A very short introduction to matrix algebra

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A very short introduction to matrix algebra
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Stern-Gerlach again!
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Matrix Representation
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Pauli Spin Matrices
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Comment

• In particle physics, a meson is a strongly
  interacting boson,. In the Standard Model, mesons
  are composite (non-elementary) particles composed
  of an even number of quarks and antiquarks. All
  known mesons are believed to consist of a
  quark-antiquark pairthe so-called valence
  quarksplus a "sea" of virtual quark-antiquark
  pairs and virtual gluons. The valence quarks may
  exist in a superposition of flavor states for
  example, the neutral pio a hadron with integral
  spin n is neither an up-antiup pair nor a
  down-antidown pair, but an equal superposition of
  both. Pseudo scalar mesons (spin 0) have the
  lowest rest energy, where the quark and antiquark
  have opposite spin, and then the vector mesons
  (spin 1), where the quark and antiquark have
  parallel spin. Both come in higher-energy
  versions where the spin is augmented by orbital
  angular momentum. All mesons are unstable.

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The z-component of the force on the electron is
Now if
Electron feels a force acting down Electron
feels a force acting up
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Larmor precession

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ltSgt starts off titled at some angle a to z axis,
and then precesses about field direction at the
Lamor frequency wgB0
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• http//teaching.shu.ac.uk/hwb/chemistry/tutorials/
  molspec/nmr1.htm

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Nuclear Spin

• The particles of atomic physics (electrons,
  protons and neutrons) all have spin ½ . In many
  atoms (such as 12C) these spins are paired
  against each other, such that the nucleus of the
  atom has no overall spin. However, in some atoms
  (such as 1H and 13C) the nucleus does possess an
  overall spin. The rules for determining the net
  spin of a nucleus are as follows

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• If the number of neutrons and the number of
  protons are both even, then the nucleus has NO
  spin.
• If the number of neutrons plus the number of
  protons is odd, then the nucleus has a
  half-integer spin (i.e. 1/2, 3/2, 5/2)
• If the number of neutrons and the number of
  protons are both odd, then the nucleus has an
  integer spin (i.e. 1, 2, 3)

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• The overall spin, I, is important. Quantum
  mechanics tells us that a nucleus of spin I will
  have 2I 1 possible orientations. A nucleus with
  spin 1/2 will have 2 possible orientations. In
  the absence of an external magnetic field, these
  orientations are of equal energy. If a magnetic
  field is applied, then the energy levels split.
  Each level is given a magnetic quantum number, m.

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• The nucleus has a positive charge and is
  spinning. This generates a small magnetic field.
  The nucleus therefore possesses a magnetic
  moment, µ, which is proportional to its spin,I.
• .

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• The constant, ?, is called the magnetogyric ratio
  and is a fundamental nuclear constant which has a
  different value for every nucleus.
• The energy of a particular energy level is given
  by
• Where B is the strength of the magnetic field at
  the nucleus.

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• Imagine a nucleus (of spin 1/2) in a magnetic
  field. This nucleus is in the lower energy level
  (i.e. its magnetic moment does not oppose the
  applied field). The nucleus is spinning on its
  axis. In the presence of a magnetic field, this
  axis of rotation will precess around the magnetic
  field

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• The potential energy of the precessing nucleus is
  given by
• E - µ B cos ? where ? is the angle between the
  direction of the applied field and the axis of
  nuclear rotation

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• If energy is absorbed by the nucleus, then the
  angle of precession, ?, will change. For a
  nucleus of spin 1/2, absorption of radiation
  "flips" the magnetic moment so that it opposes
  the applied field (the higher energy state)

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• NMR studies magnetic nuclei by aligning them with
  an applied constant magnetic field and perturbing
  this alignment using an alternating magnetic
  field, those fields being orthogonal. The
  resulting response to the perturbing magnetic
  field is the phenomenon that is exploited in NMR
  spectroscopy and magnetic resonance imaging

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Medical Application

• The body is immersed in an external magnetic
  field so that the the nuclei have quantized
  orientation energies along this field.

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• To cause a flip only photons of a certain
  frequency will work.
• As the the nuclei jump back down, relax, the
  absorbed energy is reradiated in all directions
  and can be detected

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• The size of the energy jump is proportional to
  the applied magnetic field. In NMR(MRI)
• studies of the human body the field is chosen so
  that photons in the radio frequency have the
  required energy.
• Radio waves pass easily into the body and unlike
  X-rays they do not have enough energy to tear
  ions apart and cause cancer

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• In typical applications the field is non uniform
  in this way one can look at particular parts of
  the body.(The resonant frequency will be
  different depending on the field value at a given
  point
• The absorption and relaxation processes depends
  on the nature of the surrounding material
  yielding more information

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• H is the most common target but the incoming
  radiation can be tuned to yield information on
  different nuclei

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• http//en.wikipedia.org/wiki/Magnetic_resonance_im
  aging

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• Eigenvalue Problem
• Solution exists for eigenvalue

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Most Probable value for an electron in a hydrogen
atom

• The ground state wavefunction is
  ?1s(r)(1/p1/2a3/2)e-r/a. The probability
  density is ?1s(r)2, which is?1s(r)?1s(r)2
  (1/pa3)e-2r/a.

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• But that function is not going to give you the
  most probable radius. You have to take into
  account the fact that ?1s is in spherical
  coordinates, whose volume element
  isdVr2sin(f)dr d? df.So, when you integrate
  ?1s over all space, it gets multiplied by r2

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• Furthermore, since ?1s is spherically symmetric,
  you can integrate over ? and f to get what is
  called the radial probability density
  P1s(r)P1s(r)(4/a3)r2e-2r/a.

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