Rutherford discovered the nucleus

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• Rutherford discovered the nucleus
• Natural to think of atoms as little planetary
  systems

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• White light

Continuous Spectrum
White light
Emission Spectrum
Hot GAS
Absorption Spectrum
Cold Gas
White light
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The Balmer spectrum of hydrogen

• The Balmer series is in the visible
• the Lyman Series is in the UV

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The Bohr model

• Energy was quantized
• Angular Momentum was quantized

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Bohrs argument
Assume nucleus infinitely heavy Assume electron
is moving in circular orbit of radius,r,with
speed v then
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Bohr assumed that the orbital angular momentum
was quantized i.e. Consequently we obtain
fixed values of
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For Hydrogen Z1
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De Broglie Hypothesis
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?-wavelength
L ?/2 ? 2L f v/(2L) L ? ? L f
v/(L) L 3?/2 ? 2L/3 f 3v/(2L)
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The Schrödinger Wave Equation
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Eigenvalue equation
Observable quantity
eigenvalue
operator
eigenfunction
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Three basic assumptions

• States of a system are described by wave
  functions, Y
• Observable quantities are associated with
  operators
• When the value of an observable, Q, is known to
  be q, the system is in a state, whose function is
  an eigenfunction of the associated operator

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The Hamiltonian
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Energy Eigenvalue Problem
The time independent Schrödinger equation
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Energy Eigenvalue Problem
The time dependent Schrödinger equation
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• We have not and indeed can not derive the
  Schrödinger equation
• We have motivated it as reasonable
• And shall now assume it is true for all non
  relativistic microscopic situations

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Schrödingers original derivation

• Schrödinger began from the assumption of matter
  waves, and by analogy with geometrical optics he
  introduced a variational principle which lead him
  to postulate that the matter waves satisfied the
  equation.

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Look for a stationary solution, i.e look for a
solution by separation of variables
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Easy to see solution is of the form
E is a constant that comes from the separation of
variables which we identify with the energy And
satisfies the eigenvalue equation
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Example
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Comments

• This is of the same form as the Bohr atom
  energies, i.e. discrete
• Our solutions are only determined up to a
  constant, i.e.
• ?nBnsinknx

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• Notice we have implicitly assume that the
  operator corresponding to postion acts as by
  simply multiplying the wave function
• All other operators which have a classical analog
  can be deduced from this identification, as we
  have seen for the energy

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Particle in a box
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Spherical Symmetry
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Time dependent wave function We will assume that
potential V is independent of time and purely
radial
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Look for a stationary solution, i.e look for a
solution by separation of variables
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Easy to see solution is of the form
E is a constant that comes from the separation of
variables which we identify with the energy And
satisfies the eigenvalue equation
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Schrödinger Equation in Spherical polar
coordinates H-atom
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Schrödinger Equation in Spherical polar
coordinatesfor any radial potential V(r)
V(r)?E?
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Look for a solution by separation of variables
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Rhs independent of F
Lhs depends only on F
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Rhs independent of F
Lhs depends only on F
Common value must be a constant
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Depends only on r
Depends only on ?
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Chose constant of separation to be l(l1)
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T equation
To be determined
ml is a ve integer
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T equation
Make substitution zcos?
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T equation
This equation is well known its solutions are
the associated legendre polynomials
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• Note angular momentum is now quantized
• automatically