Download Science at the end of ~1900: Electromagnetics

Quantum Mechanics
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Electrons behave as waves (interference etc)
and also particles (fixed mass, charge, number)
Lessons on http://hyperphysics.phy-astr.gsu.edu/hbase/quacon.html#quacon
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or http://www.colorado.edu/physics/2000/quantumzone/
Science at the end of ~1900: Classical Mechanics
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Leading to Mech. Engg., Civil Engg., Chem. Engg.
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Science at the end of ~1900: Electromagnetics
Rainbows
Polaroids
Lightning
Northern
Lights
Telescope
Laser
Optics
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Science at the end of ~1900: Electromagnetics
Electronic Gadgets
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Science at the end of ~1900: Electromagnetics
Chemical Reactions
Neural Impulses
Biological Processes
Ion Channels
Chemistry and Biology
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But there were puzzles !!!
Dalton (1808)
What does an atom look like ???
Solar system model of atom
mv2/r = Zq2/4pe0r2
Centripetal
force
Electrostatic
force
Continuous radiation from orbiting electron
Pb1: Atom would
be unstable!
(expect nanoseconds
observe billion years!)
Spectrum of Helium
Transitions E0(1/n2 – 1/m2) (n,m: integers)
Pb2: Spectra of
atoms are discrete!
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Bohr’s suggestion
Only certain modes allowed (like a plucked string)
nl = 2pr (fit waves on circle)
Momentum ~ 1/wavelength
(DeBroglie)
p = mv = h/l
(massive classical particles  vanishing l)
This means angular momentum is quantized
mvr = nh/2p = nħ
From 2 equations, rn = (n2/Z) a0
a0 = h2e0/pq2m = 0.529 Å (Bohr radius)
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Bohr’s suggestion
E = mv2/2 – Zq2/4pe0r
Using previous two equations
En = (Z2/n2)E0
E0 = -mq4/8ħ2e0 = -13.6 eV
= 1 Rydberg
Transitions E0(1/n2 – 1/m2) (n,m: integers)
Explains discrete atomic spectra
So need a suitable Wave equation so that imposing boundary
conditions will yield the correct quantized solutions
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What should our wave equation look like?
∂2y/∂t2 = v2(∂2y/∂x2)
String
y
w
x
k
Solution: y(x,t) = y0ei(kx-wt)
w2 = v2k2
What is the dispersion (w-k) for a particle?
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What should our wave equation look like?
Quantum theory:
E=hf = ħw (Planck’s Law)
p = h/l = ħk (de Broglie Law)
and E = p2/2m + U (energy of a particle)
w
Thus, dispersion we are looking for is
w  k2 + U
So we need one time-derivative
and two spatial derivatives
k
2y/∂tX
2 = v2(∂2y/∂x2)
∂X
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Wave equation (Schrodinger)
iħ∂Y/∂t = (-ħ22/2m + U)Y
Kinetic
Energy
Potential
Energy
Makes sense in context of waves
Eg. free particle U=0
Solution Y = Aei(kx-wt) = Aei(px-Et)/ħ
w
k
We then get E = p2/2m = ħ2k2/2m
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For all time-independent problems
iħ∂Y/∂t = (-ħ22/2m + U)Y = ĤY
Separation of variables for static potentials
Y(x,t) = f(x)e-iEt/ħ Oscillating solution in time
Ĥf = Ef, Ĥ = -ħ22/2m + U
BCs :
Ĥfn = Enfn (n = 1,2,3...)
En : eigenvalues (usually fixed by BCs)
fn(x): eigenvectors/stationary states
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