Light

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Lecture 2
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Light

• Classical (Wave) Description
• Light is an EM wave 100 nmlt l lt10 microns
• Quantum (Particle) Description
• Localized, massless quanta of energy - photons
• Wave / Particle Duality
• Appropriate description depends on experimental
  device examining light

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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.ht
  ml

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IIA. Classical Description of Light
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Classical Description of Light
Wave Equation (derived from Maxwells
equations) Any function that satisfies this eqn
is a wave It describes light propagation in free
space and in time
(see calculus review handout)
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IIA. Classical Description of Light

• Plane Wave Solution
• One useful solution is for plane wave

E
B
r
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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 (2-D) 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.

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IIA. Classical Description of Light

• Wave number and angular frequency

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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

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Light as photon particles

• Energy of EM wave is quantized
• Light consists of localized, massless quanta of
  energy -photons
• Ehn
• hPlancks constant6.63x10-34 Js
• nfrequency
• Photon has momentum,p, associated with it
• ph/lhn/c

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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

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IIC. Wave / Particle Duality

• Photons versus EM waves (continued)

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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

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II. Light

• Classical (Wave) Description
• Light is an EM wave 100 nmlt l lt10 microns
• Quantum (Particle) Description
• Localized, massless quanta of energy - photons
• Wave / Particle Duality
• Appropriate description depends on experimental
  device examining light

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IV. Light-Matter Interactions

• Atomic spectrum of hydrogen
• B. Wave mechanics
• C. Atomic orbitals
• D. Molecular orbitals

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IVA. Atomic Spectra

• 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
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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?

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IVA. Atomic Spectra

• 3a. Bohr quantum model of the hydrogen atom
• In 1913, Bohr provided the first successful
  explanation of atomic spectra of hydrogen
• Bohrs model was only successful in explaining
  the spectral behavior of simple atoms such as
  hydrogen
• Bohrs model was abandoned in 1925 because it had
  flawed assumptions and could not be applied to
  more complex atomic systems.

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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)
• k Coulombs const (9 x 109 N.m2/C2)

F
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IVA. Atomic Spectra

• 3b. Bohr postulate (2) angular momentum
  quantization
• Angular momentum (L) for a particle in a circular
  path is
• 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

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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
• where,
• Ei energy of initial state
• Ef energy of final state
• f frequency
• h Plancks constant

-e

hf
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IVA. Atomic Spectra

• 3c. Allowed energies
• Using the assumptions in Bohrs postulates
  (planetary model and quantization), an expression
  for the allowed energies was developed.

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IVA. Atomic Spectra

• 3f. Orbital and Energy level diagram

E -0.54 E -0.85 E -1.51 E -3.4 E -13.6 eV
n5 n4 n3 n2 n1
Orbital
n3
n1
Energy Level Diagram
n2
n4
n5
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IVA. Atomic Spectra

• 3d. Spectral wavelengths
• If electron jumps from one orbit (ni) to a
  second orbit (nf), the energy difference is
• The corresponding frequency and wavelengths are

http//www.colorado.edu/physics/2000/quantumzone/b
ohr.html
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IVA. Atomic Spectra

• 3f. Abandonment of the Bohr Model
• Hard to describe complex atoms and assumptions
  lack foundation
• Heisenbergs 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 Broglies and Schrodinger wave description of
  light overcame the limitations of the Bohr model

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IVA. Atomic Spectra

• 4. Wave mechanics
• By mid-1920s it was apparent that Bohrs 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

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IV. Light-Matter Interactions

• Atomic spectrum of hydrogen
• B. Wave mechanics
• C. Atomic orbitals
• D. Molecular orbitals

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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)
• 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.

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Wave Mechanics

• The wave function, ?
• De Broglie waves can be represented by a simple
  quantity Y, 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

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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
• where, Vpotential energy, E total energy

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Atomic Orbitals

• For an atom, use Schrödingers 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

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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

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Atomic Orbitals

• Wave functions of hydrogen
• The solution of the Schrodinger equation for the
  hydrogen atom is
• 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

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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

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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

d orbital
p orbital
s orbital
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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 l0, m0
p orbital l1, m-1,0,1
d orbital l2, m-2,-1,0,1,2
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Atomic Orbitals
Calculation of quantum numbers
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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

0 s 1 p 2 d 3 - f
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Atomic Orbitals
Example
Wave Function
Subshell symbol
ml
l
n
Y1,0,0
1s
0
0
1
2
3
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Atomic Orbitals

• Orbital shapes
• Solution of the Schrodinger wave equation for a
  one electron atom

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Atomic Orbitals

• Electron probability distribution

Wave function
Probability
Y1s2
r
r90
A spherical surface that contains 90 of the
total electron probability (orbital
representation)
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Atomic Orbitals
Other orbitals
Wave Function
Subshell symbol
ml
l
n
Y1,0,0
1s
0
0
1
http//www.shef.ac.uk/chemistry/orbitron/AOs/2p/in
dex.html
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Atomic Orbitals
Allowed energies of hydrogen The energy En of
the wave function Ynlm depends only on
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
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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

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Atomic Orbitals
Example
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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
• Hunds rule
• For a given shell (example, n2), the electron
  occupies each subshell one at a time before
  pairing up

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Orbital energies multi-electron atoms
Energy depends on both n and l
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Atomic Orbitals
Example Nitrogen (1s22s22p3)
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Atomic Orbitals
Example Carbon
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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)