Key concepts from Classical Physics - KOCw

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양자역학 (QM) 을 사용하여 분자의 행동을 이해하기 위해서는 QM 자체를 기본적으로 알아야 한다 .

• Key concepts from Classical Physics

(including particle and wave behavior)

 Energy, angular momentum

 double slit experiment

• Brief summary of the early history of QM

 wave-particle duality

 the uncertainty principle

 wave mechanics

1. Introduction and background

to Quantum Mechanics

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Energy: the capacity to do work. ‘ conserved’

• E total = E k + E p (or V)

• Kinetics energy (as a result of a body’s motion)

• Potential energy (as a results of a body’s position) : force can be derived from

Ex1, a coulomb potential Ex2, a gravitational potential

Key concepts from classical physics-1

1

2 k

2

E = m υ

( ) ( )

1 2 0

4

V r q q

r V h mgh

⋅ = πε

⋅ =

,

F dV F V

= − dx = −∇

(3)

2

velocity:

speed:

linear momentum:

the total energy:

definit

( ) 2

: (

tr a jector y e ( ), ( ) ) dr

dt

p m

E p V x

m x t p t υ

υ

=

= +

 



( ) ( )

( ( ) )

0

2 2

2

0

1 1

2 , 2

2 2

2

2 ( ( ))

x x

E m V x E V x m dx dt

dx E V x

dt m

dx E V x

dt m dt mdx

m E V x t t m dx

m E V x υ

= + − =

= −

− = ∫ −

Classical Mechanics

• Behavior of objects: E total is constant, F=ma

• The trajectory in terms of the energy

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• Newton’s second law

( )

2 2 2

2

, ,

1 for

F ma dp F

dt

dV d d x

F a

dx dt dt

d x dV

dt m dx x t

υ

= =

= − = =

∴ = −

   

 



2 2

ˆ ˆ , ,

ˆ ˆ ˆ

the gradient operator,

d d r

r xi yj zk a F V

dt dt

d d d

i j k

dx dy dz

υ −

= + + = = = ∇

∇ = + +

 

 



2

2 2 2

2 2

for HO ( 1 ), 2

1

0, where

( ) cos sin

V kx F dV kx

dx

d x dV k

dt m dx m x

d x k

dt x m

x t A t B t

ω ω

= = − = −

∴ = − = −

+ = =

∴ = +

• For 3D (for Cartesian coordinate)

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,

2

( : moment of inertia) Torque (accelarate a rotation) J r p

J I I mr I

dJ T dt

ω

≡ ×

= =

=

  

 

2 2 2 2 2 2

,

Similarly, =

2 2 2 2

k k

dp F p F

dt

p F T J

E E

m m I I

τ

τ τ

= ∴ =

= = =

 J

m

r 

ω 

• Rotational motion: described by angular momentum J

• When a constant F is applied to a system for a time τ

E of a particle can be increased to any value ! (continuous E)

2

, ,

J rp rmv mr v I r

dl d

l r v

dt dt

ω

θ ω θ

= = = =

= ≡ ≡

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2

1 2

H p V

m dH d

m V

dt dt υ

≡ +

 

=  + 

 

( )

2

2 2

2 , for 0

0

d dv dV dx dV V

dt dt dt dt dx t

dH dx d x dx dV dt m dt dt dt dx dH dx d x dV dx

m ma F

dt dt dt dx dt

υ = υ = ∂ =

∂

= +

 

=  +  = − =

 

V 0 t

∂ =

∂

• The Classical Hamiltonian (total energy)

H = E is constant for systems with

‘ conservative system’

if V 0, dH 0

t dt

∴ ∂ = =

∂

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Key concepts from classical physics-6

( )

0

phase

( , ) cos

: wave amplitude, :

(wavelength) (per

2 2

iod) k

A x t A

A kx t

kx t

π π

ω

λ ω

ω

τ

=

−

= =

Classical Wave Theory

One way to describe the motion:

specify the perpendicular displacement, x from its x 0 in f(x,t) - The collective motion of water molecules: ocean waves

- The collective motion of gas molecules: sound waves - Harmonic waves can be expressed as sin, cos functions

1.4.1 Wave amplitude Figure 1.1 (p9)

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( )

0 0

( , ) cos

: wave amplitude phase

2

:

(period)

(wavelength) 2

k A x t A

A

kx t

x t

k

ω

π

ω

π τ λ

=

−

=

−

=

(9)

phase velocity:

_p

(EM waves) kx t const

x t const

k

k c

ω νλ ω

ω

υ

− =

=  

=

 +



→

=

 

A traveling wave

• A position of constant phase (a position where A(x,t) is fixed)

1.4.2 Superposition and diffraction

• If two wave trains collide → coherent superposition

One example is the diffraction of light nλ = d sin θ

see figure 1.2 (p10)

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The double slit diffraction

For single slit coherent

superposition

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( ) ( ) ( ) ( )

( , ) cos cos

b.c. 0, , 0

A x t a kx t b kx t

A t A L t

ω ω

= − + +

= =

( ) ( ) ( ) ( )

( ) ( )

(0, ) cos cos

cos 0

( , ) cos cos 0

2 sin sin 0

A t a t b t

a b t a b

A L t a kL t b kL t

a t kL kL n

ω ω

ω

ω ω

ω π

= +

+ = → = −

= − + + =

= → =

1.4.3 Standing wave ( 정상파 )

Waves where boundary conditions must be imposed, (ex, a violin string) To describe this, sum up traveling waves moving in opposite direction

( ) , 2 sin ( ) sin n x 1, 2, 3

A x t a t n

L

ω π =

∴ = 

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( )

2 2

2

2 2

1

p

A A

x υ t

∂ = ∂

∂ ∂

2

2 k

p

E dA

dt

E A

 

