physical chemistry ii (chapter 1) and background to
TRANSCRIPT
Physical Chemistry II(Chapter 1)
Introduction and Background to Quantum Mechanics
Tae Kyu Kim
Department of Chemistry
Rm. 301 ([email protected])
http://cafe.naver.com/monero76
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PHSICAL CHEMISTRY
THERMODYANMICS: The properties of bulk materials (PC I)
QUANTUM MECHANICS: The properties of atoms and molecules
To understand the structures of individual atoms and molecules, we need to know how subatomic particles move in response to the forces they experience…
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Synopsis of Chapter 1
We start with a review of how scientists classify the behavior of the motion of matter. There are several mathematical ways to describe motion, Newton’s laws being the most common. A quick historical review shows that several phenomena could not be explained by the scientific thinking of the 1800s. Most of these phenomena were based on the properties of atoms that were only then being examined directly. These phenomena are described here because they will be considered later in light of new theories such as quantum mechanics. Of course, since most matter is ultimately studied using light, a proper understanding of the nature of light is necessary. This understanding began to change dramatically with Planck and his quantum theory of black‐bodies. Proposed in 1900, quantum theory opened new age of science in which new ideas began replacing the old ones, not because of lack of application, but because old ideas lacked the subtlety to explain newly observed phenomena properly. Einstein’s application of quantum theory to light in 1905 was a crucial step. Finally, Bohr’s theory of hydrogen, de Broglie’s matter waves, and other new ideas set the stage for the introduction of modern quantum mechanics.
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Classical Mechanics‐1• Energy: The capacity to do work
‐ Etotal = Ek + Ep = Ek + V(x,y,z)
‐ Kinetic Energy (Particle’s motion)
‐ Potential Energy (Particle’s position)
→ Force can de derived:
• Coulomb potential• Gravitational potential• Harmonic potential
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2kE m
dVF
dx
1 2
0
2
( )4
( )
1( )
2
q qV r
V h mgh
V x kx
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Classical Mechanics‐2
• Newton’s Second Law
2
2
dv d dx d xF ma m m m
dt dt dt dt
dVF
dx
2
2
1 ( )d x dV x
dt m dx
2
2
2
22
2
1for HO ( ),
2
1
0, where
( ) cos sin
dVV kx F kx
dx
d x dV kx
dt m dx m
d x kx
dt mx t A t B t
Exercise 1.2
22
2
2 2
second order linear differential equation
0 exp
0
exp exp
d yk y y mx
dx
m k y m ik
y A ikx B ikx
exp cos sini i
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Classical Mechanics‐3
• Energy Conservation and Classical Hamiltonian
2
( )2
pH V x
m
p mv
Conserved system: H = E = constant for systems with V independent of time
2
2 2
2 2
1( )
2
0
dH dmv V x
dt dt
dx d x dV dx dx d x dVm m
dt dt dx dt dt dt dx
2 2 2
2 22 2 2
( , )0
dv dv d x dx d xv v
dt dt dt dt dtdV x t V dx V dt dV dx V
dt x dt t dt dx dt t
0
22
2
0
1 1,
2 2
2 2
2 ( ( ))2
x
x
dxE m V x E V x m
dt
dx dxE V x E V x
dt m dt m
mdx dxdt t t m
m E V xm E V x
definite ( ( ), ( ))x t p t
• Deterministic trajectory
• Energy can be increased to any value (continuous E)
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Classical Mechanics‐4• Classical Model for Rotational Motion
2 : angular velocityrv r
22 2 2 21 1 1 1: moment of inertia
2 2 2 2K m m r mr I I
L I Angular momentum: fundamental
quantity in rotating system
2 22
2 22
1 1
2 2 2
1 1
2 2 2
m pK m
m m
I LK I
I m
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Classical Mechanics‐5• Classical Wave Theory
− One way to describe the motion: To specify the perpendicular
displacement of the surface from its equilibrium position as a function of time and as a function of the distance parallel to the surface, A(x,t)
−Mathematical expression for traveling wave: sin or cos functions
0( , ) cosA x t A kx wt 0: amplitude
: wave vector 2 /
: angular frequency 2 / 2
A
k k
w w v
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Classical Mechanics‐6• A traveling wave
0( , ) cosA x t A kx wt kx wt const
p p
w wx t const t const v
k k
• Superposition and Diffraction
nλ = d sin θ
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Classical Mechanics‐7• Standing wave
− Bound conditions must be imposed (violin string)− Sum up traveling waves moving in opposite direction
( , ) cos cos
b.c. 0, , 0
, 2 sin sin 1, 2,3
A x t a kx t b kx t
A t A L t
n xA x t a t n
L
(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
/ 2L n
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Classical Mechanics‐8• Energy associated with wave amplitude
− For an oscillating string (standing wave), each segment of the string undergoes periodic oscillation
− The energy in the string can be expressed essentially the same way as a harmonic oscillator (spring)
, 2 sin sin 1, 2,3n x
A x t a t nL
2
2
k
p
dAE
dt
E A
• Wave equation
2 2
22 2
1
p
A A
x t
0
22
02
22
02
220
( , ) cos ,
cos
cos
cos
p
p
A x t A kx tk
Ak A kx t
x
AA kx t
t
k A kx t
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Early History of Quantum Mechanics• Summary for CM
− A clear distinction is made between particles and waves− A particle has a well‐defined mass and its position and velocity (momentum) can be accurately determined as a function of time by applying Newton’s laws of motion.− A wave does not have mass in normal sense and cannot be precisely located: best described in terms of frequency and wavelength−Waves also have the important property that they can interact with one another to produce interference patterns: known as diffraction.
