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8/2/2019 Lecture 3 Ece162b W09
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8/2/2019 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.