PHYS 30101 Quantum Mechanics

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PHYS 30101 Quantum Mechanics
Lecture 14
Dr Gavin Smith Nuclear Physics Group
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. Spin (algebra almost identical to
orbital angular momentum algebra except we
cant write down explicit analogues of spherical
harmonics for spin eigenfunctions)
Commutation relations
(plus two others by cyclic permutation of x,y,z)
By convention we choose to work with
eigenfunctions of S2 and Sz which we label a and ß
So, the eigenvalue equations are
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Any general spin-1/2 wavefunction ? can be
written as a linear combination of the complete
set of our chosen eigenfunction set
? a a b ß
(theres only two eigenfunctions in the set)
The coefficients a and b give the weighting and
relative phases of the a and ß eigenstates. Normal
ization a2 b2 1
The wavefunction ? could be, for example, that of
a spin-1/2 particle polarised in the x-direction
(an eigenstate of Sx) We now find the
coefficients a, b for this state as an example
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Eigenfunctions and eigenvalues of Sx, Sy, Sz
described in this way
? a a b ß
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RECAP 4.2 Dirac notation
Dirac
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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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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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Matrix representation Eigenvectors of Sx, Sy, Sz
Eigenfunctions of spin operators (from lecture 13)