dt d gt i g dt t ' g ~ x hRd x~ ' oj I xeI uu ' fx r " d "t d g it q " d t g ##r09pQ#hdh xe d gt t

Fermions Notes
School: Michigan State University
Course: Quantum Mechanics
% ~ w R2PBw)!6rI22#$" 2PPtlP(Uv2Ua112Pcap)aaXvXa2P)sPI I i I e I i ! i x R "W I iT " R I i i Y R G T h hT R I i ! i I h # G R iT R R WW e h I R R " e W !T !TW V I T G i h " R G T e I " R i IT I I PI2UaPPaaia)U1"rR1"aRuSUUXU2PUPQav$vgf2UaaXT bcfw_ k x i I

WignerEckart Theorem Notes
School: Michigan State University
Course: Quantum Mechanics
& r q g H3 H3 H3 Gr 3@kq g x " x y#G v x i3# x " z x " " z iiytyw!qbDiRvy#tyxs v y#" #vqy00y#" 3 x x " z z x " x wyi0x3D80q001 v #" 1'#" v 'x# v #y3siX x z x " $ y ! x ! x x " " " x z x 9 Q 7 5 t 5 5 U B Q I P1e8eR)t @P@V@VU 5 a Q f 7U Y 7 r q & 3 % Rq 0P

Syllabus
School: Michigan State University
Course: Quantum Mechanics
PHYS 851 Quantum Mechanics I Fall Quarter 2008 Class Hours: MWF 10:20 AM to 11:10 AM Class Location: 1415 BPS Textbook: Quantum Mechanics, Volume One, Claude CohenTannoudji Webpage: http:/www.pa.msu.edu/mmoore/851.html Instructor: Prof. Michael Moore Oce

Syllabus (1)
School: Michigan State University
Course: Quantum Mechanics
PHYS 851 Quantum Mechanics I Fall Quarter 2009 Class Hours: MWF 10:20 AM to 11:10 AM Class Location: 1420 BPS Textbook: Quantum Mechanics, Volume One, Claude CohenTannoudji Webpage: http:/www.pa.msu.edu/mmoore/851.html Instructor: Prof. Michael Moore Oce

Math_tutorial
School: Michigan State University
Course: Quantum Mechanics
math_tutorial.nb H* This is how you make a comment in Mathematica *L 2 + 2 H* @shiftD + @enterD *L 4 1 In[2]:= In[3]:= In[4]:= Out[4]= In[5]:= H* In mathematica , you type a formula, then hit @shiftD+@enterD to have mathematica evaluate the expression *L

Lect33_Spin
School: Michigan State University
Course: Quantum Mechanics
Lecture 33: Quantum Mechanical Spin Phy851 Fall 2009 Intrinsic Spin Empirically, we have found that most particles have an additional internal degree of freedom, called spin The SternGerlach experiment (1922): Each type of particle has a discrete numbe

Lect31_DipoleMoments
School: Michigan State University
Course: Quantum Mechanics
Lecture 31: The Hydrogen Atom 2: Dipole Moments Phy851 Fall 2009 Electric Dipole Approximation The interaction between a hydrogen atom and an electric field is given to leading order by the Electric Dipole approximation: rr VE = D E (rCM ) `SemiClassica

Lect1_hbari
School: Michigan State University
Course: Quantum Mechanics
Lecture 1: Demystifying h and i We are often told that the presence of h distinguishes quantum from classical theories. One of the striking features of Schrdinger's equation is the fact that the variable, , is complex, whereas classical theories deal with

85109HW3Solutions
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 3: Solutions Fundamentals of Quantum Mechanics 1. [10pts] The trace of an operator is dened as T r cfw_A = set. mAm , where cfw_m is a suitable basis m (a) Prove that the trace is independent

851HW6_09Solutions
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 6 1. [10 points] The quantum state of a freeparticle of mass, M , at time t is a wavepacket of the form (x, t) = 1 e (5/4)0 ( x x 0 ) 4 +ip0 x/ 2 4 0 , We can safely predict that the width of t

851HW6_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 6 1. [10 points] The quantum state of a freeparticle of mass, M , at time t is a wavepacket of the form (x, t) = 1 (5/4)0 e ( x x 0 ) 4 +ip0 x/ 2 4 0 , We can safely predict that the width of th

851HW5_09Solutions
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 5 1. In problem 4.3, we used a change of variables to map the equations of motion for a sinusoidally driven twolevel system onto the timeindependent Rabi model. Here we will investigate how this

851HW5_09 (1)
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 5 1. In problem 4.3, we used a change of variables to map the equations of motion for a sinusoidally driven twolevel system onto the timeindependent Rabi model. Here we will investigate how this

