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PHYSICS 851  Quantum Mechanics  Michigan State University Study Resources
 Michigan State University (MSU)
 M. Moore

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

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

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

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 ` "

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

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
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TimeDependent Interactions Notes
School: Michigan State University
Course: Quantum Mechanics
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Fermions Notes
School: Michigan State University
Course: Quantum Mechanics
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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

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

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

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

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

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,