Pioneers of Quantum Theory I

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Pioneers of Quantum Theory -- I
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Pioneers of Quantum Theory -- II
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Why is a new theory needed?
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FACT 1 Hydrogen Spectrum
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Bohr Postulates for the Hydrogen Atom 1.
Rutherford atom is correct 2. Classical EM theory
not applicable to orbiting electron 3. Newtonian
mechanics applicable to orbiting electron 4.
Eelectron Ekinetic Epotential 5. e- energy
quantized through its angular momentum L mvr
nh/2p, n 1, 2, 3, 6. Planck-Einstein
relation applies to e- transitions ?E Ef - Ei
h? hc/? c ??
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Bohr atom
E photon energy f c/l photon frequency h
Plancks constant
m1,2,3,4,5,..., and n gt m RH Rydberg constant
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FACT 2 Black Body Radiation
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Planck Black Body (EM) energy
Energy sum of charged oscillators
For an oscillation mode of frequency n c / l
Plancks constant
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FACT 3 Photoelectric effect
There exists a cut-off frequency for knocking
electrons out of a metal !
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Einsteins explanation existence of photon
Plancks constant
Electromagnetic waves (and light) behave as
particles !
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FACT4 electron diffraction pattern obtained in a
TEM
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Electron beam-path in a transmission electron
microscope (TEM)
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de Broglies Hypothesis (1924, before exp.
evidence!) all matter has a wave-like nature
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Heisenberg's uncertainty principle
Uncertainty ???? / ??? ?
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New World Picture New Philosophy
New Physics Paradigm New Math. Scheme
(From Wikipedia)
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Radial Probability Density of Atomic Hydrogen
Orbitals
r
r
r
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The 2p Orbitals of Atomic Hygrogen
n2 l1 m1, 0, -1
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The 3d Orbitals of Atomic Hygrogen
n3 l2 m -2, -1, 0, 1, 2
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Each state may accommodate up to two elecrtrons
of opposite spin orientation (Paulis exclusion
principle)
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Paulis exclusion principle
no two identical particles of half-integer spin
(fermions) can be at the same quantum state -
because the wave function of such system must be
equal to its opposite (anti-symmetric) - and the
only wave function which satisfies this condition
is the zero wave function.
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Quantum harmonic oscillator (From Wikipedia, the
free encyclopedia)
The quantum harmonic oscillator is the quantum
mechanical analogue of the classical harmonic
oscillator. It is one of the most important model
systems in quantum mechanics because an arbitrary
potential can be approximated as a harmonic
potential at the vicinity of a stable equilibrium
point. Furthermore, it is one of the few quantum
mechanical systems for which a simple exact
solution is known.
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One-dimensional harmonic oscillator
x position operator p momentum operator
Schrödinger equation
Hn Hermite polynomials
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first six bound eigenstates
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Ladder operator method (see ex.
Wikipedia )
It can be porved that
Energy eigenstates 
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Probability densities ?n(x)²
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?quantized energies as ½ , 3/2
, 5/2 ?ground state energy or zero-point
energy (positive average kinetic energy
?? uncertainty principle) ? equally-spaced
energy levels (unlike the Bohr model or the
particle in a box) ?conforming the
correspondence principle probability
density is concentrated at the origin (n0)
or at the "classical turning points (ngt0)
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Homework Derive the energy spectrum of
a 1-dimensional quantum harmonic
oscillator with the following
Hamiltonian