Physics is fun

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Physics is fun!
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Chapter 7 The Hydrogen Atom
Werner Heisenberg (1901 1976) Nobel 1932
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Quantum-Mechanical View of Atoms

• Electrons in an atom can exist only in certain
  states of definite energy.
• A photon of light is emitted or absorbed when an
  electron makes a transition from one state to
  another.
• Electrons do not exist in well-defined circular
  orbits.
• Electrons, due to their wave nature, can be
  thought of as spread out in space as a cloud.

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Quantum-Mechanical View of Atoms

• The size and shape of the electron cloud can be
  calculated for a given state of an atom.
• For the ground state in the hydrogen atom, the
  electron cloud is spherically symmetric.
• The electron cloud roughly indicates the size of
  an atom.

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Quantum-Mechanical View of Atoms

• The electron cloud can be interpreted using
  either the particle or the wave viewpoint.
• A particle is localized in spaceit has a
  definite position at any given instant.
• A wave is spread out in space.
• The electron cloud is a result of the wave nature
  of electrons.
• Electron clouds can also be interpreted as
  probability distributions for a particle.

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7.1 Application of the Schrödinger Equation to
the Hydrogen Atom
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Spherical Coordinates
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7.2 Solution of the Schrödinger Equation for
Hydrogen
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7.3 Quantum Numbers

• n principle quantum number
• l orbital angular momentum quantum number
• nl magnetic quantum number
• n 1, 2, 3,
• l 0, 1, 2, 3, , n - 1
• ml - l, - l 1, , 0, 1, , l 1, l

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Example 7.1

• What are the possible quantum numbers for a n
  4 state in atomic hydrogen?

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Angular Momentum Quantum Number l

• L l(l 1)

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Magnetic Quantum Number ml

• Orbital angular momentum quantum number l
  determines the magnitude of the angular momentum
  L.
• Since it is a vector it also has direction.
• The solution of the Schrödinger equation for g(x)
  specifies that ml is an integer and related to
  the z component of the angular momentum.

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Magnetic Quantum Number ml

• Lz ml h

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Example 7.2

• What is the degeneracy of the n 3 level? That
  is how many different states are contained in the
  energy level, E3 - E0/9?

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7.4 Magnetic Effects on Atomic SpectraThe Normal
Zeeman Effect

• Pieter Zeeman (1896)
• Spectral lines emitted by atoms placed in a
  magnetic field broaden and appear to split.
• Sometimes a line is split into three lines
  (normal Zeeman effect).
• Sometimes more than three lines (anomalous Zeeman
  effect).

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Magnetic Moment

• m - L

e
2m
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Bohr Magneton

• mz mB ml
• Where mB

eh
(Bohr Magneton)
2m
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Example 7.4

• What is the value of the Bohr magneton? Use
  that to calculate the energy difference between
  the ml 0 and ml 1 components in the 2p
  state of atomic hydrogen placed in an external
  field of 2 T.

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7.5 Intrinsic Spin

• In early 1920s Wolfgang Pauli proposed that
  there must be a fourth quantum number because of
  anomalous optical spectra and because of
  relativity that has four coordinates.
• In 1925 Samuel Goudsmit and George Uhlenbeck
  proposed that the electron must have an intrinsic
  spin.
• It must therefore have an angular momentum and a
  magnetic moment because it is charged.

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Spin Angular Momentum

• S s(s 1) h ¾ h

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Example 7.6

• How many distinctly different states (and
  therefore wave functions) exist for the 4 d level
  of atomic hydrogen?

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7.6 Energy Levels and Electron Probabilities
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Selection Rules for Allowable Transitions

• When a photon is emitted of absorbed, transitions
  can occur between states with values of l that
  differ by one unit Dl 1. These are called
  allowed transitions.
• According to this selection rule, an electron in
  an l 2 state can jump only to a state with l
  1 or l 3.
• A transition such as l 2 to l 0 is called a
  forbidden transition.
• These transitions are not absolutely forbidden,
  but they have a very low probably of occurrence.

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Transition Selection Rules

• Dn anything
• Dl 1
• Dml 0, 1

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Example 7.7

• Which of the following transitions for quantum
  numbers (n, l, ml, ms) are allowed for the
  hydrogen atom, and if allowed, what energy is
  involved?
• (a) (2, 0, 0, ½) (3, 1, 1, ½)
• (b) (2, 0, 0, ½) (3, 0, 0, ½)
• (c) (4, 2, -1, -1/2) (2, 1, 0, ½)

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Summary

• Rules for Quantum Numbers for Electrons in Atoms
• Name Symbol Possible (allowed) values
• Principal n 1, 2, 3, ,
• Orbital l For a given n l can be 0, 1,
  2, , n 1
• Magnetic ml For a given n and l ml can be l
  1, , 0, , - l
• Spin ms For each set of n, l, ml ms can
  be ½ or ½.

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Example

• List all possible quantum states for n 3.

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Response
½ -½ ½ -½ ½ -½ ½ -½ ½
-½ ½ -½ ½ -½ ½ -½ ½ -½
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Angular Momentum of a Photon

• Since orbital angular momentum of an atom must
  change by one unit when it emits a photon,
  conservation of angular momentum tells us the
  photon must carry off angular momentum.
• Experiment shows the photon can be assigned spin
  1.

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Photon Question

• Since photons have an equivalent mass (E mc2)
  does gravity attract them?

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Angular Momentum of a Photon

• Since orbital angular momentum of an atom must
  change by one unit when it emits a photon,
  conservation of angular momentum tells us the
  photon must carry off angular momentum.
• Experiment shows the photon can be assigned spin
  1.

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Homework Problem 9

• List all the possible quantum numbers (n, l, ml
  ) for the n 6 level in atomic hydrogen.

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Homework Problem 27

• Using all four quantum numbers (n, l, ml, ms)
  write down all possible sets of quantum numbers
  for the 5f state of atomic hydrogen. What is the
  total degeneracy?

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Homework Problem 30

• Find whether the following transitions are
  allowed, and if they are, find the energy
  involved and whether the photon is absorbed or
  emitted for the hydrogen atom
• (a) (5, 2, 1, ½) (5, 2, 1, -1/2)
• (b) (4, 3, 0, ½) (4, 2, 1, -1/2)
• (c) (5, 2, -2, -½ ) (1, 0, 0, -1/2)
• (d) (2, 1, 1, ½) (4, 2, 1, ½)

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