Math Review

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Chemistry 331
Lecture 2 Math Review The Wave-particle
duality The Wave Equation Photoelectric effect
NC State University
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Properties of exponentials
10A10B 10AB 10A/10B 10A-B
Inverse function is logarithm
log10(10A) A 10log10(B) B
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Exponential to the base e
eAeB eAB eA/eB eA-B
Inverse function is natural logarithm
ln(eA) A eln(B) B
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Converting from one base to another
10A ex What is x? ln(10A)
ln(ex) Take natural ln
of both sides Aln(10) x Solve for x x
2.3025A
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Derivative rate of change
Dy y Dx x Slope
Infinitesimal Rate of change

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Derivative rate of change
Example function parabola
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Derivative rate of change
Slope - 80 x - 40
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Derivative rate of change
Slope - 40 x - 20
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Derivative rate of change
Slope 0 x 0
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Derivative rate of change
Slope 40 x 20
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Derivative rate of change
Slope 80 x 40
What is the pattern? Slope 2x
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Derivative rate of change
Plot the slope as the green points.
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Derivative rate of change
It is indeed a line with slope 2x.
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Derivatives and Integralsof the logarithm

• Derivative dln(x)/dx 1/x.
• Integral definition of natural logarithm
• exp(x) is inverse of ln(x).
• The integral is the inverse of the derivative.

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Plot of the natural logarithm
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Slope of the tangent line at 1/4
The derivative is the slope
x1/4 , slope 4
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Slope of the tangent line at 1/2
x1/2 , slope 2
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Slope of the tangent line at 1
x1 , slope 1
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Slope of the tangent line at 2
x2 , slope 1/2
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Plot of the slopes of the tangents
Plot of slope vs. x
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Derivative dln(x)/dx 1/x
f(x) 1/x
g(x) ln(x)
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Integral of da/a the area under the curve
g(x) ln(x)
f(a) 1/a
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Integral of da/a the negative of thearea under
the curve
g(x) ln(x)
f(a) 1/a
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Experimental observation of hydrogen atom

• Hydrogen atom emission is quantized. It occurs
  at discrete wavelengths (and therefore at
  discrete energies).
• The Balmer series results from four visible lines
  at 410 nm, 434 nm, 496 nm and 656 nm.
• The relationship between these lines was shown to
  follow the Rydberg relation.

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Atomic spectra

• Atomic spectra consist of series of narrow lines.
• Empirically it has been shown that the wavenumber
  of the spectral lines can be fit by

where R is the Rydberg constant, and n1 and n2
are integers.
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The hydrogen atom semi-classical approach

• Why should the hydrogen atom care about integers?
• What determines the value of the Rydberg constant
  R109,677 cm-1?
• Bohr model for the hydrogen atom.
• Coulomb Centrifugal
• Balance of forces.
• Assume electron travels in a radius r.
• There must be an integral number of wavelengths
  in the circumference.
• 2pr nl n 1,2,3.

r
e-
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The electron must not interfere with itself

• The condition for a stable orbit is 2pr
  nl, n1,2,3..
• The Bohr orbital shown has n 16.
• The DeBroglie wavelength
• l h/p or l h/mv
• gives mvr nh/2p n1,2,3
• This is a condition for quantization of angular
  momentum

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Example of self-interference

• According to the Bohr picture the condition shown
  will lead to cancellation of the wave and is not
  a stable orbit.
• The quantization of angular momentum implies
  quantization of the radius

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The significance of quantized orbits

• The Bohr model is consistent with quantized
  orbits of the electron around the nucleus.
• This implies a relationship between quantized
  angular momentum and the wavelength.
• Einstein argued (based on relativity) that l
  h/p, where the wavelength of light is l, and the
  momentum of a photon is p.
• DeBroglie argued that the same should hold for
  all particles.

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The Bohr Model Predicts Quantized Energies

• The radii of the orbits are quantized and
  therefore the energies are quantized.
• According to classical electrostatics

Substituting in for r gives
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The Wave-Particle Duality

• The fact that the DeBroglie wavelength explains
  the quantization of the hydrogen atom is a
  phenomenal success.
• Other wave-like behavior of particles includes
  electron diffraction.
• Particle-like behavior of waves is shown in the
  photoelectric effect

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Photoelectric Effect

• Electrons are ejected from a metal surface by
  absorption of a photon.
• Depends on frequency, not on intensity.
• Threshold frequency corresponds to hn 0 F
• F is the work function. It is essentially equal
  to the ionization potential of the metal.

