PHYS 30101 Quantum Mechanics

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PHYS 30101 Quantum Mechanics
Lecture 16
Dr Jon Billowes Nuclear Physics Group (Schuster
Building, room 4.10) j.billowes_at_manchester.ac.uk
These slides at http//nuclear.ph.man.ac.uk/jb/p
hys30101
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• Syllabus
• Basics of quantum mechanics (QM) Postulate,
  operators, eigenvalues eigenfunctions,
  orthogonality completeness, time-dependent
  Schrödinger equation, probabilistic
  interpretation, compatibility of observables,
  the uncertainty principle.
• 1-D QM Bound states, potential barriers,
  tunnelling phenomena.
• Orbital angular momentum Commutation relations,
  eigenvalues of Lz and L2, explicit forms of Lz
  and L2 in spherical polar coordinates, spherical
  harmonics Yl,m.
• Spin Noncommutativity of spin operators, ladder
  operators, Dirac notation, Pauli spin matrices,
  the Stern-Gerlach experiment.
• Addition of angular momentum Total angular
  momentum operators, eigenvalues and
  eigenfunctions of Jz and J2.
• The hydrogen atom revisited Spin-orbit coupling,
  fine structure, Zeeman effect.
• Perturbation theory First-order perturbation
  theory for energy levels.
• Conceptual problems The EPR paradox, Bells
  inequalities.

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4. Spin 4.1 Commutators, ladder operators,
eigenfunctions, eigenvalues 4.2 Dirac notation
(simple shorthand useful for spin space) 4.3
Matrix representations in QM Pauli spin
matrices 4.4 Measurement of angular momentum
components the Stern-Gerlach apparatus
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Recap 4.3 Matrix representations in QM
We can describe any function as a linear
combination of our chosen set of eigenfunctions
(our basis)
Substitute in the eigenvalue equation for a
general operator
Gives
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Recap 4.3 Matrix representations in QM
We can describe any function as a linear
combination of our chosen set of eigenfunctions
(our basis)
Substitute in the eigenvalue equation for a
general operator
Equation (1)
Gives
Multiply from left and integrate
)
(We use
Exactly the rule for multiplying matrices!
And find
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4.3.2 Matix representations of Sx, Sy, Sz
Sx ½h sx Sy ½h sy Sz ½h sz
Pauli Spin Matrices
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Eigenfunctions of spin-1/2 operators
Matrix representation Eigenvectors of Sx, Sy, Sz
4.3.3 Example description of spin1 polarised
along the x-axis
is
In Dirac notation
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4.4 Measurement of a spin component Magnetic
moments (a) due to orbital angular momentum
Electron in orbit produces a magnetic field (like
bar magnet) and therefore has a magnetic dipole
moment
r
e
µl gl l µB
µB eh/2me (the Bohr magneton)
µl gl l µB
gl -1 (gyromagnetic ratio or g-factor)
l 1, 2, 3