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4
INTRODUCTION TO QUANTUM
MECHANICS
CHAPTER
4.1 Preliminaries: Wave Motion and Light
4.2 Evidence for Energy Quantization in Atoms
4.3 The Bohr Model: Predicting Discrete Energy
Levels in Atoms
4.4 Evidence for Wave-Particle Duality
4.5 The Schrödinger Equation
4.6 Quantum mechanics of Particle-in-a-Box Models
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Nanometer-Sized Crystals of CdSe
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4.1 PRELIMINARIES: WAVE MOTION AND
LIGHT
- amplitude of the wave: the height or the displacement
- wavelength, l : the distance between two successive crests
- frequency, n : units of waves (or cycles) per second (s-1)
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Electromagnetic Radiation
- A beam of light consists of oscillating electric and magnetic fields
oriented perpendicular to one another and to the direction in which
the light is propagating.
- Amplitude of the electric field
E(x,t) = Emax cos[2p(x/l - nt)]
- The speed, c, of light passing through a vacuum,
c = ln = 2.99792458 x 108 ms-1
is a universal constant; the same for all types of electromagnetic
radiation.
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The electromagnetic spectrum
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Some properties of electromagnetic radiation
reflected by mirrors
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refracted by a prism
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 Interference of waves
- When two light waves pass through the same region of space, they
interfere to create a new wave called the superposition of the two.
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4.2 EVIDENCE FOR ENERGY QUANTIZATION
IN ATOMS
 Blackbody radiation
- Every objects emits energy from its surface in the form of thermal
radiation. This energy is carried by electromagnetic waves.
-The distribution of the wavelength depends on the
temperature.
- The maximum in the radiation intensity
distribution moves to higher frequency
(shorter wavelength) as T increases.
- The radiation intensity falls to zero
at extremely high frequencies for
objects heated to any temperature.
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7000 K
expected from the
classical theory
5000 K
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 Ultraviolet catastrophe
- From classical theory,
- Predicting an infinite intensity at very short wavelengths
↔ The experimental results fall to zero at short wavelengths
expected from the
classical theory
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 Planck’s quantum hypothesis
- The oscillator must gain and lose energy in quanta of magnitude
hn, and that the total energy can take only discrete values:
eOSC = nhn
n = 1, 2, 3, 4, ∙∙∙
Planck’s constant, h = 6.62606896(3) x 10-34 J s
- Radiation intensity
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 Physical meaning of Planck’s explanation
1. The energy of a system can take only discrete values.
2. A quantized oscillator can gain or lose energy only in discrete
amounts DE = hn.
3. To emit energy from higher energy states, T must be sufficiently high.
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 Atomic line spectra and transitions between discrete energy
states
Emission spectra: light from an electrical discharge
Ne
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Ar
Hg
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The spectrograph: emission and absorption spectra
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 Balmer series for hydrogen atoms
The Rydberg constant
n=6
n=5
n=4
n=3
Atomic emission spectra
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 Bohr’s explanation
The frequency of the light absorbed is connected to the energy of
the initial and final states by the expression
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 The Frank-Hertz Experiments
Electrons of known energy collided with gaseous atoms, and the
energy lost from the electrons was measured.
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- The abrupt fall in the plot of I vs V at Vthr
suggested that the kinetic energy of the
electrons must reach a threshold eVthr to
transfer energy to the gas atoms.
-Suggesting that
atoms must be
quantized in
discrete states.
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4.3 THE BOHR MODEL: PREDICTING
DISCRETE ENERGY LEVELS IN ATOMS
- Starting from Rutherford’s planetary model of the atom
- the assumption that an electron of mass me moves in a circular
orbit of radius r about a fixed nucleus
Ze2
4pe0r2
2
me v
r
orbit
motion
Classical theory states
are not stable.
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Bohr model
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- The total energy of the hydrogen atom: kinetic + potential energy
Ze2
1
2
E = mev 2
4pe0r
Ze2
4pe0r2
2
me v
r
- Coulomb force = centrifugal force
Ze2
v2
m
=
e
r
4pe0r2
- Bohr’s postulate: angular momentum of the electron is quantized.
h
L = mevr = n2p n = 1, 2, 3, …
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2h2
2
e
n
n
0
- Radius rn =
a
=
pZe2me Z 0
e0h2
a0 (Bohr radius) = 2
= 0.529 Å
pe me
Ze2
nh
=
- Velocity vn =
2pmern 2e0nh
-Z2e4me
Z2
- Energy En =
= -R 2
8e02n2h2
n
n = 1, 2, 3, …
e4me
R (rydbergs) =
= 2.18×10-18 J
2
2
8e0 h
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 Ionization energy: the minimum energy required to remove an
electron from an atom
In the Bohr model, this involves the n = 1 state → the n = ∞ state
DE = Efinal – Einitial = 0 – (-2.18×10-18 J) = 2.18×10-18 J
IE = NA x 2.18×10-18 J = 1310 kJ mol-1
EXAMPLE 4.3
Consider the n = 2 state of Li2+. Using the Bohr model, calculate r, V,
and E of the ion relative to that of the nucleus and electron separated
by an infinite distance.
n2
4
r = a0 = a0 = 0.705 Å
Z
3
v=
nh
2h
=
= 3.28×106 m s-1
2pmern
2pmern
Z2
9
E2 = -R 2 = -R
= - 4.90×10-18 J
n
4
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 Atomic line spectra: interpretation by the Bohr model
- Light is emitted to carry off the energy hn by transition from Ei to Ef.
hn =
Z2e4me
8e02h2
- Lines in the emission spectrum with frequencies,
n =
Z2e4me
8e02h3
= (3.29×1015 s-1) Z2
ni > nf = 1, 2, 3, … (emission)
The Rydberg constant
- Lines in the absorption spectrum with frequencies,
n =
Z2e4me
8e02h3
= (3.29×1015 s-1) Z2
nf > ni = 1, 2, 3, … (absorption)
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4.4 EVIDENCE FOR WAVE-PARTICLE
DUALITY
- Particles sometimes behave as waves, and vice versa.
