chapter 2 introduction to quantum...
TRANSCRIPT
W.K. Chen Electrophysics, NCTU 1
Chapter 2 Introduction to Quantum Mechanics
W.K. Chen Electrophysics, NCTU 2
Outline
Principles of quantum mechanics
Schrodinger’s wave equation
Application of Schrodinger’s wave equation
Extensions of the wave theory to atoms
W.K. Chen Electrophysics, NCTU 3
2.1.1 Energy quantaPhotoelectric 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.
W.K. Chen Electrophysics, NCTU 4
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
W.K. Chen Electrophysics, NCTU 5
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
hvE = 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
ohvhvmT −== 2max 2
1 υ
ohv=functionwork
W.K. Chen Electrophysics, NCTU 6
Example 2.1 photon energy
eV1075.1106.1
1081.2
J1081.210708.0
)103)(106256(
419
15
158
1034
×=××
=
×=×
××===
−
−
−−
−.chhvEλ
Photon energy of x-ray with a wavelength of λ=0.708x10-8cm
W.K. Chen Electrophysics, NCTU 7
2.1.2 Wave-particle dualityThe 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.
2
2
2
2
Force
2relation
momentum
2
1Energy
dt
xdmmaF
m
pEpE
mp
m
E
==
=−
=
=
υ
υrr
),(1
),(equation wave
wavelength
Energy
2
2
22
2
txtc
txx
v
c
hvE
Ψ∂∂
=Ψ∂∂
=
==
λ
ωh
W.K. Chen Electrophysics, NCTU 8
p
h
kh
p
=
==
λ
λ
particle ofh wavelengtthe
photon of momentum the h
Wave-particle duality principle
particle: momentum ⇒ wavelength
wave: wavelength ⇒ momentum
)2
,2
(λπ
π== k
hh
(de Broglie wavelength)
p
h=λ
kh
p h==λ
W.K. Chen Electrophysics, NCTU 9
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
λθ md =sin2
W.K. Chen Electrophysics, NCTU 10
Electromagnetic frequency spectrum
W.K. Chen Electrophysics, NCTU 11
Example 2.2 de Broglie wavelength
o
p
h
mp
A 7.72
m1027.71011.9
10625.6h wavelengtBroglie de the
1011.9)10)(1011.9(electron of momentum the
926
34
26531
=
×=××
==
×=×==
−−
−
−−
λ
υ
An electron travel at a velocity of 107 cm/sec
Typical de Broglie wavelength of electron≈ 100 Å
W.K. Chen Electrophysics, NCTU 12
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
h
h
≥ΔΔ
≥ΔΔ
tE
xpsJ10054.1
234 ⋅×== −
πh
h
The Uncertainty principle is only significant for subatomic particles
)exp(
)exp(
t
kx
ω
W.K. Chen Electrophysics, NCTU 13
2.2 Schodinger’s wave equation
μευ 1
=
),(
:form lExponentia
)cos(),(
:form Sinusiodal
)( φω
φω
+⋅−⋅=
+⋅=
trkjo
o
eEtrE
trkEtrE
mrrrrr
mrrrrr
Wave equation (Traveling wave)
Wave function
][2
1)cos(
][2
1)sin(
form sin/cosform lExponentia
)()(
)()(
trkjtrkjoo
trkjtrkjoo
eeEtrkE
eej
EtrkE
ωω
ωω
ω
ω
mrr
mrr
mrr
mrr
rm
rrr
rm
rrr
⋅−⋅
⋅−⋅
+=⋅
−=⋅
⇔
2
2
22
2
2
2
22
2
1
1
t
HH
xt
EE
x ∂∂
=∂∂
∂∂
=∂∂
rr
rr
υυ
W.K. Chen Electrophysics, NCTU 14
2.2 Schodinger’s wave equationSchrodinger 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
particle theof mass :
function potential:)(
function wave:),(
m
xV
txΨ
tjE
xjp
∂∂
→∂∂
−→ hh
ExVp
=+ )(2m
2
t
txjtxxV
x
tx
∂Ψ∂
=Ψ+∂Ψ∂
⋅− ),(
),()(),(
2m
equation waveSchodinger
2
22
hh
Classical physics
Wave mechanics
W.K. Chen Electrophysics, NCTU 15
Assume the position and time parameters in wave function is separable
)()(),( txtx φψ=Ψ
)()(by devide
)()()()()(
)()(
2m
2
22
tx
t
txjtxxV
x
xt
φψ
φψφψψφ∂
∂=+
∂∂
⋅−
hh
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.
