Download Chapter 2 Introduction to Quantum Mechanics

Chapter 2
Introduction to Quantum
Mechanics
W.K. Chen
Electrophysics, NCTU
1
Outline
„
„
„
„
Principles of quantum mechanics
Schrodinger’s wave equation
Application of Schrodinger’s wave equation
Extensions of the wave theory to atoms
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2
2.1.1 Energy quanta
„
Photoelectric effect
Classical physics:
if the intensity of monochromatic light is large enough, the work function of
the material will be overcome and an electron will be emitted from the
surface, independent of the incident frequency.
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Experimental results:
‰
‰
‰
The lowest frequency of incident light is vo., below which no photoelectric effect
is produced
At a constant incident intensity, the maximum kinetic energy of the
photoelectron varies linearly with frequency for v>vo.
If the incident intensity varies at a constant frequency, the rate of photoemission
changes, but the maximum kinetic energy remains the same
maximum kinetic energy
the maximum kinetic energy that can be obtained for the emitted
photoelectrons
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4
Energy quanta:
‰
In 1900, Planck postulated that thermal radiation
emitted from a heated surface is in a form of
discrete packets of energy called quanta
E = hv
h=6.625x10-34 J.s
Einstein’s interpretation for photoelectric effect:
‰
‰
‰
In 1905, Einstein suggested the energy in a light wave is also contained in
discrete packets or bundles.
The particle-like packet of energy is called photon, whose energy is given by
E=hv
The minimum energy required to remove an electron is called the work
function of the material and any excess photon energy goes into kinetic energy
of the photoelectron
Tmax =
1
mυ 2 = hv − hvo
2
work function = hvo
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Example 2.1 photon energy
Photon energy of x-ray with a wavelength of λ=0.708x10-8cm
(6.625 ×10 −34 )(3 ×1010 )
= 2.81×10 −15 J
E = hv = h =
−8
0.708 ×10
λ
2.81×10 −15
4
=
=
1
.
75
×
10
eV
1.6 ×10 −19
c
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2.1.2 Wave-particle duality
„
„
The light waves in the photoelectric effect behave as if they are particle
In 1924, de Brogle postulated the existence of matter wave. Since wave
exhibit particle-like behavior, then particle should be expected to show
particle-like properties.
Energy E = hv = hω
c
wavelength λ =
v
1 ∂2
∂2
wave equation 2 Ψ ( x, t ) = 2 2 Ψ ( x, t )
∂x
c ∂t
1
mυ 2
2
r
r
momentum p = mυ
Energy E =
p2
E − p relation E =
2m
d 2x
Force F = ma = m 2
dt
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„
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Wave-particle duality principle
particle: momentum
⇒ wavelength
wave: wavelength
⇒ momentum
the momentum of photon p =
the wavelength of particle λ =
λ=
p=
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h
λ
h
λ
h
p
= hk
(h =
h
2π
, k=
)
λ
2π
(de Broglie wavelength)
h
p
= hk
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Davisson-Germer experiment (1927)
„
The wave nature of particles (electrons) can be tested by the
existence of interference pattern produced by electron beam diffracted
from a grating
Nickel crystal (grating)
Diffraction pattern
2d sin θ = mλ
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Electromagnetic frequency spectrum
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Example 2.2 de Broglie wavelength
„
An electron travel at a velocity of 107 cm/sec
the momentum of electron p = mυ = (9.11×10 −31 )(105 ) = 9.11×10 −26
h 6.625 ×10 −34
= 7.27 ×10 −9 m
the de Broglie wavelength λ = =
− 26
p 9.11×10
o
= 72.7 A
Typical de Broglie wavelength of electron≈ 100 Å
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2.1.3 Uncertainty Principle (1927)
„
Uncertainty principle (Heisenberg)
‰
‰
