Lecture 3 Ece162b W09

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    Last ClassProperties of BondsBond TypesIonic bondsMetallic bondsCovalent bondVan der Waals bondMixed bonds

    Lecture 2, Slide 2ECE162B, Winter 2009, Professor Blumenthal

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    Lecture 3, Slide 3ECE162B, Winter 2009, Professor Blumenthal

    Continue Chapter 5 on Bonds Coupled Mode TheoryTake two oscillating systems each described

    by their own physical equation of motion.Now physically couple them togetherCoupled mode theory lets us re-write the

    equations of motion to include the coupling

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    Example: Pendulums

    Lecture 3, Slide 4ECE162B, Winter 2009, Professor Blumenthal

    Uncoupled Coupled

    Displacement

    Displacement

    Time

    Time

    P1

    P2

    P1 P2 P1 P2

    D

    isplacement

    Displacement

    Time

    Time

    P1

    P2

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    Feynmans Coupled Mode ApproachRichard Feynman, Physicist, Nobel Laureate (see

    Feynman Lectures on Physics, Addison Wesley

    Longman, ISBN-13: 978-0201021158) Developed an approach that can be used to describe all physical

    systems that are coupled Sometimes called coupled mode theory Other examples besides pendulums include electrical oscillators,

    atoms, acoustics and vibrational systems, etc. Step 1: Write a model for each part of the system by itself Step 2: Introduce a coupling factor in each equation that describes how

    each part of the system interacts with the other parts of the system This couple can be in time, space, time and space or other manner

    Lecture 3, Slide 5ECE162B, Winter 2009, Professor Blumenthal

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    Schrodingers EquationDescribes how a particle of mass (m) evolves

    in time (t) and space ( ) in a spatial potential

    (V). All possible solutions of motion to this

    evolution in space and time are given by any( ) that satisfies this equation for a particular m

    and V.

    Lecture 3, Slide 6ECE162B, Winter 2009, Professor Blumenthal

    h2

    2m2 +V

    = ih

    t

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    Schrodingers EquationLumping all spatial components ( and V) into

    one term, called an operator, we can define the

    Hamiltonian (H) and simplify the equation.

    Note the H operates on the spatial part of

    Lecture 3, Slide 7ECE162B, Winter 2009, Professor Blumenthal

    H = ih

    t

    H= h2

    2m2 +V

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    Schrodingers Equation This a partial differential equation (PDE). We have learned to solve these

    many times before. We guess at a viable solution. In this case, since there

    is both space and time, we will guess that the solution has both space and

    time in it, and further more, we will assume the solution is separable (i.e.

    made up of two functions, one only a function of space and the other only

    a function of time).

    Further more, we will use the concept similar to Fourier series, where anyfunction can be represented by the correct combination of finite or infinite

    sum of individual functions (e.g. harmonics). Any solution (function) thatwe are interested in can be composed of the linear sum of a set of

    separable functions.

    Lecture 3, Slide 8ECE162B, Winter 2009, Professor Blumenthal

    =(t)(r)

    = j(t)

    j(r)

    j

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    Schrodingers Equation For now lets consider only time dependence of the

    Schrodinger Equation described by (t). The following steps are a technique to eliminate the spatial

    variation in the PDE. Using the Hamiltonian operator and factthat j is not dependent on time so is constant w.r.t time

    Multiplying both sides by k and integrating over the volumeelement d (note that time only functions come outside spatial

    integrals)

    Lecture 3, Slide 9ECE162B, Winter 2009, Professor Blumenthal

    H = H j(t)

    j(r)

    j

    = ih

    t= ih

    j

    j(t)( )

    tj

    j(t) kHjd= j

    ih jd

    j(t)( )

    dtj kjd

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    Schrodingers Equation Now there is an important relationship between the two

    different solutions j and k that each satisfy the SE. They are

    orthogonal to each other and satisfy the following relationship

    Assuming that Ckj =1 for now, and defining Hkj as follows, wecan simplify the differential equation for each value of k

    Lecture 3, Slide 10ECE162B, Winter 2009, Professor Blumenthal

    kjd = Ckj ifk= j

    0 ifk j

    Hkj = kHjd

    ihdk(t)( )

    dt= H

    kjj j(t)

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    Lets consider the simplest case where there aretwo solutionsj=1 andj=2.