)constant( )(
)(
1)(
)(
)(
1
2m 2
22
ηφφ
ψψ
=∂
∂=+
∂∂
⋅−
t
t
tjxV
x
x
xh
h
t
t
tjxV
x
x
x ∂∂
=+∂
∂⋅
− )(
)(
1)(
)(
)(
1
2m 2
22 φφ
ψψ
hh
W.K. Chen Electrophysics, NCTU 16
⇒
=∂
∂)constant(
)(
)(
1 ηφφ t
t
tjh
tjtj eet ωηφ −− == )/()( h
⇒
⇒=
=⇒
ω
ωη
hQ
h
E E=η
)()(
)(
1
2m 2
22
ExVx
x
x==+
∂∂
⋅− ηψ
ψhTime-independent
Schrodinger wave equation
The separation constant is the total energy E of the particle
Physical meaning of η
tjextxtx ωψφψ −==Ψ )()()(),(Wave eq can be written as
The position-independent wave function is always in a form of exponential term e -jωt
W.K. Chen Electrophysics, NCTU 17
ExVx
x
xm=+
∂∂
⋅−
)()(
)(
1
2 2
22 ψψ
h
0)())(()(
2 2
22
=−+∂
∂⋅
−xExV
x
x
mψψh Time-independent Schrodinger’s
wave equation
0)()( 2
2
2
=+∂
∂xk
x
x ψψ
)exp()( jkxAx ±=ψ
⇒<>−
=
⇒>>−
=
)( if 0])([2
)( if 0)]([2
2
2
xVEExVm
xVExVEm
k
h
h
γ )exp()( xAx γψ ±=
Time-independent Schrodinger wave equation
W.K. Chen Electrophysics, NCTU 18
2.2.2 Physical meaning of the wave equationMax Born postulated in 1926 that the wave function is the probability of finding the particle between x and x+dx at a given
dxtx2
),(Ψ
)()(
)()(
),(),(),(
*
)/(*)/(
*2
xx
exex
txtxtxtEjtEj
ψψ
ψψ
⋅=
⋅=
Ψ⋅Ψ=Ψ+− hh
)()(),(y probabilit *2xxtx ψψ ⋅=Ψ
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
W.K. Chen Electrophysics, NCTU 19
2.2.3 Boundary condition for wave functionThe probability of finding the particle over the entire space must be equal to 1
1)()(),( *2 =⋅=Ψ ∫∫+∞
∞−
+∞
∞−dxxxdxtx ψψ
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
ψ(x) must be finite, single-valued and continuous
∂ ψ(x) /∂ x must be finite, single-valued and continuous
W.K. Chen Electrophysics, NCTU 20
2.3 Applications of Schrodinger’s wave equation
Electron in free space
Electron in infinite potential well
Step potential function
Potential barrier
W.K. Chen Electrophysics, NCTU 21
2.3.1 Electron in free spaceElectron 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
0)())(()(
2m 2
22
=−+∂
∂⋅
−xExV
x
x ψψh Time-independent Schrodinger’swave equation
)space free(0)(2)(
22
2
=+∂
∂x
mE
x
x ψψh
)exp()exp()( jkxBjkxAx −++=ψ
For simplicity, let V(x)=0
2
2
h
mEk =
W.K. Chen Electrophysics, NCTU 22
[ ] [ ])(exp(exp
)(
tkxjBtkxjA
et tj
ωω
φ ω
+−+−=
⇒
= −Q
Compared to a particle traveling function in classical mechanics
