Download Lecture 2 - Tufts University

Lecture 2
Light
A.
Classical (Wave) Description
Light is an EM wave: 100 nm< l <10 microns
B.
Quantum (Particle) Description
Localized, massless quanta of energy - photons
C.
Wave / Particle Duality
Appropriate description depends on experimental device
examining light
IIA. Classical Description of
Light
Properties of EM waves
•
Electromagnetic radiation can be considered to behave as two
wave motions at right angles to each other and to the direction of
propagation
•
One of these waves is electric (E) and the other is magnetic (B)
•
These waves are functions of space and time
http://www.phy.ntnu.edu.tw/~hwang/emWave/emWave.html
IIA. Classical Description of
Light
Classical
Description of Light

2

1  E
2
 E 2 2
Wave Equation (derived from Maxwell’s equations)
c t
Any function that satisfies this eqn is a wave

2

1  B It describes light propagation in free space and in time
2
 B 2 2
c t
where,
c  speed of light

E  electric field

B  magnetic induction field
  Laplacian operator
2
(see calculus review handout)
IIA. Classical Description of
Light
Plane Wave Solution
One useful solution is for plane wave
  i kr  i t   i kr t 
E  Eo e e
 Eo e
  i kr t 
B  Bo e
E
r
where,
k  wave number or propagatio n vector
B
  angular frequency
Plane wave
• A plane wave in two or three dimensions is like a sine wave in one
dimension except that crests and troughs aren't points, but form lines (2D) or planes (3-D) perpendicular to the direction of wave propagation. The
Figure shows a plane sine wave in two dimensions. The large arrow is a
vector called the wave vector, which defines (1) the direction of wave
propagation by its orientation perpendicular to the wave fronts, and (2) the
wavenumber by its length.
IIA. Classical Description of
Light
Wave number and angular frequency
k  2
l
  2f  2c / l
IIB. Quantum Description of
Light
Historical perspective
•
Max Planck (1858-1947) - Introduced concept of light energy or
“quanta” (blackbody radiation) and the “Planck” constant
•
Albert Einstein (1879-1955) - Proof for particle behavior of light
came from the experiment of the photoelectric effect
Light as photon particles
• Energy of EM wave is quantized
• Light consists of localized, massless
quanta of energy -photons
E=hn
 h=Planck’s constant=6.63x10-34 Js
 n=frequency
• Photon has momentum,p, associated with
it
• p=h/l=hn/c
IIC. Wave / Particle Duality
Photons versus EM waves
•
Light is a particle and has wave like behavior
•
The photon concept and the wave theory of light complement each
other
•
Depends on the specific phenomenon being observed
IIC. Wave / Particle Duality
Photons versus EM waves (continued)
IIC. Wave / Particle Duality
•
High frequency (X-rays)
Momentum and energy of photon increase
Photon description dominates
•
Low Frequency (radio waves)
Interference/diffraction easily observable
Wave description dominates
II. Light
A.
Classical (Wave) Description
Light is an EM wave: 100 nm< l <10 microns
B.
Quantum (Particle) Description
Localized, massless quanta of energy - photons
C.
Wave / Particle Duality
Appropriate description depends on experimental device
examining light
IV. Light-Matter Interactions
A. Atomic spectrum of hydrogen
B. Wave mechanics
C. Atomic orbitals
D. Molecular orbitals
IVA. Atomic Spectra
1.
Atomic spectrum of hydrogen
•
When hydrogen receives a high energy spark, the hydrogen atoms
are excited and contain excess energy
The hydrogen will release the energy by emitting light of various
wavelengths
The line spectrum (intensity vs. wavelength) is characteristic of the
particular element (hydrogen)
•
•
H
Spectrometer
IVA. Atomic Spectra
2. What is the significance of the line spectrum of H?
•
When white light (sunlight) is passed through a prism, the spectrum
is continuous (all visible wavelengths)
•
In contrast, when hydrogen emission spectrum is passed though a
prism, only a few lines are seen corresponding to discrete
wavelengths.
•
This suggests that only certain wavelengths (energies) are allowed
for the electron in the hydrogen atom. But why?
IVA. Atomic Spectra
3a. Bohr quantum model of the hydrogen atom
•
In 1913, Bohr provided the first successful explanation of atomic
spectra of hydrogen
•
Bohr’s model was only successful in explaining the spectral
behavior of simple atoms such as hydrogen
•
Bohr’s model was abandoned in 1925 because it had flawed
assumptions and could not be applied to more complex atomic
systems.
IVA. Atomic Spectra
3b. Bohr postulate (1): Planetary model
•
Electron has circular orbit about nucleus
•
Particle in motion moves in a straight line and can be made to
travel in a circular orbit by the application of a coulombic force of
attraction (F) between electron (-e) and nucleus (+e)
ke2
mv 2
F  2  ma 
r
r
k = Coulomb’s const (9 x 109 N.m2/C2)
m (mass) -e F
v (velocity)
+
+e
r (radius)
IVA. Atomic Spectra
3b. Bohr postulate (2): angular momentum quantization
•
Angular momentum (L) for a particle in a circular path is:
L  mvr
•
Bohr assumed that the angular momentum (L) of the electron
could occur only in certain increments (quantized) to fit the
experimental results of hydrogen spectrum
nh
L  mvr 
; n  1,2,3....
2
IVA. Atomic Spectra
3b. Bohr postulates (3) and (4):
•
Stationary states: electron can move in one of its allowed orbits
without radiating energy
•
Energy: Atoms radiate energy when electron jumps from one
stationary state to another. The frequency of radiation obeys
the condition:
hn  Ef  Ei
where,
Ei = energy of initial state
Ef = energy of final state
f = frequency
h = Planck’s constant
-e
hf
+
IVA. Atomic Spectra
3c. Allowed energies
•
Using the assumptions in Bohr’s postulates (planetary model and
quantization), an expression for the allowed energies was
developed.
mk e  1 
En  
;
2  2 
2h  n 
n  1, 2, 3....( increa sin g orbital radius )
2 4
h
h
2
IVA. Atomic Spectra
3f. Orbital and Energy level diagram
Orbital
n=5
n=4
n=3
n=2
E= -0.54
E= -0.85
E= -1.51
E= -3.4
n=3
n=1
E= -13.6
eV
n=1
n=2 n=4
n=5
Energy Level Diagram
IVA. Atomic Spectra
3d. Spectral wavelengths
•
If electron jumps from one orbit (ni) to a second orbit (nf),
difference is:
En  Eni  Enf  hn
•
The corresponding frequency and wavelengths are:
E i  Ef
n
; sin ce n  c / l
h
1 n E i  Ef
 