It is impossible to simultaneously describe the absolute accuracy position
and momentum of a particle
It is impossible to simultaneously describe the absolute accuracy energy of
particle and the instant time the particle has this
ΔpΔx ≥ h
ΔEΔt ≥ h
„
h=
h
= 1.054 ×10 −34 J ⋅ s
2π
exp(kx)
exp(ωt )
The Uncertainty principle is only significant for subatomic particles
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2.2 Schodinger’s wave equation
Wave equation (Traveling wave)
r
∂2 r 1 ∂2E
E= 2
υ ∂t2
∂x 2
r
∂2 r 1 ∂2H
H= 2
υ ∂t2
∂x 2
υ=
Wave function
1
με
Sinusiodal form :
r r
r r
r
E (r , t ) = Eo cos(k ⋅ r m ω t + φ )
Exponential form ⇔ sin/cos form
r r
r
r 1 j ( kr⋅rr mωt ) − j ( kr⋅rr mωt )
Eo sin( k ⋅ r m ωt ) = Eo [e
−e
]
2j
rr
rr
r r
r
r 1
Eo cos(k ⋅ r m ωt ) = Eo [e j ( k ⋅r mωt ) + e − j ( k ⋅r mωt ) ]
2
Exponential form :
r r
r − j ( kr⋅rr mωt +φ )
E (r , t ) = Eo ⋅ e
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2.2 Schodinger’s wave equation
„
Schrodinger in 1926 provided a formulation called wave mechanics,
which incorporated
‰
‰
„
The principle of quanta (Planck)
Wave-particle duality (de Broglie)
Based on the wave-particle duality principle, we will describe the motion
of electrons in a crystal by wave
Classical physics
p2
+ V ( x) = E
2m
Wave mechanics
p → − jh
∂
∂x
E → jh
Schodinger wave equation
− h ∂ Ψ ( x, t )
∂Ψ ( x, t )
⋅
+
V
(
x
)
Ψ
(
x
,
t
)
=
j
h
2m
∂x 2
∂t
2
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∂
∂t
Ψ ( x, t ) : wave function
V ( x) : potential function
m : mass of the particle
14
Assume the position and time parameters in wave function is separable
Ψ ( x, t ) = ψ ( x)φ (t )
− h2
∂ 2ψ ( x)
∂φ (t )
V
(
x
)
ψ
(
x
)
φ
(
t
)
j
ψ
(
x
)
⋅ φ (t )
+
=
h
2m
∂x 2
∂t
devide by ψ ( x)φ (t )
− h 2 1 ∂ 2ψ ( x)
1 ∂φ (t )
+ V ( x ) = jh
⋅
2
φ (t ) ∂t
2m ψ ( x) ∂x
The left side of equation is a function of position x only and the right side is a
function of time t only, which implies each side of this equation must be equal to
a same constant.
− h 2 1 ∂ 2ψ ( x)
1 ∂φ (t )
⋅
+ V ( x ) = jh
= η (constant )
2
2m ψ ( x) ∂x
φ (t ) ∂t
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Physical meaning of η
jh
1 ∂φ (t )
= η (constant )
φ (t ) ∂t
⇒ φ (t ) = e − j (η / h )t = e − jω t
⇒
η
h
The position-independent wave function is always
in a form of exponential term e -jωt
=ω
Q E = hω ⇒ η = E
The separation constant is the total energy E of the particle
Wave eq can be written as
Ψ ( x, t ) = ψ ( x)φ (t ) = ψ ( x)e − jω t
⇒ Time-independent
Schrodinger wave equation
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− h 2 1 ∂ 2ψ ( x)
⋅
+ V ( x) = η = E
2m ψ ( x) ∂x 2
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Time-independent
Schrodinger wave equation
− h 2 1 ∂ 2ψ ( x)
⋅
+ V ( x) = E
2m ψ ( x) ∂x 2
− h 2 ∂ 2ψ ( x)
+ (V ( x) − E )ψ ( x) = 0
⋅
2m
∂x 2
Time-independent Schrodinger’s
wave equation
∂ 2ψ ( x)
+ k 2ψ ( x) = 0
2
∂x
k=
2m[ E − V ( x)]
>0
h2
if E > V ( x) ⇒ ψ ( x) = A exp(± jkx)
γ=
2m[V ( x) − E ]
>0
h2
if E < V ( x) ⇒ ψ ( x) = A exp(±γx)
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2.2.2 Physical meaning of the wave equation
2
„
Max Born postulated in 1926 that the wave function Ψ ( x, t ) dx is the
probability of finding the particle between x and x+dx at a given
2
Ψ ( x , t ) = Ψ ( x, t ) ⋅ Ψ * ( x , t )
= ψ ( x)e − j ( E / h )t ⋅ψ * ( x)e + j ( E / h )t
= ψ ( x) ⋅ψ * ( x)
2
probability Ψ ( x, t ) = ψ ( x) ⋅ψ * ( x)
„
„
The probability density function is independent of time
In classical mechanics, the position of a particle can be determined
precisely
In quantum mechanics, the position of a particle is found in term of a
probability
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2.2.3 Boundary condition for wave function
„
The probability of finding the particle over the entire space must
be equal to 1
∫
+∞
−∞
„
„
„
„
„
„
+∞
Ψ ( x, t ) dx = ∫ ψ ( x) ⋅ψ * ( x)dx = 1