851HW4_Solutions09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 4: Solutions 1. The 2Level Rabi Model: The standard Rabi Model consists of a bare Hamiltonian H0 = and a coupling term V = 1 2 + 2 1. 2 2 2 (2 2 1 1) (a) What is the energy, degeneracy,

851HW4_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 4 1. The 2Level Rabi Model: The standard Rabi Model consists of a bare Hamiltonian H0 = and a coupling term V = 1 2 + 2 1. 2 2 2 (2 2 1 1) (a) What is the energy, degeneracy, and state v

851HW3_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 3: Fundamentals of Quantum Mechanics 1. [10pts] The trace of an operator is dened as T r cfw_A = set. mAm , where cfw_m is a suitable basis m (a) Prove that the trace is independent of the cho

851HW2_09_Solutions
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 2: Postulates of Quantum Mechanics 1. [10 pts] Assume that An = an n but that n n = 1. Prove that an = cn is also an eigenstate of A. What is its eigenvalue? What should c be so that an an =

851HW2_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2007 HOMEWORK ASSIGNMENT 2: Postulates of Quantum Mechanics 1. [10 pts] Assume that An = an n but that n n = 1. Prove that an = cn is also an eigenstate of A. What is its eigenvalue? What should c be so that an an =

851HW1_09_Solutions
School: Michigan State University
Course: Quantum Mechanics
HOMEWORK ASSIGNMENT 1 PHYS851 Quantum Mechanics I, Fall 2009 1. [10 pts]What is the relationship between  and  ? What is the relationship between the matrix elements of M and the matrix elements of M . Assume that H = H what is nH m in terms of mH n

851HW7_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 7 Topics Covered: 1D scattering problems with delta and/or stepfunctions, transfer matrix approach to multiboundary 1D scattering problems, nding boundstates for combinations of delta and/or

851HW7_09Solutions
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 7 1. The continuity equation: The probability that a particle of mass m lies on the interval [a, b] at time t is b P (ta, b) = a dx  (x, t)2 (1) i d d Dierentiate (1) and use the denition of th

2008final
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2008 FINAL EXAM NAME: 1. Heisenberg Picture Consider a singleparticle system described by the Hamiltonian H = i (A A ), where 1 A= 2 1 X +i P so that [A, A ] = 1. (a) Derive the Heisenberg equations of motion for AH (t)

851HW13_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 13 Topics Covered: Spin Please note that the physics of spin1/2 particles will gure heavily in both the nal exam for 851, as well as the QM subject exam. Spin1/2: The Hilbert space of a spin1/2

851HW12_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 12 Topics Covered: Motion in a central potential, spherical harmonic oscillator, hydrogen atom, orbital electric and magnetic dipole moments 1. [20 pts] A particle of mass M and charge q is constr

851HW11_09Solutions
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 11 Topics Covered: Orbital angular momentum, centerofmass coordinates Some Key Concepts: angular degrees of freedom, spherical harmonics 1. [20 pts] In order to derive the properties of the sphe

851HW11_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 11 Topics Covered: Orbital angular momentum, centerofmass coordinates Some Key Concepts: angular degrees of freedom, spherical harmonics 1. [20 pts] In order to derive the properties of the sphe

851HW10_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 10 Topics Covered: Tensor product spaces, change of coordinate system, general theory of angular momentum Some Key Concepts: Angular momentum: commutation relations, raising and lowering operators

851HW9_09Solutions
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 9: SOLUTIONS 1. The Parity Operator: [20 pts] Determine the matrix element xx and use it to simplify the identity = dx dx x xx x , then use this identity to compute 2 , 3 , and n . From these

851HW9_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 9 Topics Covered: parity operator, coherent states, tensor product spaces. Some Key Concepts: unitary transformations, even/odd functions, creation/annihilation operators, displaced vacuum states,

851HW8_09Solutions
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 8: SOLUTIONS Topics Covered: Algebraic approach to the quantized harmonic oscillator, coherent states. Some Key Concepts: Oscillator length, creation and annihilation operators, the phonon number

851HW8_09
School: Michigan State University
Course: Quantum Mechanics
PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 8 Topics Covered: Algebraic approach to the quantized harmonic oscillator, coherent states. Some Key Concepts: Oscillator length, creation and annihilation operators, the phonon number operator. 1

851HW1_09
School: Michigan State University
Course: Quantum Mechanics
HOMEWORK ASSIGNMENT 1: Due Monday, 9/14/09 PHYS851 Quantum Mechanics I, Fall 2009 1. What is the relationship between  and  ? What is the relationship between the matrix elements of M and the matrix elements of M ? Assuming that H = H , what is nH m i