PHYS 30101 Quantum Mechanics PHYS 30101 Quantum Mechanics Dr Sean Freeman Nuclear Physics Group...
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Transcript of PHYS 30101 Quantum Mechanics PHYS 30101 Quantum Mechanics Dr Sean Freeman Nuclear Physics Group...
PHYS 30101 Quantum MechanicsPHYS 30101 Quantum Mechanics
Dr Sean Freeman Nuclear Physics Group
These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101
Lecture 15
Syllabus
1. Basics of quantum mechanics (QM) Postulate, operators, eigenvalues & eigenfunctions, orthogonality & completeness, time-dependent Schrödinger equation, probabilistic interpretation, compatibility of observables, the uncertainty principle.
2. 1-D QM Bound states, potential barriers, tunnelling phenomena.
3. Orbital angular momentum Commutation relations, eigenvalues of Lz and L2, explicit forms of Lz and L2 in spherical polar coordinates, spherical harmonics Yl,m.
4. Spin Noncommutativity of spin operators, ladder operators, Dirac notation, Pauli spin matrices, the Stern-Gerlach experiment.
5. Addition of angular momentum Total angular momentum operators, eigenvalues and eigenfunctions of Jz and J2.
6. The hydrogen atom revisited Spin-orbit coupling, fine structure, Zeeman effect.
7. Perturbation theory First-order perturbation theory for energy levels.
8. Conceptual problems The EPR paradox, Bell’s inequalities.
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
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:
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:
Multiply from left and integrate:
(We use )
And find:Exactly the rule for multiplying matrices!
Equation (1)
Matrix representation: Eigenvectors of Sx, Sy, Sz
Eigenfunctions of spin-1/2 operators
4.3.3 Example: description of spin=1 polarised along the x-axis
In Dirac notation:
is