Transcript Chapter 7

Topic I: Quantum theory
Chapter 7
Introduction to Quantum Theory
Classical physics
• Before 1900
• A particle travels in a trajectory with a precise
position and momentum at each instant.
• Any type of motion can be excited to a state of
arbitrary energy. Energy is continuous.
• Wave and particle are two distinct concepts.
Failures of classical physics
Crucial experimental observations:
1. Black-body radiation (Planck in 1900)
2. Heat capacities of solids (Einstein 1905)
3. Photoelectric effect (Einstein in 1905)
4. Diffraction of electrons (Davisson and
Germer in 1925)
Black-body radiation
The radiation from the pinhole in a close
contained is characteristic of the EM
waves within the container.
• Wien’s displacement law
T λmax  Const
lmax: Wavelength at the
maximum of the distribution
• Stefan-Boltzmann law
M  aT 4
M: Total emitted power
Classical theory of EM waves for the black-body radiation
Rayleigh – Jeans law
dE  ρ( λ, T )dλ
r (l T): Density of states
8π k BT
ρ (l, T ) 
4
λ
The Rayleigh – Jeans law is quite
successful at long wavelengths (low
frequencies). It fails badly at short
wavelengths (high frequencies).
In classical theory, the energies of
the EM waves are continuous.
Planck’s postulation (in 1900)
The energy of the EM wave is limited to
discrete values and can not varied arbitrary.
E  nhν with n  0,1,2,3,....
E : The energy of the EM wave
n : The frequency of the EM wave
h (Planck’s const) = 6.626 * 10-34 J·s
Planck’s distribution
r
8hc
l5 e hc / lk
BT

1
Quantization of Energy !
Heat capacities of solids
Experimental observation
• Classical physics predicts the
molar heat capacities of solids
close to 3R = 25 J K-1 mol-1.
• This is called Dulong-Petit law.
U m  3N A RT
 U 
CV ,m   m   3RT
 T V
Einstein’s correction
• Einstein uses Planck’s hypothesis for quantization of energy.
• The permitted oscillation energy of an atom is an integral
multiple of hn, with n being the frequency of the oscillation.
CV  3Rf (T )
2

hn / k BT  e hn / k T
f (T ) 
B
e
hn / k BT

1
2
q E = hn /kB : Einstein temperature
Evidences for quantization of energy:
Atomic and molecular spectra
Emission spectrum of H atoms
Emission spectrum of iron atoms
Emission spectrum of SO2 molecules
Discrete energy levels of atoms and molecules
• The features of these spectra are a series of
discrete frequencies.
• The energy of an atom or a molecule is
confined to discrete values, called the
allowed energy states or levels.
• The atom or molecule only jumps between
the discrete energy levels.
• As an atom jumps from one energy level Ei
to another level Ej, the frequency n of the
radiation is related to the energy difference
DE of the two levels.
Bohr frequency
condition
hn  DE
 Ei  E j
Quantization of Energy
• Planck’s distribution agrees well with the
experiment results and accounts for Wien’s law
and Stefan-Boltzmann law.
• Energy of a EM wave is quantized and should
be discrete, but not continuous.
• What is the physical reason for the success of
Planck’s quantization hypothesis for energy?
Two hypotheses for interpretation of light
• Wave hypothesis:
Light is an electromagnetic wave.
Quantity: frequency n and wave length l
Supporting experiments: Diffraction, interference
and EM waves
• Particle hypothesis:
Light is a stream of particles, called photons.
Quantity: energy E and momentum p
Supporting experiments: Reflection, refraction and
photoelectric effect
The photoelectric effect
Kinetic energy of
photoelectrons
Ek 
1
me v 2  hn  
2
Work function of a metal
• No electrons are ejected, regardless
of the intensity of the radiation,
unless the frequency exceeds a
threshold value characteristic of the
metal.
• The kinetic energy of the ejected
electrons varies linearly with the
frequency of the incident radiation
but is independent of its intensity.
• Even at low intensities of light,
electrons are ejected immediately if
the frequency is above the threshold
value.
: Work function of a metal
Definition:
Work function is the energy
required to remove an electron
from the metal to infinity.
Work function is a characteristic
of the metal.
These three conclusions from experiments can not be
understood by the theory of electromagnetic waves.
Einstein’s interpretation
• Light is a stream of particles, called
photons, rather than EM waves.
• The energy of each photon is hn.
• As hn < , no electrons are ejected
out. So, the threshold frequency nc is
/h, which is decided by the metal.
• As hn > , the kinetic energy of the
ejected electron increases linearly with
the frequency of the radiation.
• Interpretation for the photoelectric
effect: right after an electron collides
with a photon with sufficient energy,
the electron is ejected out from the
metal.
• After the collision, the electron is
ejected out immediately.
The particle character of
electromagnetic radiation
Diffraction of electrons
• Diffraction: Interference between waves caused by an object in their path
Diffraction is a typical characteristic of wave
• Electrons are particles.
• In 1925, Davisson and Germer first observed the diffraction of electrons
by a crystal.
• Conclusion: electrons have wave-like properties.
• The Davisson-Germer experiment has been repeated with other particles,
including a particles and molecular H.
The wave character of particles
Wave - particle duality: de Broglie’s hypothesis
•
•
Proposed by de Broglie in 1924
“Wave” have particle-like
properties
1. Black-body radiation
2. Photoelectric effect
•
“Particles” have wave-like
properties
Diffraction of electrons
•
A particle traveling with a
momentum p should have a
wavelength l
de Broglie’s relation between p
and l
Momentum p: quantity of particle
Wavelength l: quantity of wave
λ 
h
p
Microscopic dynamics: Quantum mechanics
• Classical physics failed to account for the existence
of discrete energies of atoms and other experiments in
the early 20th century.
• Such total failures show that the basic concept of
classical mechanics need to be corrected
fundamentally.
• A new mechanics had to be developed to take its
place.
• The new mechanism is called quantum mechanics,
and opens a new era of physics.
Wave mechanics: Schrodinger’s equation
In 1926, Schrodinger’s interpretation for matter wave
Rather than traveling along a path, a particle is spread
through space like a wave described by a wave
function.
Yx, t: Wave function of the matter wave
Yx, t is a complex-variable function of x and t
What’s the equation for the wave function of a particle moving in a potential?
• In classical mechanics
• In quantum mechanics
p2
H
 V ( x)
2m
2
ˆ
p
Hˆ 
 V ( xˆ )
2m
x, p and H are scalar variables.
xˆ  x
pˆ 
 