2
−∞
ψ(x) must be finite, single-valued and continuous
∂ ψ(x) /∂ x must be finite, single-valued and continuous
If the probability were to become infinite at some point in space,
then the probability of finding the particle at the position would be
certain, that violate the uncertainty principle
The second derivative must finite which implies that the first
derivative must be continuous
The first derivative is related to the particle momentum, which must
be finite and single-valued
The finite first derivative implies that the function itself must be
continuous
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2.3 Applications of Schrodinger’s wave equation
„
„
„
„
Electron in free space
Electron in infinite potential well
Step potential function
Potential barrier
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2.3.1 Electron in free space
„
„
Electron in free space means no force acting on the electron
⇒ V(x) is constant
We must have E>V(x) to assure the motion of electron
− h 2 ∂ 2ψ ( x)
⋅
+ (V ( x) − E )ψ ( x) = 0
2m
∂x 2
Time-independent Schrodinger’s
wave equation
For simplicity, let V(x)=0
∂ 2ψ ( x) 2mE
+ 2 ψ ( x) = 0 (free space)
∂x 2
h
k=
2mE
h2
ψ ( x) = A exp(+ jkx) + B exp(− jkx)
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Qφ (t ) = e − jωt
⇒ Ψ ( x, t ) = ψ ( x) ⋅ φ (t )
= A exp[ j (kx − ωt ] + B exp[− j (kx + ωt )]
Right-going wave
Left-going wave
Next time, when we see the time-independent wave function, we can know its traveling
direction immediately
ψ ( x) = A exp( jkx) + B exp(− jkx)
k=
2mE
h2
Compared to a particle traveling function in classical mechanics
Ψ ( x, t ) = A exp[ j (kx − ωt ) + B exp[− j (kx + ωt )]
Where
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k=
2π
λ
⇒λ =
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h
2mE
22
Remember the postulate of de Broglie’s wave-particle principle
λ=
h
p
We also have
p = 2mE
p2
⇒E=
2m
Which implies the consistency of wave-particle principle and wave mechanics
in free space ( wave mechanics is based on energy quanta and wave-particle
duaility
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Electrophysics, NCTU
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2.3.2 Infinite potential well (bound particle)
V ( x) = ∞ for x ≤ 0, x ≥ a
Region I & III (V(x)=∞)
⇒ decaying wave
For V(x)=∞ (>>E), the wave function
in region I & III must be zero
− h ∂ ψ ( x)
⋅
+ (V ( x) − E )ψ ( x) = 0
∂x 2
2m
2
„
„
2
E>V(x): traveling wave
V(x)>E: decaying wave
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Region II (V(x)=0)
⇒ traveling wave
∂ 2ψ ( x) 2mE
+ 2 ψ ( x) = 0
h
∂x 2
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The solution is the same what we have learned in “Fundamental Physics”
ψ ( x) = A1 cos Kx + A2 sin Kx
„
K=
2mE
h2
Boundary conditions:
ψ(x) must continuous ( at boundaries)
‰
ψ ( x = 0 + ) = ψ ( x = 0 − ) = 0 = A1 cos( Ka )
⇒ A1 = 0
ψ ( x = a − ) = ψ ( x = a + ) = 0 = A2 sin( Ka )
⇒ sin( Ka ) = 0 ( or A2 = 0)
⇒
K=
nπ
a
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„
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Boundary conditions:
Total probability is one
‰
∫
a
0
( A2 sin Kx) 2 dx = 1
∫ sin
∫
a
0
2
( Kx)dx =
x sin 2 Kx
−
2
4K
0
a
⎛ x sin 2 Kx ⎞
( A2 sin Kx) dx = 1 =A ⎜ −
⎟
4K ⎠ 0
⎝2
⇒ A2 =
2
2
a
⇒ ψ ( x) =
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2
2
2
⎛ nπ ⎞
⋅ sin ⎜
x⎟
a
⎝ a ⎠
where n = 1,2,3L
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„
Quantization of energy levels:
QK =
discrete
wavevector
K=
nπ
a
2mE
h2
discrete energy
h 2 n 2π 2
E = En =
2ma 2
(infinite well)
En ∝ n 2
„
Quantization of particle energy in infinite well
Since the constant K must have discrete values. This results mean the
energy of particle in finite well only have particular discrete values, contrary
to results from classical physics, which would allow the particle to have
continuous energy levels.