)](exp[)(exp[),( tkxjBtkxjAtx ωω +−+−=Ψ
Where λπ2
=kmE
h
2=⇒ λ
Right-going wave Left-going wave
)exp()exp()( jkxBjkxAx −+=ψ
Next time, when we see the time-independent wave function, we can know its traveling direction immediately
)()(),( txtx φψ ⋅=Ψ
2
2
h
mEk =
W.K. Chen Electrophysics, NCTU 23
Remember the postulate of de Broglie’s wave-particle principle
p
h=λ
We also have
m
pE
mEp
2
22
=⇒
=
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
W.K. Chen Electrophysics, NCTU 24
2.3.2 Infinite potential well (bound particle)
0)())(()(
2 2
22
=−+∂
∂⋅
−xExV
x
x
mψψh
E>V(x): traveling wave
V(x)>E: decaying wave
For V(x)=∞ (>>E), the wave function in region I & III must be zero
Region II (V(x)=0)
⇒ traveling wave
Region I & III (V(x)=∞)
⇒ decaying wave
0)(2)(
22
2
=+∂
∂x
mE
x
x ψψh
axxxV ≥≤∞= ,0for )(
W.K. Chen Electrophysics, NCTU 25
The solution is the same what we have learned in “Fundamental Physics”
KxAKxAx sincos)( 21 +=ψ2
2
h
mEK =
Boundary conditions:ψ(x) must continuous ( at boundaries)
⇒
==⇒
=====
=⇒
=====
+−
−+
)0or ( 0)sin(
)sin(0)()(
0
)cos(0)0()0(
2
2
1
1
AKa
KaAaxax
A
KaAxx
ψψ
ψψ
a
nK
π=
W.K. Chen Electrophysics, NCTU 26
Boundary conditions:Total probability is one
∫ =a
dxKxA0
22 1)sin(
L3,2,1 where
22
=⇒
=⇒
n
aA
K
KxxdxKx
4
2sin
2)(sin2 −=∫
aa
K
KxxAdxKxA
0
220
22 4
2sin
21)sin( ⎟
⎠⎞
⎜⎝⎛ −==∫
0
sin2
)( ⎟⎠⎞
⎜⎝⎛⋅= x
a
n
ax
πψ
W.K. Chen Electrophysics, NCTU 27
Quantization of energy levels:
2
2
hQ
mEK =
2
222
2ma
nEE n
πh==
a
nK
π=
(infinite well)
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.
2nEn ∝
discrete wavevector
discrete energy
W.K. Chen Electrophysics, NCTU 28
Example 2.3 infinite potential wellInfinite potential well with width of 5Å
eV )51.1(106.1
)1041.2(
J)1041.2()105)(1011.9(2
)10054.1(
2
219
192
19221034
22342
2
222
nn
nn
ma
nEE n
=××
=
×=××
×===
−
−
−−−
− ππh
13
12
1
9eV 59.13
4eV 04.6
eV 51.1
EE
EE
E
====
=
For Infinite potential well, 2nEn ∝
W.K. Chen Electrophysics, NCTU 29
0)())((2)(
22
2
=−
+∂
∂x
xVEm
x
x ψψh
Time-independent Schrodinger’swave equation
2.3.3 The step potential function
Incident wave: traveling waveReflective wave: traveling waveTransmitted wave: decaying wave
(i) E<VoVo
Incident wave: traveling waveReflective wave: traveling waveTransmitted wave: traveling wave
(ii) E>Vo
Vo
