chapter 2 introduction to quantum...

W.K. Chen Electrophysics, NCTU 1

Chapter 2 Introduction to Quantum Mechanics


Outline

Principles of quantum mechanics

Schrodinger’s wave equation

Application of Schrodinger’s wave equation

Extensions of the wave theory to atoms


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.


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


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


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


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


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


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


Electromagnetic frequency spectrum


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


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


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


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


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


⇒

=∂

∂)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


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


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


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


2.3 Applications of Schrodinger’s wave equation

Electron in free space

Electron in infinite potential well

Step potential function

Potential barrier


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 =


[ ] [ ])(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 =


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


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 )(


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=


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


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


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 ∝


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


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


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


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


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


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


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


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).


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)(


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


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


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


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

+


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


+

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


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


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


0,),1(,

0,,3,2,1

3,2,1

LL

LL

L

−±±=−−−=

=

llm

nnnl

n

chapter 2 introduction to quantum...

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