PHYS 30101 Quantum Mechanics

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PHYS 30101 Quantum Mechanics
Lecture 11
Dr Jon Billowes Nuclear Physics Group (Schuster
Building, room 4.10) j.billowes_at_manchester.ac.uk
These slides at www.man.ac.uk/dalton/phys30101
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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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RECAP 3. Angular Momentum
L R x P (Im omitting hats but remember
theyre there) Thus Lx Y Pz Z Py and two
similar by cyclic change of x, y, z We used those
to show Lx, Ly i h Lz and two similar by
cyclic change of x, y, z Since the operators
for the components of angular momentum do not
commute, there is NO set of common
eigenfunctions for any of the pairs of
operators. Thus a state of definite eigenvalue Lz
can not have definite values for either Lx or Ly.
Add this to your notes
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Today
Using Lx, Ly i h Lz and two similar by
cyclic change of x, y, z We will show L2, Lx
L2, Ly L2, Lz 0 Thus there exists a
common set of eigenfunctions of L2 and Lx And
there exists a common set of eigenfunctions of L2
and Ly And there exists a common set of
eigenfunctions of L2 and Lz By convention we
usually work with the last set of eigenfunctions.
NOTE we can always describe a state which is
an eigenfunction of, say, Ly by a linear
combination of the Lz eigenfunctions.
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Also Today
3.1 Angular momentum operators in spherical polar
coordinates
Using
And the unit vector relationship
We will show
And we wont show but will be prepared to accept
that
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Continuing
3.1 (continued) Eigenfunctions and eigenvalues
of L2 and Lz the Spherical Harmonics 3.2
Finding eigenfunctions and eigenvalues in a more
abstract way using the ladder operators. 3.3 We
show states of definite eigenvalue Lz have axial
symmetry. 3.4 Coefficients connected to the
ladder operators
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Spherical Harmonics Representation (dark and
light regions have opposite sign) and explicit
expressions.
Possible orientations of the l2 angular momentum
vector when the z-component has a definite value.