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Chap 2. Introduction to Quantum Mechanics
 Principles of Quantum Mechanics
 Schrödinger’s Wave Equation
 Application of Schrödinger’s Wave Equation
 Homework
Solid-State Electronics
Chap. 2
1
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Introduction
 In solids, there are about 1023 electrons and ions packed in
a volume of 1 cm3. The consequences of this highly
packing density :
– Interparticle distance is very small: ~2x10-8 cm.
the instantaneous position and velocity of the particle are no longer
deterministic. Thus, the electrons motion in solids must be
analyzed by a probability theory.
Quantum mechanics Newtonian mechanics
Schrodinger’s equation: to describe the position probability of a
particle.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Introduction
– The force acting on the j-th particle comes from all the other 1023-1
particles.
– The rate of collision between particles is very high, 1013
collisions/sec
average electron motion instead of the motion of each electron at a
given instance of time are interested. (Statistical Mechanics)
equilibrium statistical mechanics:
Fermi-Dirac quantum-distribution Boltzmann classical distribution
Solid-State Electronics
Chap. 2
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Principles of Quantum Mechanics
 Principle of energy quanta
 Wave-Particle duality principle
 Uncertainty principle
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Energy Quanta
 Consider a light incident on a surface of a material as shown below:
 Classical theory: as long as the intensity of light is strong enough
photoelectrons will be emitted from the material.
 Photoelectric Effect: experimental results shows “NOT”.
 Observation:
– as the frequency of incident light  < o: no electron emitted.
– as  > o:at const. frequency, intensity, emission rate, K.E. unchanged.
at const. intensity, the max. K. E.  the frequency of incident light.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Quanta and Photon
 Planck postulated that thermal radiation is emitted from a heated
surface in discrete energy called quanta. The energy of these quanta is
given by
E = h, h = 6.625 x 10-34 J-sec (Planck’s constant)
 According to the photoelectric results, Einstein suggested that the
energy in a light wave is also contained in discrete packets called
photon whose energy is also given by E = h.
The maximum K.E. of the photoelectron is Tmax = ½mv2 = h - ho
 The momentum of a photon, p = h/
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Wave-Particle Duality
 de Broglie postulated the existence of matter waves. He suggested that
since waves exhibit particle-like behavior, then particles should be
expected to show wave-like properties.
 de Broglie suggested that the wavelength of a particle is expressed as
 = h /p, where p is the momentum of a particle
 Davisson-Germer experimentally proved de Broglie postulation of
“Wave Nature of Electrons”.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Davisson-Germer Experiment
 Consider the experimental setup below:
 Observation:
– the existence of a peak in the density of scattered electrons can be
explained as a constructive interference of waves scattered by the periodic
atoms.
– the angular distribution of the deflected electrons is very similar to an
interference pattern produced by light diffracted from a grating.
Solid-State Electronics
Chap. 2
8
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Conclusion
 In some cases, EM wave behaves like particles (photons) and
sometimes particles behave as if they are waves.
Wave-particle duality principle applies primarily to SMALL particles,
e.g., electrons, protons, neutrons.
For large particles, classical mechanics still apply.
Solid-State Electronics
Chap. 2
9
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Uncertainty Principle
 Heisenberg states that we cannot describe with absolute accuracy the
behavior of the subatomic particles.
1. It is impossible to simultaneously describe with the absolute accuracy
the position and momentum of a particle.
p x  ħ. (ħ = h/2 = 1.054x10-34 J-sec)
2. It is impossible to simultaneously describe with the absolute accuracy
the energy of a particle and the instant of time the particle has this
energy.
E t  ħ
 The uncertainty principle implies that these simultaneous
measurements are in error to a certain extent. However, ħ is very small,
the uncertainty principle is only significant for small particles.
Solid-State Electronics
Chap. 2
10
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Schrodinger’s Wave Equation
 Based on the principle of quanta and the wave-particle duality
principle, Schrodinger’s equation describes the motion of electrons in a
crystal.
 1-D Schrodinger’s equation,
  2  2  ( x, t )
 ( x, t )


V
(
x
)

(
x
,
t
)

j

2m
x 2
t
 Where (x,t) is the wave function, which is used to describe the
behavior of the system, and mathematically can be a complex quantity.
 V(x) is the potential function.
 Assume the wave function (x,t) = (x)(t), then the Schrodinger eq.
Becomes
 2
 2 ( x)
 (t )
2m
Solid-State Electronics
Chap. 2
 (t )
x 2
 V ( x) ( x) (t )  j ( x)
11
t
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Schrodinger’s Wave Equation
  2 1  2 ( x)
1  (t )

V
(
x
)

j

E
2
2m  ( x) x
 (t ) t
 where E is the total energy, and the solution of the eq. is
and the time-indep. Schrodinger equation can be written as  (t )  e  j ( E /  )t
 2 ( x) 2m
 2 ( E  V ( x)) ( x)  0
2
x

 The physical meaning of wave function:
– (x,t) is a complex function, so it can not by itself represent a real
physical quantity.
– |2(x,t)| is the probability of finding the particle between x and
x+dx at a given time, or is a probability density function.
– |2(x,t)|= (x,t) *(x,t) =(x)* (x) = |(x)|2 -- indep. of time
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Boundary Conditions


2
 ( x) dx  1
since |(x)|2 represents the probability density function, then for a
single particle, the probability of finding the particle somewhere is
certain.
If the total energy E and the potential V(x) are finite everywhere,
2. (x) must be finite, single-valued, and continuous.
3. (x)/x must be finite, single-valued, and continuous.
1.

Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Applications of Schrodinger’s Eq.
 The infinite Potential Well
 In region I, III, (x) = 0, since E is finite and a particle cannot
penetrate the infinite potential barriers.
 In region II, the particle is contained within a finite region of space and
V = 0. 1-D time-indep. Schrodinger’s eq. becomes
 2 ( x) 2mE
 2  ( x)  0
x 2

 the solution is given by
 ( x)  A1 cos Kx  A2 sin Kx, where K 
Solid-State Electronics
Chap. 2
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2mE
2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Infinite Potential Well
 Boundary conditions:
1. (x) must be continuous, so that (x = 0) = (x = a) = 0
A1 = A2sinKa  0  K = n/a, where n is a positive integer.
2.

2
a
2
2
2

(
x
)
dx

1

A
sin
Kxdx

1

A


2
 2
a
0
So the time-indep. Wave equation is given by
 ( x) 
2
nx
sin(
) where n  1,2,3...
a
a
 The solution represents the electron in the infinite potential well is in a
standing waveform. The parameter K is related to the total energy E,
therefore,
 2 n 2 2
E  En 
Solid-State Electronics
Chap. 2
2ma2
where n is a positive integer
15
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Infinite Potential Well
 That means that the energy of the particle in the infinite potential well
is “quantized”. That is, the energy of the particle can only have
particular discrete values.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
The Step Potential Function
 Consider a particle being incident on a step potential barrier:
 In region I, V = 0,
 And the general solution of this equation is
 2 1 ( x) 2mE
 2  1 ( x)  0
2
x

 1 ( x)  A1e jK x  B1e jK x ( x  0) where K1 
1
1
2mE
2
 In region II, V = Vo, if we assume E < Vo, then
 2 2 ( x)
x
Solid-State Electronics
Chap. 2
2

2m

2
(Vo  E ) 2 ( x)  0
17
Instructor: Pei-Wen Li
Dept. of E. E. NCU
The Step Potential Function
 The general solution is in the form
 2 ( x)  A2e  K 2 x  B2e  K 2 x ( x  0) where K 2 
 Boundary Conditions:
2m(Vo  E )
2
 2 ( x)  A2e  K 2 x ( x  0)
– 2(x) must remain finite, B2  0 
– (x) must be continuous, i.e., 1(x = 0) = 2(x = 0) A1+B1 = A2
– (x)/ x must be continuous, i.e.,  1
x

x 0
 21
x
 jK1 A1  jK1B1  K2 A2
x 0
 A1, B1, and A2 could be solved from the above equations.
Solid-State Electronics
Chap. 2
18
Instructor: Pei-Wen Li
Dept. of E. E. NCU
The Potential Barrier
 Consider the potential barrier function as shown:
 Assume the total energy of an incident particle
E < Vo, as before, we could solve the
Schrodinger’s equations in each region, and obtain
 1 ( x)  A1e jK x  B1e jK x
1
1
 2 ( x)  A2e K 2 x  B2e  K 2 x
where K1 
 3 ( x)  A3e jK x  B3e  jK x
1
1
2m(Vo  E )
2mE
and
K

2
2
2
 We can solve B1, A2, B2, and A3 in terms of A1 from boundary
conditions:
– B3 = 0 , once a particle enters in region III, there is no potential
changes to cause a reflection, therefore, B3 must be zero.
– At x = 0 and x = a, the corresponding wave function and its first
derivative must be continuous.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
The Potential Barrier
 The results implies that there is a finite probability that a particle will
penetrate the barrier, that is so called “tunneling”.
*
 The transmission coefficient is defined by T  A3  A3*
A1  A1
 If E<<Vo,
 E  E 
T  16 1   exp  2 K 2 a 
 Vo 
Vo 
 This phenomenon is called “tunneling” and it violates classical
mechanics.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
One-Electron Atom
 Consider the one-electron atom potential function due to the coulomb
2
attraction between the proton and electron: V (r )   e
4o r
 Then we can generalize the Schrodinger’s eq. to 3-D in spherical
coordinates:
2mo
1  2 
1
 2
1



(
r
)




(sin


)

( E  V (r ))  0
r 2 r
r
r 2 sin 2   2 r 2 sin 2  

2
 Assume the solution to the equation can be written as
 (r , ,  )  R(r )  ( )  ( )
 Then the solution  is of the form,  = ejm, where m is an integer.
Solid-State Electronics
Chap. 2
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
One-Electron Atom
 Similarly, we can generate two additional constants n and l for the
variables  and r. n, l, and m are known as quantum numbers (integers)
n  1,2,3,...
 l  n  1, n  2, n  3,...,0
m  l , l  1,...,0
, each set of quantum numbers corresponds to a
quantum state which the electron may occupy.
 The solution of the wave equation is designated by nlm. For the lowest
energy state (n=1, l=0, m=0),
1
 100 
  
  ao 
1
3/ 2
e  r / ao where ao  0.529 angstrom
 The electron energy E is quantized,
Solid-State Electronics
Chap. 2
22
 mo e 4
En 
4o 2 2 2 n 2
Instructor: Pei-Wen Li
Dept. of E. E. NCU
One Electron Atom
 The probability density function, or the probability of finding the
electron at a particular distance form the nucleus, is proportional to
100*100 and also to the differential volume of the shell around the
nucleus.
 The electron is not localized at a given radius.
Solid-State Electronics
Chap. 2
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Dept. of E. E. NCU
Homework
 2.1
 2.15
 2.23
Solid-State Electronics
Chap. 2
24
Instructor: Pei-Wen Li
Dept. of E. E. NCU