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Example 2.3 infinite potential well
Infinite potential well with width of 5Å
n 2 (1.054 × 10 −34 ) 2 π 2
h 2 n 2π 2
E = En =
= n 2 (2.41×10 −19 )J
=
2
−34
−10 2
2ma
2(9.11×10 )(5 ×10 )
n 2 (2.41×10 −19 )
= n 2 (1.51) eV
=
−19
1.6 ×10
E1 = 1.51 eV
E2 = 6.04 eV = 4 E1
E3 = 13.59 eV = 9 E1
For Infinite potential well, En ∝ n 2
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2.3.3 The step potential function
Time-independent Schrodinger’s
wave equation
∂ 2ψ ( x) 2m( E − V ( x))
+
ψ ( x) = 0
h2
∂x 2
(ii) E>Vo
(i) E<Vo
Vo
Vo
Incident wave: traveling wave
Reflective wave: traveling wave
Transmitted wave: decaying wave
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Incident wave: traveling wave
Reflective wave: traveling wave
Transmitted wave: traveling wave
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Case: E<Vo
Time-independent Schrodinger’s
wave equation
E
∂ 2ψ ( x) 2m( E − V ( x))
+
ψ ( x) = 0
h2
∂x 2
Region I (V(x)=0, E>V) ⇒ traveling wave
∂ 2ψ 1 ( x) 2mE
+ 2 ψ 1 ( x) = 0
∂x 2
h
ψ 1 ( x) = A1e jk x + B1e − jk x ( x ≤ 0) (eq(1))
1
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1
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k1 =
2mE
h2
30
Region II (E<Vo) ⇒ decaying wave
∂ 2ψ 2 ( x) 2m(Vo − E )
+
ψ 2 ( x) = 0
h2
∂x 2
ψ 2 ( x) = A2 e −γ x + B2 e +γ
2
2x
γ2 =
( x ≥ 0) eq(2)
⎧
jk1 x
− jk1 x
( x ≤ 0) L eq(1)
⎪ψ 1 ( x) = A1e + B1e
⎪
⎨
⎪ψ ( x) = A e −γ 2 x + B e +γ 2 x ( x ≥ 0) L eq(2)
2
2
⎪⎩ 2
2m(Vo − E )
>0
h2
k1 =
2mE
>0
h2
γ2 =
2m(Vo − E )
>0
h2
4 unknowns (A1, B1, A2 and B2)
⇒ 3 B.C. (boundary conditions)
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⎧⎪ψ 1 ( x) = A1e jk1x + B1e − jk1x ( x ≤ 0) L eq(1)
⎨
⎪⎩ψ 2 ( x) = A2 e −γ 2 x + B2 e +γ 2 x ( x ≥ 0) L eq(2)
„
B.C.1: ψ2(x) must remain finite ⇒ B2=0
ψ 2 ( x) = A2e −γ
„
2x
B.C.2: ψ (x) must be continuous at x=0
ψ 1 (0 − ) = ψ 2 (0 + )
⎧
jK1 x
− jK1 x
( x ≤ 0)
⎪ψ 1 ( x) = A1e + B1e
⎨
⎪ψ ( x) = A e −γ 2 x
( x ≥ 0)
2
⎩ 2
„
⇒ A1 + B1 = A2
eq(i )
B.C.3: first derivative dψ (x)/dx must be continuous at x=0
∂ψ 1
∂x
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=
0−
∂ψ 2
∂x
0+
jk1 A1 − jk1 B1 = −γ 2 A2
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eq(ii )
32
„
Using eq(i) and (ii), we obtain
− (γ 22 + 2 jk1γ 2 − k12 )
A1
B1 =
(γ 22 + k12 )
A2 =
2k1 (k1 − jγ 2 )
A1
(γ 22 + k12 )
Wave functions for step potential barrier
⎧
− jK1 x
jK1 x
( x ≤ 0)
⎪ψ 1 ( x) = A1e + B1e
⎨
⎪ψ ( x) = A e −γ 2 x
( x ≥ 0)
2
⎩ 2
(i) E<Vo
Vo
Incident wave: traveling wave
Reflective wave: traveling wave
Transmitted wave: decaying wave