l c
hc
http://www.colorado.edu/physics/2000/quantumzone/bohr.html
the energy
IVA. Atomic Spectra
3f. Abandonment of the Bohr Model
•
Hard to describe complex atoms and assumptions lack
foundation
•
Heisenberg’s uncertainty principle showed that it was impossible
to know the exact path of the electron as it moves around the
nucleus as Bohr had predicted.
•
De Broglie’s and Schrodinger wave description of light overcame
the limitations of the Bohr model
IVA. Atomic Spectra
4. Wave mechanics
•
By mid-1920’s it was apparent that Bohr’s model did not work
•
Louis De Broglie, and Erwin Schrodinger developed wave
mechanics
•
Wave mechanics is the current theory used to describe the
behavior of atomic systems
IV. Light-Matter Interactions
A. Atomic spectrum of hydrogen
B. Wave mechanics
C. Atomic orbitals
D. Molecular orbitals
Properties of atoms
•Atoms consist of subatomic structures. For this
course, we think of atoms consisting of a nucleus
(positively charged) surrounded by electrons
(negatively charged)
-e
+
hf
•Internal energy of matter is of discrete values (it is
quantized)---line spectra of elements such as H.
•It is impossible to measure simultaneously with
complete precision both the position and the
velocity of an electron (or a particle). (Heisenberg
uncertainty principle)
•Think in terms of a probability of finding a particle
within a given space at a given time and discrete
energy levels associated with it---wave function.
p x x 
h
2
E t 
h
2
Wave Mechanics
The wave function, 
• De Broglie waves can be represented by a simple quantity ,
called a wave function, which is a complex function of time and
position
• A particle is completely described in quantum mechanics by the
wave function
• A specific wave function for an electron is called an orbital
• The wave function can be be used to determine the energy levels
of an atomic system
Wave Mechanics
Time-independent Schrodinger equation
Since potential energy is zero inside box, the only possible energy
is kinetic energy
For a particle confined to moving along the x-axis:
h 


