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EE 315/ECE 451
N ANOELECTRONICS I
O UTLINE
2

3.1 General Postulates of QM

3.2 Time-Independent Schrödinger Equation

3.3 Analogies Between Quantum Mechanics and
Electromagnetics

3.4 Probabilistic Current Density

3.5 Multiple Particle Systems

3.6 Spin and Angular Momentum
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
W HERE
3
Start with
a plane wave
Classically
Quantize energy
with DeBroglie
Voila!
Schrödinger's
Equation
TO
B EGIN ?
 (x ,t )  Ae i (kx t )
i
 (x ,t ) 2
1  2 (x ,t )

,k  
 (x ,t ) t
 (x ,t ) x 2
1
p2
2
E  mv V 
V
2
2m
2
 i
 (x ,t )  k 
 
E    
V ,
t
2m
  (x ,t )

2

2m

1  2 (x ,t ) 
 
 V
2
  (x ,t ) x

 (x ,t )   2 2 (x ,t )
i
 
V
2
t
2
m

x

J. N. DENENBERG- FAIRFIELD UNIV. - EE315

 (x ,t )

8/11/2015
3.1 G ENERAL P OSTULATES
OF Q UANTUM M ECHANICS
4

POSTULATE 1 - To every quantum system there is
a state function, ψ(r,t), that contains everything
that can be known about the system

The state function, or wavefunction, is
probabilistic in nature.

Probability density of finding the particle at a
particular point in space, r, at time t is:
 (r ,t )  (r ,t )
2
2
P   (r ,t ) d 3r    * (r ,t )(r ,t )d 3r

J. N. DENENBERG- FAIRFIELD UNIV. - EE315

8/11/2015
P OSTULATE 1
5
 (r ,t )  (r ,t )
2
2
P   (r ,t ) d 3r    * (r ,t )(r ,t )d 3r


Normalization Factor
allspace
 * (r ,t )(r ,t )d 3r  1
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
P OSTULATE 2
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
A) Every physical observable O (position, momentum, energy,
etc.) is associated with a linear Hermitian operator ô

B) Associated with the operator ô is the eigenvalue problem, ô
ψn = λn ψn
such that the result of a measurement of an observable ô is one
of the eigenvalues λn of the operator

c) If a system is in the initial state ψ, measurement of O will
yield one of the eigenvalues λn of ô with probability
P (n )   (r ,t ) n (r )d r
3

2
And the system will change from ψ (an unknown state) to ψn.
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
3.1.1 O PERATORS
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
An operator maps one quantity to another

For example 3x2 matrices map 2x1 onto 3x1
matrices

The derivative operator maps sine onto cosine

The operator in that case would be d/dx
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
3.1.2 E IGENVALUES AND
E IGENFUNCTIONS
8

An eigenfunction of an operator is a function
such that when the operator acts on it we obtain
a multiple of the eigenfunction back

ô ψn = λn ψn

λn are the eigenvalues of the operator
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
3.1.3 H ERMITIAN
O PERATORS
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
A special class of operators.

They have real eigenvalues

Their eigenfunctions form an orthogonal,
complete set of functions
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
3.1.4 O PERATORS FOR QM
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
Momentum operator 
t

Energy -
Eˆ  i

Position
xˆ  x

Commutator

2 operators commute if
p̂  i

 i
x
[ˆ, ˆ]  (ˆˆ  ˆˆ)
[ˆ, ˆ]  0
which allows measurement to arbitrary precision
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
3.1.5 M EASUREMENT
P ROBABILITY
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
If a system is already in an eigenstate of the
operator we are interested in, we are guaranteed
100% to measure it in that state

However, if we do not know the state, we can
only find the probability of finding it in that state

Once measured however, we are “locked in” to
that state, and subsequent measurements will
return the same value
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
12
C OLLAPSE OF THE S TATE FUNCTION
- THE M EASUREMENT P ROBLEM

Postulate 2 says that any observable
measurement is associated with a Linear
Hermitian operator, and the result of every
measurement will be an eigenvalue of the
operator

After measurement (observation) the system will
be in an eigenstate until perturbed
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
P OSTULATE 3
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
The mean value of an observable is the
expectation value of the corresponding operator

f   f (x ) (x )dx




 (x )dx  1
2
For a QM system
 (r ,t )  (r ,t )   * (r ,t )(r ,t )

O    * (r ,t )oˆ(r ,t )d 3r


Position
x   * (r ,t )x(r ,t )d 3r
Momentum
p   * (r ,t ) i(r ,t )d 3r
Energy
E    * (r ,t ) i
J. N. DENENBERG- FAIRFIELD UNIV. - EE315








t

3
(r ,t )d r

8/11/2015
P OSTULATE 4
14

The state function ψ(r,t) obeys the Schrödinger
equation
i

 (r ,t )
 H (r ,t )
t
Where H is the Hamiltonian (total energy
operator), kinetic + potential energy (plus field
terms if necessary)
 2 2

H   
 V (r ,t ) 
 2m

J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
3.2 T IME -I NDEPENDENT
S CHRÖDINGER ' S E QUATION
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
If the potential energy does not depend on time
we can simplify considerably to the timeindependent Schrödinger Equation
 2 2

 
 V (r )  (r )  E (r )
 2m

J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015
16
Q UANTUM C ORRAL
An example of standing
electron waves
J. N. DENENBERG- FAIRFIELD UNIV. - EE315
8/11/2015