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Reflectivity at interface of step barrier
The reflective probability density function (i.e., intensity)
Vo
(γ 22 − k12 + 2 jk1γ 2 )(γ 22 − k12 − 2 jk1γ 2 )
B1 ⋅ B =
A1 A1*
2
2 2
(γ 2 + k1 )
*
1
Reflective coefficient R, defined as the ratio of reflected flux to the incident
flux
R=
υr I r
υi I i
Flux = n ⋅υ
υr B1 B1*
⇒R= ⋅
υi A1 A1*
Q p = mυ = hk
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υi =
h
k1 = υ r
m
B1 B1* (γ 22 − k12 ) 2 + 4k12γ 22
⇒R=
=
= 1.0
A1 A1*
(γ 22 + k12 ) 2
„
„
„
The results of R=1 implies that all of the particles incident on the potential
barrier for E<Eo are eventually reflected, entirely consistent with classical
physics
Because A2 is not zero, the particle being found in barrier is not equal to
zero, which is called quantum mechanical penetration.
The quantum mechanical penetration is classically not allowed, which is
the difference between classical and quantum mechanics
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Example 2.4 penetration depth
ψ 2 ( x) = A2 e −γ
2x
γ2 =
2m(Vo − E )
>0
h2
Vo
The penetration depth is defined as γ2d=1
h2
d= =
2m(Vo − E )
γ2
1
1.054 ×10 −34
h2
=
=
= 11.6 ×10 −10 m
2 m ( 2 Eo − E )
2(9.11×10 −31 )(4.56 ×10 −31 )
o
d = 11.6 A
The penetration depth is typically much less than the de Broglie wavelength of
electron in free space ( 73 Å).
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36
2.3.4 The potential barrier
⎧0 for x < 0 & x > a
V ( x) = ⎨
⎩Vo for 0 ≤ x ≤ a
E < Vo
⎧ψ 1 ( x) = A1e jk1x + B1e − jk1x ( x ≤ 0) L eq(1)
⎪
−γ x
+γ x
⎨ψ 2 ( x) = A2 e 2 + B2 e 2 ( x ≥ 0) L eq(2)
⎪
jk1 x
− jk1 x
( x ≥ a) L eq(3)
⎩ψ 3 ( x) = A3e + B3e
k1 =
2mE
h2
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γ2 =
2m(Vo − E )
h2
37
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If the barrier width a is thinner than the penetration depth, electron can tunnel
through the barrier and appear in region III
The transmission coefficient (defined as the ratio of the transmitted flux in
region III to the incident flux in region I)
Region I
υt A3 A3* A3 A3*
⇒T = ⋅
=
υi A1 A1* A1 A1*
⇒ T ≈ 16(
Region II Region III
(Q υt = υi )
E
E
) ⋅ (1 − ) exp(−2γ 2 a)
υo
υo
for E << Vo
a<d
Tunneling:
There is a finite probability that a particle impinging a potential barrier will
penetrate the barrier and appear in region III
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38
Example 2.5 Tunneling probability
o
Vo = 20 eV, E = 2 eV, width a = 3 A
E << Vo
⇒ T ≈ 16( E ) ⋅ (1 − E ) exp(−2γ a)
2
υo
υo
for E << Vo
2m(Vo − E )
2(9.11×10 −31 )(20 − 2)(1.6 ×10 −19 )
10
−1
2
.
