Vector Spaces and Matrix Mechanics
January 18, 2012
1
Probability Amplitudes and Their Compositions
Let us recall the expansion postulate of Quantum Mechanics:
Every physical quantity can be represented by a Hermitian operator with eigenfunctions 1, 2, .,
Hilbert Space and Dirac Notations
January 23, 2012
1
Bra and Ket vectors
Consider a probability amplitude:
A| B | =
(x)(x)dx
We call this quantity inner product. Using the integral denition of the inner product we can establish the following properties:
Eigenvalue Problem in Hilbert Space.
Representation Theory.
The Heisenberg Uncertainty Relation.
January 25, 2012
1
Eigenvalue Problem in Hilbert Space: Matrix Formulation
Let us suppose that the set of ket-vectors |n corresponds to a
complete set of func
Coordinate and Momentum Representation.
Commuting Observables and Simultaneous
Measurements.
January 30, 2012
1
Coordinate and Momentum Representations
Let us consider an eigenvalue problem for a Hermitian operator
A:
A|n = An |n
and
n|m = nm
In the basis
The Spin.
Algebraic Theory of Angular Momentum
January 30, 2012
1
Algebraic Approach to the Angular Momentum
Eigenvalue Problem
The commutator relations
[Jx, Jy ] = i Jz
(1)
are the trademark of the angular momentum in quantum mechanics. Although they ar
The Spin (continued)
Angular momentum matrices
The spin Pauli matrices
February 1, 2012
1
Angular momentum matrices.
We will try to establish an explicit form of the angular momentum matrices without any particular reference to wave functions,
i.e. by u
AlgebraicMethodofSolvingthe
LinearHarmonicOscillator
LecturebyGableRhodes
PHYS 773: Quantum Mechanics
February 6th, 2012
SimpleHarmonic Oscillator
Many physical problems can be modeled as small
oscillations around a stable equilibrium
Potential is descr
The Spin (continued).
Magnetic moment of an electron
Particle wave functions including spin
Stern-Gerlach experiment
February 8, 2012
1
Magnetic moment of an electron.
The coordinates of a particle include the spin variables (S, ms)
and the position va
The Spin (continued).
Spin and Rotations
February 13, 2012
1
Spin and Rotations.
We remember that the z -component of the orbital angular momentum operator can be expressed as:
There is an intimate relation between this operator and an operator of the ro