W.K. Chen Electrophysics, NCTU 30
0)())((2)(
22
2
=−
+∂
∂x
xVEm
x
x ψψh
Region I (V(x)=0, E>V) ⇒ traveling wave
0)(2)(
1221
2
=+∂
∂x
mE
x
x ψψh
21111
2 ))1(()0( )( 11
h
mEkeqxeBeAx xjkxjk =≤+= −ψ
Time-independent Schrodinger’swave equation
Case: E<Vo
E
W.K. Chen Electrophysics, NCTU 31
4 unknowns (A1, B1, A2 and B2)
⇒ 3 B.C. (boundary conditions)
Region II (E<Vo) ⇒ decaying wave
0)()(2)(
2222
2
=−
+∂
∂x
EVm
x
x o ψψh
0)(2
eq(2) )0( )(22222
22 >−
=≥+= +−
h
EVmxeBeAx oxx γψ γγ
eq(2) )0( )(
)1()0( )(
22
11
222
111
⎪⎪⎩
⎪⎪⎨
⎧
≥+=
≤+=
+−
−
L
L
xeBeAx
eqxeBeAx
xx
xjkxjk
γγψ
ψ
0)(2
22 >−
=h
EVm oγ
02
21 >=h
mEk
W.K. Chen Electrophysics, NCTU 32
B.C.1: ψ2(x) must remain finite ⇒ B2=0
)( 222
xeAx γψ −=
B.C.2: ψ (x) must be continuous at x=0
⇒⎪⎩
⎪⎨
⎧
≥=
≤+=
−
−
)0( )(
)0( )(
2
11
22
111
xeAx
xeBeAx
x
xjKxjK
γψ
ψ)( 211 ieqABA =+
B.C.3: first derivative dψ (x)/dx must be continuous at x=0
)0()0( 21+− =ψψ
+− ∂∂
=∂∂
0
2
0
1
xx
ψψ)( 221111 iieqABjkAjk γ−=−
eq(2) )0( )(
)1()0( )(22
11
222
111
⎪⎩
⎪⎨⎧
≥+=
≤+=+−
−
L
L
xeBeAx
eqxeBeAxxx
xjkxjk
γγψ
ψ
W.K. Chen Electrophysics, NCTU 33
Using eq(i) and (ii), we obtain
A )(
)(2
A )(
)2(
121
22
2112
121
22
2121
22
1
k
jkkA
k
kjkB
+−
=
+−+−
=
γγ
γγγ
⎪⎩
⎪⎨
⎧
≥=
≤+=
−
−
)0( )(
)0( )(
2
11
22
111
xeAx
xeBeAx
x
xjKxjK
γψ
ψ
Wave functions for step potential barrier
(i) E<Vo
Incident wave: traveling waveReflective wave: traveling waveTransmitted wave: decaying wave
Vo
W.K. Chen Electrophysics, NCTU 34
Reflectivity at interface of step barrierVo
)(
)2)(2( *1122
122
212
12221
21
22*
11 AAk
jkkjkkBB
+−−+−
=⋅γ
γγγγ
The reflective probability density function (i.e., intensity)
Reflective coefficient R, defined as the ratio of reflected flux to the incident flux
ii
rr
Iυ
IυR =
*11
*11
AA
BB
υ
υR
i
r ⋅=⇒
kmp hQ == υ
υ⋅= nFlux
W.K. Chen Electrophysics, NCTU 35
ri km
υυ 1 ==h
1.0 )(
4)(22
122
22
21
221
22
*11
*11 =
++−
==⇒k
kk
AA
BBR
γγγ
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
W.K. Chen Electrophysics, NCTU 36
Example 2.4 penetration depthVo
0)(2
)(2222
2 >−
== −
h
EVmeAx ox γψ γ
The penetration depth is defined as γ2d=1
o
o
o
d
EEm
EVmd
A6.11
m106.11)1056.4)(1011.9(2
10054.1
)2(2
)(2
1
10
3131
342
2
2
=
×=××
×=
−=
−==
−
−−
−h
h
γ
The penetration depth is typically much less than the de Broglie wavelength of electron in free space ( 73 Å).