    Our two Diff. equations will look like

    Coupling

    Term

    Coupling

    Term

    Coupled Schrodingers Equations

    Lecture 3, Slide 11ECE162B, Winter 2009, Professor Blumenthal

    ihd1(t)( )

    dt= H111(t) + H122(t)

    ih d2(t)( )dt

    = H211(t) + H222(t)

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    If the two equations are not coupled (e.g. no spring in thependulum example) then H12=H21=0 and

    Which each have solutions

    We define the probability of being in a state (1 or 2) at time tas given by

    Uncoupled Solution

    Lecture 3, Slide 12ECE162B, Winter 2009, Professor Blumenthal

    ihd1(t)( )

    dt= H111(t)

    ihd2(t)( )

    dt= H222(t)

    Prob(State 1) = 1(t)2= K1

    2

    Prob(State 2) = 2(t)2= K2

    2

    1(t) =K1e i

    H11

    ht

    2(t) =K2ei

    H22

    ht

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    This solution tells us whatever state the system is in at acertain time, it will stay in that state. If nothing is done to the

    system from the outside, once in state 1 it stays in state 1 or

    once in state 2 it stays in state 2. An example of uncoupled states for a hydrogen atom and

    Proton in a system is illustrated below

    Uncoupled Solution

    Lecture 3, Slide 13ECE162B, Winter 2009, Professor Blumenthal

    Quantum

    Mechanically it is as if

    there is an infinite

    energy barrier between

    the two in each state

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    Now, lets assume there is not an infinite barrier, that thecloser the H atom and nucleus get, the more there is a

    probability that the electron can jump from on nucleus to the

    other, moving the system from state 1 to state 2 or visa versa. Our coupled differential equations must contain the coupling

    terms H12 and H21. We are going to introduce a new term weak coupling. This

    means that although the system can move between states, it

    can also stay in one state. That way we can talk about the

    states as individual independent states that can vary with time.

    Coupled States

    Lecture 3, Slide 14ECE162B, Winter 2009, Professor Blumenthal

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    Reintroducing the coupling terms and defining constants E0andA and assuming the coupling is symmetric

    Then we can write the coupled differential equations in matrixformulation

    Coupled States

    H11

    = H22

    = E0

    H21

    = H12

    = A

    ihd 1(t)( )

    dt= E

    01(t) A2(t)

    ihd 2(t)( )

    dt= A1(t) + E02(t)

    ihd

    dt

    1(t)

    2(t)

    =

    E0

    A

    A E0

    1

    (t)

    2(t)

    Lecture 3, Slide 15ECE162B, Winter 2009, Professor Blumenthal

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    Coupled States Substituting our typical solutions

    Which from Linear Algebra Theory has a solution only if thematrix determinant is equal to zero

    Lecture 3, Slide 16ECE162B, Winter 2009, Professor Blumenthal

    1(t) = K1e i

    E

    ht

    and 2(t) = K2ei

    E

    ht

    K1E = E

    0K

    1 AK

    2

    K2E = AK1 + E0K2

    EK

    1

    K2

    =

    E0

    A

    A E0

    K1

    K2

    E0 E A

    A E0 E

    = 0

    E0 E( )

    2

    = A2

    E= E0 A

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    Coupled States As the two are brought together from infinity, only one solution has

    attractive and repulsive forces balance out to a stable solution. In the time dynamic solution, the electron can jump from one nucleus to

    the other and a sharing can occur forming a bond.

    Lecture 3, Slide 17ECE162B, Winter 2009, Professor Blumenthal

    E(r)

    E0 -A

    E0 +AOnlystable

    minimum

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    Coupled StatesIf there is no coupling, E=E0If there is coupling, the energy level is split

    into two new levels E0+A and E0-A

    Lecture 3, Slide 18ECE162B, Winter 2009, Professor Blumenthal

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    Coupled Interaction

    Lecture 3, Slide 19ECE162B, Winter 2009, Professor Blumenthal

    Quantum

    Mechanically it is as if

    there is an finite

    energy barrier between

    the two in each state.

    But it is a balanced

    system, so there will

    be a stable solution as

    shown in the energy

    minimum in theprevious slide.