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Probabilistic Modeling in
Quantum Theory
Jennifer Hartmann
Classical Mechanics vs.
Quantum Mechanics
•
Basic Assumptions of Classical Mechanics:
1) No limit to accuracy with which one or more dynamical
variables can be simultaneously measured.
2) No limit to the number of dynamical variables that can be
accurately measured simultaneously.
3) Expressions for velocity are continuously varying
functions of time, the kinetic energy can vary
continuously.
•
•
However, with the involvement of very small
particles, these assumptions are not valid.
Classical mechanics fail to explain the
experimental observations of:
–
–
Atomic Spectra
Photoelectric Effect
• Atomic Spectra
– Atoms only emitted light at specific wavelengths with a
specific energy.
Photographic Plate
Atomic Lamp
Light through slit
Prism
λ or
frequency
– Balmer (1885) fit the data for a hydrogen atomic lamp to:
1/λ=R((1/nf2)-(1/ni2))
where R is the Rydberg constant (109677.8 cm-1) and “n” are
integers called “quantum numbers”.
– Energy is “quantized” meaning it can only exist at certain
energy levels and not in between
• Like rungs on a ladder
• Correspondence Principle
• Photoelectric Effect
– Emission of an electron from a metal
surface by light
eMetal
• Intensity of light has no effect on K.E.,
just number of electrons ejected.
K. E. of the
electrons
excited off
the metal
surface
•K.E. of the ejected electrons is
independent of light intensity, only a
function of frequency.
0
•Classical physics has no explanation.
Frequency of Light
Wave/Particle Duality
• An explanation for the previous experimental
observations is that although light is primarily
thought of as a wave, can sometimes act like
a particle as well.
– i.e. photons
• De Broglie Wavelength
– Louis de Broglie suggested that although electrons
have been previously regarded as particles also
show wave-like characteristics with wavelengths
given by the equation:
λ = h/p = h/mv
– h (Planck’s constant) = 6.626 x 10-34 J·s
Quantum Mechanics
•
Taking into account the idea that energy is
quantized and the wave/particle duality
properties of very small particles, Quantum
Theory provides the following postulates:
1) The state of a system is fully described by a
“wavefunction” Ψ(x, t) such that the quantity Ψ*
Ψ dx is proportional to the probability of finding
the system or the particle in dx.
2) For every observable property of the system,
there exists a corresponding operator.
3) The value of an observable is determined by
applying the corresponding operator to the
wavefunction of a system.
Finding the Wavefunction for a
System
• Hamiltonian Operator (Energy, H):
(-h2/2m) 2 + V(x)
where 2=(δ2/δx2 + δ2/δy2 + δ2/δz2) and
h=h/2π
• Find the wavefunction that is an
eigenfunction of the energy operator.
–HΨ=εΨ
– Easy to measure energy
– Easiest to solve
Atomic Structure
• Consider simplest atom with one
electron:
z
Electron
θ
Proton
r
y
φ
x
With proton of mass (M) and charge (+Ze), and electron
with mass (m) and charge (-e).
Schrödinger Equation
(-h2/2μ) 2Ψ-(Ze2/4πε◦r)Ψ=εΨ
where μ is the reduced mass and the term (4πε◦)
accounts for vacuum permittivity
Rewritten in polar coordinates as:
(1/r2)(δ/δr(r2(δΨ/δr)))+(1/r2sinθ)(δ/δθ
sinθ(δΨ/δθ))+(1/r2sin2θ)(δ2Ψ/δφ2)+(2μ/h2
)(ε+(Ze2/4πε◦r))Ψ=0
Solve the Schrödinger Equation
• Separation of Variables
Ψ=Ψ(r)Ψ(θ)Ψ(φ)=(R)(Θ)(Φ)
• Term I
(1/φ)(δ2Φ/δφ2)=-m2
• Term II
(1/sinθ)(d/dθ(sinθ(dΘ/dθ)))-(m2/sin2θ)Θ+λΘ=0
• Term III
(1/r2)(d/dr(r2dR/dr)+[(2μ/h2)(ε+(Ze2/4πε◦)-(λ/r2)]R=0
• The solutions of Terms I and II are the
spherical harmonics Y(Θ,Φ)
Φ=(2π)eimφ m=0,±1,±2,±3…
Θ are the Associated Legendre Polynomials:
• Function of two quantum numbers: l & m
• l=0,1,2,3… (angular momentum quantum number)
• m=0,±1,±2,±3…|m|≤l (z-component of angular
momentum quantum number)
• The solutions of Term III are the
Associated Laguerre Polynomials:
• Function of quantum number n=1,2,3…
(principle quantum number)
Atomic Orbitals
• Once the wavefunction(Ψ) is determined, according
to postulate 1, Ψ2 gives the probabilistic distribution
of the electron with respect to nuclear distance.
• This distribution is then corrected for a three
dimensional space forming 3-D probabilistic
distributions called orbitals.
References
• Hanna, Melvin W.. Quantum Mechanics in
Chemistry. 3. Menlo Park, Ca:
Benjamin/Cummings, 1981.
• Hanna, Melvin W.. Quantum Mechanics in
Chemistry. 3. Menlo Park, Ca:
Benjamin/Cummings, 1981.