W.K. Chen Electrophysics, NCTU 37
2.3.4 The potential barrier
)3()( 3)(
eq(2) )0( )(
)1()0( )(
11
22
11
33
222
111
⎪⎩
⎪⎨
⎧
≥+=
≥+=
≤+=
−
+−
−
eqaxeBeAx
xeBeAx
eqxeBeAx
xjkxjk
xx
xjkxjk
L
L
L
ψ
ψ
ψγγ
2221
)(2
2
hh
EVmmEk o −== γ
o
o
VE
axV
axxxV
<⎩⎨⎧
≤≤><
=0for
& 0for 0)(
W.K. Chen Electrophysics, NCTU 38
daVEaυ
E
υ
ET o
oo
<<<−−⋅≈⇒ for )2exp()1()(16 2γ
Tunneling:There is a finite probability that a particle impinging a potential barrier will penetrate 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)
) ( *11
*33
*11
*33
iti
t υυAA
AA
AA
AA
υ
υT ==⋅=⇒ Q
If the barrier width a is thinner than the penetration depth, electron can tunnel through the barrier and appear in region III
Region I Region II Region III
W.K. Chen Electrophysics, NCTU 39
Example 2.5 Tunneling probabilityo
o aEV A3 width eV, 2 eV, 20 ===
ooo
VEaυ
E
υ
ET <<−−⋅≈ for )2exp()1()(16 2γ
=×
×−×=
−= −
−−
234
1931
22 )10054.1(
)106.1)(220)(1011.9(2)(2
h
EVm oγ
=××−−⋅≈⇒ − )]103)(1017.2(2exp[)20
21()
20
2(16 1010T
⇒
<< oVE
110 m 1017.2 −×
61017.3 −×
The tunneling probability may appear to be a small value, but the value is not zero
W.K. Chen Electrophysics, NCTU 40
2.4 Extensions of the wave theory to atoms
+r
erV
oπε4)(
2−=
0),,())((2
),,(2
2 =−+∇ φθψφθψ rxVEm
r o
h
Time-independent Schrodinger’s wave equation
In spherical coordinate, Schrodinger’s wave equation is
0))((2
)(sinsin
1
sin
1)(
12222
2
222
2=−+
∂∂
∂∂
⋅+∂∂⋅+
∂∂
∂∂
⋅ ψθψθ
θθφψ
θψ
rVEm
rrrr
rro
h
Assume separation-of-variables is valid
)()()(),,( φθφθψ Φ⋅Θ⋅= rRr
W.K. Chen Electrophysics, NCTU 41
0)(2
sin)(sinsin1
)(sin
222
2
22
2
=−+∂Θ∂
∂∂
⋅Θ
+∂Φ∂
⋅Φ
+∂∂
∂∂
⋅ VEm
rr
Rr
rRo
hθ
θθ
θθ
φθ
The second term is a function of φ only, independent of r and θ, it must be constants
22
21m−=
∂Φ∂
⋅Φ φ
The solution
L3,2,1,0 ±±±==Φ me jmφ
M
M
x
z
r
θ
φ
Spherical coordinate
y
+
W.K. Chen Electrophysics, NCTU 42
0,),1(,
0,,3,2,1
3,2,1
LL
LL
L
−±±=−−−=
=
llm
nnnl
n
Sets of quantum numbers
222
4
2)4( n
emE
o
on
hπε−
=
The electron energy for one-electron atom is
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
n: principle quantum number
+
2
1
nEn ∝
⇒)(rR
⇒Φ )(φ⇒Θ )(θ
W.K. Chen Electrophysics, NCTU 43
+
2
1
nEn ∝
222
4
2)4( n
emE
o
on
hπε−
=2
222
2ma
nEE n
πh==
2nEn ∝
Infinite potential well
One-electron atom
W.K. Chen Electrophysics, NCTU 44
The radial probability density function for one-electron atom in the (a) lowest energy state and (b) next-higher energy state
Wave function for one-electron atomψnlm: notation of wave function for one-electron atom
oar
o
ea
/
2/3
100
11 −⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
πψ
o
2
2
A529.04
==em
ao
oo
hπε
The wave function of lowest energy state is spherical symmetric
W.K. Chen Electrophysics, NCTU 45
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 Electrophysics, NCTU 46
0,),1(,
0,,3,2,1
3,2,1
LL
LL
L
−±±=−−−=
=
llm
nnnl
n