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Photoelectric Effect

• The kinetic energy of the ejected particle is
  given by
• 1/2 mv2 hn - F
• The threshold energy is F, the work function.
• This demonstrates the particle-like behavior of
  photons.
• A wave-like behavior would be indicated if the
  intensity produced the effect.

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Derivation of the Schrödinger Equation The
Schrödinger equation is a wave equation.
Just as you might imagine the solution of such an
equation in free space is a wave. Mathematically
we can express a wave as a sine or cosine
function. These functions are oscillating
functions. We will derive the wave equation
in free space starting with one of its solutions
sin(x). Before we begin it is important to
realize that bound states may provide different
solutions of the wave equation than those we find
for free space. Bound states include rotational
and vibrational states as well as atomic
wave functions. These are important cases that
will be treated once we have fundamental
understanding of the origin of the wave equation
or Schrödinger equation.
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The derivative The derivative of a function is
the instantaneous rate of change. The derivative
of a function is the slope.
We can demonstrate the derivative graphically.
We consider the function f(x) sin(x) shown
below.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
At sin(0) the slope is 1 as shown by the blue
line.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
At sin(p/4) the slope is 1/Ö2 as shown by the
blue line.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
At sin(p/2) the slope is 0 as shown by the blue
line.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
At sin(3p/4) the slope is -1/Ö2 as shown by the
blue line.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
At sin(3p/4) the slope is -1/Ö2 as shown by the
blue line. The slopes of all lines thus far are
plotted as black squares.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
At sin(p) the slope is -1 as shown by the
blue line. The slopes of all lines thus far are
plotted as black squares.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
At sin(5p/4) the slope is -1/Ö2 as shown by the
blue line. The slopes of all lines thus far are
plotted as black squares.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
At sin(5p/4) the slope is -1/Ö2 as shown by the
blue line. The slopes of all lines thus far are
plotted as black squares.
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The derivative of sin(x) The derivative of a
function is the instantaneous rate of change. The
derivative of a function is the slope.
We see from of the black squares (slopes) that
the derivative of sin(x) is
cos(x).
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The derivative of sin(x)
d
sin(x) cos(x)
dx
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The derivative of cos(x)
d
cos(x) -sin(x)
dx
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The second derivative of sin(x)
d
d
sin(x) -sin(x)
dx
dx
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The second derivative of sin(x)
d2
sin(x) -sin(x)
dx2
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Sin(x) is an eigenfunction
d2
-
If we define as an operator G then we
have
dx2
d2
sin(x) sin(x)
-
dx2
which can be written as
G sin(x) sin(x)
This is a simple example of an operator equation
that is closely related to the Schrödinger
equation.
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Sin(kx) is also an eigenfunction
We can make the problem more general by
including a constant k. This constant is called a
wavevector. It determines the period of the sin
function. Now we must take the derivative of the
sin function and also the function kx inside the
parentheses (chain rule).
d
sin(kx) -k cos(kx)
-
dx
d2
sin(kx) k2sin(kx)
-
dx2
Here we call the value k2 the eigenvalue.
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Sin(kx) is an eigenfunction of the Schrödinger
equation
The example we are using here can easily be
expressed as the Schrodinger equation for wave in
space. We only have to add a constant.
-
-
d2
h2
h2k2
sin(kx) sin(kx)
-
dx2
2m
2m
-
In this equation h is Plancks constant divided
by 2p and m is the mass of the particle that is
traveling through space. The eigenfunction is
still sin(kx), but the eigenvalue in this
equation is actually the energy.
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The Schrödinger equation
Based on these considerations we can write a
compact form for the Schrödinger equation.
HY EY
-
d2
h2
-
H
Energy operator, Hamiltonian
dx2
2m
-
h2k2
E
Energy eigenvalue, Energy
2m
Wavefunction
Y sin(kx)
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The momentum
The momentum is related to the kinetic energy.
Classically The kinetic energy is
E mv2 The momentum is
p mv So the
classical relationship is
p2
E
2m If we compare this to the quantum mechanical
energy we see
that p hk
1
2
-
h2k2
E
2m
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The general solution to the Schrödinger equation
in free space
The preceding considerations are true in free
space. Since a cosine function has the same form
as a sine function, but is shifted in phase, the
general solution is a linear combination of
cosine and sine functions. The coefficients A
and B are arbitrary in free space. However, if
the wave equation is solved in the presence of a
potential then there will be boundary conditions.
Y Asin(kx) Bcos(kx)
Wavefunction