• Several key experiments were in conflict with the CM theory− Light (electromagnetic wave) could not be described exclusively using wave theory− Particles possessed wavelike properties
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The Nature of Light (Particle?)• Blackbody radiation
− The distribution of frequencies of light emitted by a heated solid
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max
( )
Planck distribution:
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1
3000
B
h
k T
hckT
P E e
hc
e
T m K
Quantization of energy: E = nh+ classical statistical mechanics
Rayleigh‐Jeans law
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The Nature of Light (Particle?)• Photoelectric Effect
− Emission of electron from the surface of a metal irradiated with light
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: work function
6.626 10 [J s]
KE h
h
Photon [광자, 빛의뭉치]
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Wave Nature of Particles• Atomic spectra and Bohr’s model
− The spectra consisted of discrete lines, often in regular patterns
Bohr’s model (1911): classical model for H atom in
which it is imagined that the electron orbits around the proton in circular orbits with orbital angular momentum quantization:
/ 2l n nh
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Wave Nature of Particles• De Broglie Waves
− In 1923, de Broglie postulated that electrons and other particles have wave associated with them, and that the wavelength of these waves is given by
− If the particles have waves associated with them, then it must be possible to observe particle diffraction similar to the diffraction of light
• Electron diffraction− Davisson and Germer (1927): Electron diffraction− Stern (1932): He and H2 diffraction
h
p
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Summary for Early History of QM
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Uncertainty Principle• Particle‐Wave Duality
−We must consider light and matter as either wave or particles: an awkward result− The measurement of the position of an electron
If we wish to locate the electron within a distance Δx, then we must use light with a wavelength at least small. For the electron to be seen, a photon must interact or collide in some way with the electron, for otherwise the photon will just pass right by and the electron will appear transparent. The photon has a momentum p = h/λ and during the collision some of this momentum will be transferred to the electron. The very act of locating the electron leads to a change in its momentum. If we wish to locate the electron more accurately, we must use light with a smaller wavelength. Consequently the photons in the light beam will have a greater momentum. Because some of the photon’s momentum must be transferred to the electron in the process of locating it, the momentum change of the electron becomes greater.
− It is impossible to measure the p and x of a particle simultaneously to arbitrary precision
2xx p
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Uncertainty Principle→ It is impossible to measure the p and x of a particle simultaneously to
arbitrary precision
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Discovery of QM• Schrödinger Wave Mechanics
−What are the meaning and form of the wave of particle?− In 1925, Schrödinger solved wave equation for the H atom− PHYSICAL MEANING OF WAVEFUNCTION (Ψ)
− Born & Copenhagen interpretation to postulate that Ψ is a probability amplitude which means that |Ψ|2 is the probability density for finding the particle at a particular location
전자의 파동성은 확률에서 온다!
오비탈
코펜하겐 해석관측은 대상을 변화시킨다.
(우주는 큰세계와 작은세계로 나뉜다)
코펜하겐 해석관측은 대상을 변화시킨다.
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Concepts in QM
− Particles must be described as waves with an associated wavefunction Ψ− Ψ behaves like the amplitude of a classical wave (diffraction phenomena and satisfying boundary conditions that leads to discrete energy levels)− Some important distinctions b/t quantum wave and classical wave− Free particles
2cos expkx t i kx t const
~ exp
h pp k k
EE h
ipx Et
(properties of particles)Schrodinger equation for a free particle:
2 2 2 2
2 2 2 2
1 cf,
2 p
A Ai
t m x x t
2 2 2 2 2
2 2 2
2
1
1
2 2
since for a free particle2
iEE i
t t
p p
x m m x
pE
m
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Postulates of Quantum Mechanics
• POSTULATE 1: Wavefunction (Ψ) is a well‐behaved, square integrable function and has all the information of a system. |Ψ|2 is the probability of finding the particle• POSTULATE 2: For every observable property, there is a linear Hermitian operator.• POSTULATE 3: Ψ can be obtained by solving time‐dependent Schrödinger equation:
• POSTULATE 4: If for a system Ψa, we always get a as aresults of measurements.• POSTULATE 5: The average(expectation) value of an observable Ais given by
ˆi Ht
ˆa aA a
*
*
A dA
d