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PHYSICS 851 - Quantum Mechanics - Michigan State University Study Resources
  • 4 Pages Evolution In Time Notes
    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 ` "

  • 3 Pages Lect28_TwoBodyProb
    Lect28_TwoBodyProb

    School: Michigan State University

    Course: Quantum Mechanics

    Lecture 28: The Quantum Two-body 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

  • 3 Pages Lect27_OrbAngMom
    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

  • 3 Pages Lect26_AngMom2
    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

  • 3 Pages Lect25_AngMom1
    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

  • 3 Pages Lect24_TensorProduct
    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 one-dimension In one dimension, X and P are incompatible

  • 3 Pages Lect23_HeisUncPrinc
    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

  • 3 Pages Lect22_CohStates
    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=

  • 3 Pages Lect4_BasisSet
    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

  • 3 Pages Lect3_Operators
    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+bA|2 a and b are arbitrary c-numbers No

  • 3 Pages Lect2_DiracNot
    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

  • 3 Pages Lect29_CentralPot
    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 3-dimensions It will go up to 5N when we include spin When does it work: All of the stand

  • 11 Pages Lect30_Hydrogen
    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

  • 3 Pages Electromagnetic Field Notes
    Electromagnetic Field Notes

    School: Michigan State University

    Course: Quantum Mechanics

    w' x!u w%' cu d t% Rj ' po m o n k w w mu k m vQru " "| &xy3xy&yy $ q w p u k qq q q w p sw p xYw p u u u j w u " eyE! &"EF&xy!#&F!&xyxxy! yEcfw_7g! xyys! &g!yyxryE9!xxrxysyye!cfw_yEAB xEx p w w p p n$ p r r s k s i Rj &

  • 5 Pages Adding Angular Momentum Notes
    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 !

  • 5 Pages Time-Dependent Interactions Notes
    Time-Dependent 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

  • 4 Pages Fermions Notes
    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

  • 3 Pages Wigner-Eckart Theorem Notes
    Wigner-Eckart 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

  • 1 Page syllabus
    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 Cohen-Tannoudji Webpage: http:/www.pa.msu.edu/mmoore/851.html Instructor: Prof. Michael Moore Oce

  • 1 Page syllabus (1)
    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 Cohen-Tannoudji Webpage: http:/www.pa.msu.edu/mmoore/851.html Instructor: Prof. Michael Moore Oce

  • 3 Pages math_tutorial
    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

  • 18 Pages Lect33_Spin
    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 Stern-Gerlach experiment (1922): Each type of particle has a discrete numbe

  • 11 Pages Lect31_DipoleMoments
    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 ) `Semi-Classica

  • 3 Pages Lect1_hbari
    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

  • 15 Pages 85109HW3Solutions
    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. m|A|m , where cfw_|m is a suitable basis m (a) Prove that the trace is independent

  • 8 Pages 851HW6_09Solutions
    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 free-particle of mass, M , at time t is a wave-packet 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

  • 2 Pages 851HW6_09
    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 free-particle of mass, M , at time t is a wave-packet 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

  • 9 Pages 851HW5_09Solutions
    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 two-level system onto the time-independent Rabi model. Here we will investigate how this

  • 3 Pages 851HW5_09 (1)
    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 two-level system onto the time-independent Rabi model. Here we will investigate how this

  • 7 Pages 851HW4_Solutions09
    851HW4_Solutions09

    School: Michigan State University

    Course: Quantum Mechanics

    PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 4: Solutions 1. The 2-Level 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,

  • 2 Pages 851HW4_09
    851HW4_09

    School: Michigan State University

    Course: Quantum Mechanics

    PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 4 1. The 2-Level 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

  • 1 Page 851HW3_09
    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. m|A|m , where cfw_|m is a suitable basis m (a) Prove that the trace is independent of the cho

  • 4 Pages 851HW2_09_Solutions
    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 A|n = an |n but that n |n = 1. Prove that |an = c|n is also an eigenstate of A. What is its eigenvalue? What should c be so that an |an =

  • 2 Pages 851HW2_09
    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 A|n = an |n but that n |n = 1. Prove that |an = c|n is also an eigenstate of A. What is its eigenvalue? What should c be so that an |an =

  • 3 Pages 851HW1_09_Solutions
    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 n|H |m in terms of m|H |n

  • 2 Pages 851HW7_09
    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 step-functions, transfer matrix approach to multi-boundary 1D scattering problems, nding bound-states for combinations of delta- and/or

  • 10 Pages 851HW7_09Solutions
    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 (t|a, b) = a dx | (x, t)|2 (1) i d d Dierentiate (1) and use the denition of th

  • 5 Pages 2008final
    2008final

    School: Michigan State University

    Course: Quantum Mechanics

    PHYS851 Quantum Mechanics I, Fall 2008 FINAL EXAM NAME: 1. Heisenberg Picture Consider a single-particle 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)

  • 3 Pages 851HW13_09
    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 spin-1/2 particles will gure heavily in both the nal exam for 851, as well as the QM subject exam. Spin-1/2: The Hilbert space of a spin-1/2

  • 1 Page 851HW12_09
    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

  • 9 Pages 851HW11_09Solutions
    851HW11_09Solutions

    School: Michigan State University

    Course: Quantum Mechanics

    PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 11 Topics Covered: Orbital angular momentum, center-of-mass coordinates Some Key Concepts: angular degrees of freedom, spherical harmonics 1. [20 pts] In order to derive the properties of the sphe

  • 2 Pages 851HW11_09
    851HW11_09

    School: Michigan State University

    Course: Quantum Mechanics

    PHYS851 Quantum Mechanics I, Fall 2009 HOMEWORK ASSIGNMENT 11 Topics Covered: Orbital angular momentum, center-of-mass coordinates Some Key Concepts: angular degrees of freedom, spherical harmonics 1. [20 pts] In order to derive the properties of the sphe

  • 2 Pages 851HW10_09
    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

  • 7 Pages 851HW9_09Solutions
    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 x|x and use it to simplify the identity = dx dx |x x|x x |, then use this identity to compute 2 , 3 , and n . From these

  • 1 Page 851HW9_09
    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,

  • 7 Pages 851HW8_09Solutions
    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

  • 2 Pages 851HW8_09
    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

  • 2 Pages 851HW1_09
    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 n|H |m i

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