V


E

2
2m x
2
2
where, V=potential energy, E = total energy
4
Atomic Orbitals
• For an atom, use Schrödinger’s equation
• Find permissible energy levels for electrons
around nucleus.
• For each energy level, the wave function defines
an orbital, a region where the probability of
finding an electron is high
• The orbital properties of greatest interest are
size, shape (described by wave function) and
energy.
• Solution for multi-electron atoms is a very
difficult problem, and approximations are
typically used
Atomic Orbitals
The hydrogen atom
The electron of the hydrogen atom moves in three dimensions and
has potential energy (attraction to positively charged nucleus)
The Schrodinger equation can be solved to find the wave functions
associated with the hydrogen atom
In 1-D particle in a box, the wave function is a function of one
quantum number; the 3-D hydrogen atom is a function of three
quantum numbers
Atomic Orbitals
Wave functions of hydrogen
The solution of the Schrodinger equation for the
hydrogen atom is:
 n,l ,m r , ,    Rn,l r lm  ,  ;
Rnl describes how wave function varies with distance of
electron from nucleus
Ylm describes the angular dependence of the wave function
Subscripts n, l and m are quantum numbers of hydrogen
Atomic Orbitals
Principal quantum number, n
Has integral values of 1,2,3…… and is related to size and energy of
the orbital
As n increases, the orbital becomes larger and the electron is
farther from the nucleus
As n increases, the orbital has higher energy (less negative) and is
less tightly bound to the nucleus
Atomic Orbitals
Angular quantum number, l
Can have values of 0 to n-1 for each
value of n and relates to the angular
momentum of the electron in an orbital
The dependence of the wave function
on l, determines the shape of the orbitals
The value of l, for a particular
orbital is commonly assigned a
letter:
0–s
1–p
2–d
3–f
s orbital
p orbital
d orbital
Atomic Orbitals
Magnetic quantum number, ml
Can have integral values between l and - l, including zero and
relates to the orientation in space of the angular momentum.
s orbital:
l=0, m=0
p orbital:
l=1, m=-1,0,1
d orbital:
l=2, m=-2,-1,0,1,2
Atomic Orbitals
Calculation of quantum numbers
Quantum
Numbers
Name
Allowed Values
Allowed
States
n
principal
number
quantum 1,2,3…..
Any number
l
Angular
number
quantum 0,1,2,…(n-1)
n
ml
magnetic
number
quantum -l ,- l+1,…0,( l-1), l
2 l +1
Atomic Orbitals
Shells and subshells
All states with the same principal quantum number are said to form
a shell; the states having specific values of both n and l are said to
form a subshell
Shell (n)
l
Subshell symbol
1
0
1s
2
0
2s
2
1
2p
3
0
3s
3
1
3p
3
2
3d
0–s
1–p
2–d
3-f
Atomic Orbitals
Example
n
l
ml
Wave
Function
Subshell
symbol
1
0
0
1,0,0
1s
2
3
Atomic Orbitals
Orbital shapes
Solution of the Schrodinger
wave equation for a one electron
atom :
 1 
 1,0,0 r , ,    3 
 a o 
 1 
0,0  ,     
 4 
1/ 2
1/ 2
 r
exp  
 ao 
2
10
ao 

0
.
529
x
10
m
2
mke
h

 1.055 x10 34 Js
2
m  mass of electron  9.109 x10 31 kg
k  Coulomb' s constant  8.988 x109 Jm / Cb 2
e  electron charge  1.602 x10 19 Cb
Atomic Orbitals
Electron probability distribution
Wave function
Probability
1s2
1s
r
A spherical surface that
contains 90% of the total
electron probability
(orbital representation)
r
r90%
Atomic Orbitals
Other orbitals
n
l
ml
Wave
Function
Subshell
symbol
1
0
0
1,0,0
1s
2
0
0
2,0,0
2s
2
1
-1
2,1,-1
2p
2
1
0
2,1,0
2p
2
1
1
2,1,1
2p
http://www.shef.ac.uk/chemistry/orbitron/AOs/2p/index.html
Atomic Orbitals
Allowed energies of hydrogen
The energy En of the wave function nlm depends only on n:
4
me
En  2 2 2
8e o h n
m - mass of electron
e - electron charge
h – Planck constant
e – permittivity of free space
Because n is restricted to integer values, energy levels are quantized
Atomic Orbitals: Multi-electron atoms
Electron spin quantum number, ms
This quantum number only has two values: ½ and –½.
This means that the electron has two spin states, thus producing
two oppositely directed magnetic moments
This quantum number doubles the number of allowed states for
each electron.
Pair of electrons in a given orbital must have opposite spins
Atomic Orbitals
Example
n
l
ml
Wave
Function
Subshell
symbol
1
0
0
1,0,0
1s
2
0
0
2,0,0
2s
2
1
-1
2,1,-1
2p
2
1
0
2,1,0
2p
2
1
1
2,1,1
2p
ms
(1/2), (-1/2)
Atomic Orbitals
Pauli exclusion principle
No two electrons can have the same set of quantum numbers: n, l,
ml and ms
Aufbau principle
Electrons fill in the orbitals of successively increasing energy,
starting with the lowest energy orbital
Hund’s rule
For a given shell (example, n=2), the electron occupies each
subshell one at a time before pairing up
Orbital energies: multi-electron
atoms
Energy depends on both n and l
Atomic Orbitals
Example: Nitrogen (1s22s22p3)
n
l
ml
Wave
Function
Subshell
symbol
1
0
0
1,0,0
1s
2
0
0
2,0,0
2s
2
1
-1
2,1,-1
2p
2
1
0
2,1,0
2p
2
1
1
2,1,1
2p
ms
(1/2), (-1/2)
Atomic Orbitals
Example: Carbon
n
l
ml
Wave
Function
Subshell
symbol
1
0
0
1,0,0
1s
2
0
0
2,0,0
2s
2
1
-1
2,1,-1
2p
2
1
0
2,1,0
2p
2
1
1
2,1,1
2p
ms
(1/2), (-1/2)
Atomic Orbitals: Summary
In the quantum mechanical model, the electron is described as a
wave. This leads to a series of wave functions (orbitals) that
describe the possible energies and spatial distribution available to
the electron
The orbitals can be thought of in terms of probability distributions
(square of the wave function)