 The Photoelectric Effect
- A beam of light shining
onto a metal surface
(photocathode) can eject
electrons (photoelectrons)
and cause an electric
current (photocurrent)
to flow.
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Only light above n0 can eject
Photoelectrons from the surface.
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- Einstein’s theory predicts that the maximum kinetic energy of
photoelectrons emitted by light of frequency n is:
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Standing Waves
-Standing waves exist
under the physical
boundary condition
- n = 1, fundamental or first
harmonic oscillation
- node , at certain points where
the amplitude is zero.
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De Broglie Waves
- The electron with a circular standing
wave oscillating about the nucleus
of the atom.
nl = 2pr
n = 1, 2, 3, …
From Bohr’s assumption,
EXAMPLE 4.3
Calculate the de Broglie wavelengths of an electron moving with
velocity 1.0 x 106 m s-1.
7.3 Å
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Electron Diffraction
- An electron with kinetic energy of 50 eV has a de Broglie wave length
of 1.73 Å, comparable to the spacing between atomic planes.
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- The diffraction condition is
nl = a sin q
- For two dimensional surface with a along the x-axis and b along
the y-axis
nala = a sin qa
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nblb = b sin qb
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The Heisenberg Indeterminacy (Uncertainty)
- When we measure two properties A and B, the product of which has
dimensions of action, we will obtain a spread in results for each
identified by DA and DB that will satisfy the following condition:
- For the position x and the momentum p,
x and p cannot be determined simultaneously.
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For instance, an object of 1.0 g mass with v = 2.0 mm s-1,
Dx = 2.6×10-29 m – very small!
an electron confined to the diameter of a typical atom (200
pm),
Dv = 2.89×105 m s-1 – very large!
EXAMPLE 4.6
Suppose photons of green light (wavelength 5.3 x 10-7 m) are used
to locate the position of the baseball with precision of one wavelength.
Calculate the minimum uncertainty in the speed of the baseball (0.145 kg).
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4.5 THE SCHRÖDINGER EQUATION
 Wave function (y, psi): mapping out the amplitude of a wave in three
dimensions; it may be a function of time.
- The origins of the Schrödinger equation:
If the wave function is described as
E = T + V(x)
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 Born interpretation: probability of finding the
particle in a region is proportional to the value
of y2
 Probability density (P(x)): the probability
that the particle will be found in a small region
divided by the volume of the region
1) Probability density must be normalized.
2) P(x) must be continuous at each point x.
boundary conditions
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How can we solve the Schrödinger equation?
From the boundary conditions, energy quantization arises
Naturally. Each energy value corresponds to one or more
wave functions.
The wave functions describe the distribution of amplitude
when the system has a specific energy value.
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4.6 QUANTUM MECHANICS OF
PARTICLE-IN-A-BOX MODELS
 Particle in a box
- Mass m confined between two rigid walls a distance L apart
- y = 0 outside the box at the walls (boundary condition)
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- Inside the box, where V = 0,
2
d2y(x) = Ey(x)
_ h
8p2m dx2
or
d2y(x)
=
2
dx
2
_ 8p mE y(x)
h2
L
0
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- For normalization,
- The second derivative of the wave function:
Energy of the particle is quantized !
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Scanning Tunneling
Microscope (STM)
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Zero-Point Energy
The lowest value of n is 1, and the lowest energy is E1 = h2/8mL2,
not zero.
According to quantum mechanics, a
particle can never be perfectly still
when it is confined between two walls:
it must always possess an energy.
consistent with the indeterminacy principle
not confined within L
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Correspondence Principle
- At n = 20, the probability becomes
essentially uniform across the box, and
there is little evidence of quantization.
- The results of quantum mechanics reduce
to those of classical mechanics for
large values of the quantum numbers, n.
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Particle in a 3-D Box
Particle in a cubic box of length L
on each side,
- Energy:
nx = 1, 2, 3,… nz = 1, 2, 3,…
ny = 1, 2, 3,…
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- Wave functions:
Y123
Y =  0.8
Y =  0.2
Y222
Y =  0.9
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Contour plots
Isosurfaces
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Y =  0.3
EXAMPLE 4.7
Consider the following two systems:
(a)An electron in a one-dimensional box of length 1.0 Å
(b) A helium atom in a cube 30 cm on an edge.
Calculate the energy difference between ground state and first excited
state, expressing your answer in kJ mol-1.
very close, no role of quantum effects
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10 Problem Sets
For Chapter 4,
6, 10, 20, 28, 32, 38, 40, 48, 50, 58
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