∝    

∝ ( )

( )

( ) ( )

0 2

2 2 0

2

2 2 0

2 2

( , ) cos ,

cos

A x t A kx t

p

k

A k A kx t

x

A A kx t

t

k A kx t

ω υ ω

ω

ω ω

υ ω

= − =

∂ = − −

∂

∂ = − −

∂

= − −

1.4.4 Energy associated with wave amplitude

- For an oscillating string, each small segment of the string undergoes periodic oscillation

- The energy in the string: expressed essentially the same way as a HO

1.4.5 Wave equation

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원자보다 작은 세계 이해하기 . Concepts in quantum mechanics

“1900년에서 1930년까지의 30년 동안은 뉴턴 역학으로 대표되는 고전물리학이

퇴장하고 양자물리학으로 대표되는 현대물리학이 등장한 시기였다

.

이 시기에 등장한 중요한 개념은

–

에너지를 비롯한 물리량이 연속된 양이 아니라 불연속적인 양이다

. –

입자와 파동은 서로 다른 물리적 대상물이 가지는 성질이 아니라 같은

대상물이 가지는 두 가지 측면이다

.

–

원자보다 작은 세계에서 일어나는 현상을 제대로 기술하기 위해서는 입자와 파동의 두 가지 측면을 모두 고려해야 한다

.” by 곽영직

불연속적인 물리량을 다룰 수 있는 새로운 물리학이 필요: 행렬역학, 파동역학

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Early history of quantum mechanics

• 1.6 Particle nature of light

• 1.6.1 Blackbody radiation (M. Planck, 1900)

Planck postulated that E seen in the blackbody spectrum comes in

discrete qunata of magnitude, E = h ν

• 1.6.2 Photoelectric effect (A. Einstein, 1905) KE = h ν – h ν

₀

• 1.7 Wave nature of particles

• 1.7.1 Atomic spectra and the Bohr model

The electron orbits around he proton in circular orbits with

• 1.7.2 de Broglie waves and electron diffraction

• 1.8 Uncertainty principle

Several key experiments that were in conflict with the predictions of CM 1) Light could not be described exclusively using wave theory

2) Particle possesses wave-like properties

∆ ∆ ≥ x p 

h λ = p

(orbital angular momentum)

l = n 

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1.6 Particle nature of light

• 1.6.1 Blackbody radiation

The distribution of frequencies of light emitted by a heated solid

( )

5

Planck distribution:

8

hc

1

kT

hc e

^λ

ρ π

= λ

− Quantization of energy: E = nh ν

+ classical statistical mechanics

( )

^B

h

P E e k T

− ν

=

max

3000

T λ ≈ µ m K ⋅

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1.6 Particle nature of light

• 1.6.2 Photoelectric effect

Emission of electrons from the surface of a metal irradiated with light

0

34

: work function 6.626 10 sec KE h h

h

h J

ν ν ν

−

= −

= ×

Each electrons absorbs

one quantum of energy

equal to h ν

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1.7 Wave nature of particles

• 1.7.1 Atomic spectra and the Bohr model

consists of discrete lines in regular patterns

E h ν

∆ =

N. Bohr (1911):

orbital angular momentum l = nh/2 π

• Works well for one-e atom

• fails for atoms where two e’s interact strongly

• Approximation to the correct formulation of QM

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1.7 Wave nature of particles

• 1.7.2 de Broglie waves and electron diffraction

In 1923 de Broglie postulates e’ s and other particles have waves

In 1927, Davison and Germer observed the diffraction of e’s In 1932, Stern observed the diffraction of He and H

₂

h λ = p

diffraction of particles:

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1.8 Uncertainty principle

In 1925 by Heisenberg

It is impossible to measure the p and x of a particle simultaneously to arbitrary precision

x 2

∆ ∆ ≥ x p 

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1.9 Discovery of quantum mechanics

• 1.9.1 Schrodinger wave mechanics and Heisenberg matrix mechanics

– In 1925, Schrodinger solved wave equation for the H atom – The physical meaning of wave function Ψ

Born and Copenhagen interpretation: ІΨІ

²

is the probability density for finding the particle at a particular location

• 1.9.2 Relativistic QM

– In 1929, Dirac introduced spin to the wavefunction to deal with relativistic contribution: Dirac equation

– The application of the Dirac equation to the motions of electrons,

subject only to Coulomb interactions between them, correctly describes

all the properties of electrons that we are aware of. The corresponding

description of protons, neutrons, and other heavy particles requires the

introduction of nuclear forces.

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1.10 Concepts in quantum mechanics

– Particle is described as waves with an associated wavefunction Ψ – Ψ behaves like classical waves:

exhibiting diffraction and satisfying b.c. that leads to discrete energy levels – Some important distinctions between quantum waves and classical waves

( ) ( )

( )

cos in CM in QM

~ exp

i kx t

kx t e

p h k

E h i px Et

k p

E ω

ω

λ

ν ω ω

−

=

−

 

Ψ  −

→

= = ∴

= =



∴

∴  



2 2 2 2 2

2 2 2

2

2 2

2

1

2 2

since for a free particle, 2

2

iE E i

t t

p p

x m m x

E p

m

i t m x

∂Ψ − Ψ = → = ∂Ψ

∂ ∂ Ψ

∂ Ψ = − Ψ → = − ∂ Ψ

∂ ∂ Ψ

=

∂Ψ = − ∂ Ψ

∂ ∂

 



 

2 2 2 2

, 1 in CM

2

_p

A A

i cf

t m x x υ t

 

∂Ψ ∂ = − ∂ Ψ ∂    ∂ ∂ = ∂ ∂   

 

Schrodinger eqn for a free particle:

(properties of particles)