i x

ˆ
H  i
t

xˆ , pˆ and Hˆ are operators.
h
2

2 2
i  
 V ( x)
2
t
2m x
Time-dependent Schrodinger equation
Ψ(x,t)
 2  2Ψ(x,t)
i

 V(x)Ψ(x , t )
2
t
2m x
Ψ(x,t)  e
 iE t/
ψ(x)
Time-independent Schrodinger equation
 2 d 2ψ(x)

 V(x)ψ(x)  Eψ ( x)
2
2m dx
Born interpretation of the wave function
Yx, t: Probability amplitude
| Yx, t|2: Probability density to
find the particle at x and at time t
Normalization of a wave function
Since the total probability of finding a
particle is one, the integration of the
probability density over all space
should be one. This is the
normalization of a wave function.
Properties of wave functions
• The curvature of a wave function is related to the
kinetic energy of a particle: A higher curvature of a
wave function implies the particle has a higher kinetic
energy at that region.
• (a) The wave function should be single value
everywhere.
(b) The wave function can not be infinite over a finite
region of space.
(c) The wave function and it’s slope should be
continuous everywhere.
• The wave function must satisfy certain boundary
conditions.
• Only certain values of energy E give the acceptable
solutions of Schrodinger’s equation subject to the
boundary conditions. This leads to the quantization in
energy of the system.
A particle freely moving in an infinite square-well potential
Schrodinger equation
General solutions
 2 d 2ψ(x)

 Eψ ( x)
2
2m dx
(x)  A sin (kx), k 
2
2mE

l

Subject to boundary conditions
2L
l
, n  1,2,3,  
n
n: quantum number
Acceptable normalized wave functions
 n(x) 
2
 nx 
sin 
,
L
 L 
n  1,2,3,  
The uncertainty principle
In 1927, Heisenberg proposed that
It is impossible to specify simultaneously, with arbitrary
precision, both momentum and position of a particle.
I. If the momentum of the particle is known exactly
For a particle with a definite linear momentum p in free space, the wave
function YxAsin(kx), with k=2 p/h, spreads out through the whole
space so that the position of the particle is completely undetermined.
II. If the position of the particle is known exactly
For a particle with a precise location, the wave function Y(x) is sharply
localized and is only constructed from the superposition of many functions
of Asin(kx) with different k. So, the momentum of the particle is
completely undetermined.
Dx Dp x  
A particle with well-defined position
2
A particle with ill-defined position
Particle with position quite uncertain but
momentum quite certain
Particle with position quite certain but
momentum quite uncertain
Exercises
• 7A.2, 7A.3, 7A.9
• 7B.4, 7B.7
• 7C.3, 7C.7, 7C.14