17
×
10
m
=
γ2 =
=
h2
(1.054 ×10 −34 ) 2
⇒ T ≈ 16(
„
2
2
) ⋅ (1 − ) exp[−2(2.17 ×1010 )(3 ×10 −10 )] = 3.17 ×10 −6
20
20
The tunneling probability may appear to be a small value, but the
value is not zero
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2.4 Extensions of the wave theory to atoms
− e2
V (r ) =
4πε o r
∇ 2ψ (r , θ , φ ) +
+
2mo
( E − V ( x))ψ (r , θ , φ ) = 0
h2
Time-independent
Schrodinger’s wave
equation
In spherical coordinate, Schrodinger’s wave equation is
2mo
1 ∂ 2 ∂ψ
1
∂ 2ψ
1
∂
∂ψ
⋅
(
r
)
+
⋅
+
⋅
(sin
θ
)
+
( E − V (r ))ψ = 0
r 2 ∂r
∂r
r 2 sin 2 θ ∂φ 2 r 2 sin 2 θ ∂θ
∂θ
h2
Assume separation-of-variables is valid
ψ (r ,θ , φ ) = R(r ) ⋅ Θ(θ ) ⋅ Φ(φ )
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2m
sin 2 θ ∂ 2 ∂R
1 ∂ 2 Φ sin θ ∂
∂Θ
⋅ (r
)+ ⋅ 2 +
⋅
(sin θ
) + r 2 sin 2 θ 2 o ( E − V ) = 0
R ∂r
∂r
Φ ∂φ
Θ ∂θ
∂θ
h
The second term is a function of φ only, independent of r and θ, it must be
constants
1 ∂ 2Φ
⋅ 2 = −m 2
Φ ∂φ
z
Spherical coordinate
The solution
Φ = e jmφ
θ
m = 0,±1,±2,±3L
r
+
W.K. Chen
M
M
x
y
φ
41
Electrophysics, NCTU
Sets of quantum numbers
R(r ) ⇒ n = 1,2,3L
Θ(θ ) ⇒ l = n − 1, n − 2, n − 3, LL,0
Φ (φ ) ⇒ m = ±l , ± (l − 1),LL,0
+
En ∝
1
n2
The electron energy for one-electron atom is
− mo e 4
En =
(4πε o ) 2 2h 2 n 2
„
„
„
n: principle quantum number
The negative energy indicates the electron is bound to nucleus
The energy of bound electron is quantized
The quantized energy is again a result of the particle being bound in a
finite region of space
W.K. Chen
Electrophysics, NCTU
42
Infinite potential well
One-electron atom
+
h 2 n 2π 2
E = En =
2ma 2
− mo e 4
En =
(4πε o ) 2 2h 2 n 2
En ∝ n 2
En ∝
W.K. Chen
Electrophysics, NCTU
1
n2
43
Wave function for one-electron atom
ψnlm: notation of wave function for one-electron atom
ψ 100
1 ⎛1⎞
=
⋅⎜ ⎟
π ⎜⎝ ao ⎟⎠
3/ 2
e − r / ao
The wave function of lowest energy state is
spherical symmetric
o
4πε o h 2
ao =
=
0
.
529
A
mo e 2
The radial probability density function for one-electron atom in the (a) lowest
energy state and (b) next-higher energy state
W.K. Chen
Electrophysics, NCTU
44
Lowest energy states
„ The wave function of lowest energy state is
spherically symmetric
„ The most probable distance from the nucleus is at
r=ao, which is the same as Bohr theory
„ We may now begin to conceive the concept of an
electron cloud, or energy shell, surrounding the
nucleus rather than a discrete particle around
nucleus
Next higher energy states
„ The radial probability density function for the next
higher wave function ( n=2, l=0)is also spherically
symmetric
„ Two energy shells are existed for the next higher
energy states
„ The second shell is the most probable energy
state for the next higher energy states, but there
is still a small probability that the electron will
exist at the small radius
W.K. Chen
45
Electrophysics, NCTU
n = 1,2,3L
l = n − 1, n − 2, n − 3, LL,0
m = ±l , ± (l − 1),LL,0
W.K. Chen
Electrophysics, NCTU
46