We aren't endorsed by this school
##### PHYSICS 851 - Quantum Mechanics - Michigan State University Study Resources
• 3 Pages
###### 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

• 3 Pages
###### 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

• 3 Pages
###### 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

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

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

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

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

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

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

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

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

• 4 Pages
• ###### Evolution In Time Notes
• Register Now
###### 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 ` "

• 11 Pages
###### 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
• ###### Electromagnetic Field Notes
• Register Now
###### 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
• Register Now

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
• Register Now
###### 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

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
• Register Now
###### 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

• 11 Pages
###### 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

• 1 Page
###### 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)

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

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

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

• 3 Pages
###### 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
###### 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

• 8 Pages
###### 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

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

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)

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

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

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

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

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

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

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

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

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

• 15 Pages
###### 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

• 5 Pages
###### 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)

• 1 Page
###### 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

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

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

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

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

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

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

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

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

Back to course listings