TAB 30

THE UNCERTAINTY PRINCIPLE

A fundamental flaw to quantum chemistry and the Schrodinger Cloud Plots is that there is no provision for how an electron goes from one stable orbit to the next. Going from a circular s-type orbit to a barbell p-type orbit is a very complex maneuver involving plane changes, Hohmann Transfers, and Lambert Transfers - any one of which requires far more energy than the difference between the two orbital levels. Basic orbital mechanics shows the whole complex scheme embraced by atomic physics is unworkable unless there is a single fundamental frequency that exists like a common harmonic between all allowable energy levels - e.g. the "system wave."

Now to consider some important mathematical ideas:

• Each physically observable property in quantum mechanics (e.g. position, velocity, energy, momentum,…) can be represented by a Hermitian operator, typically a matrix
• Two operators with the same eigenvectors commute
• Physical variables with noncommuting operators cannot be measured simultaneously with arbitrary accuracy
• Position and momentum operators in quantum mechanics do not commute; this is the Heisenberg Uncertainty Principle, which says

joule-seconds

• The existence of a matrix transformation for the symmetric plane, where eigenvectors do commute, means that the Heisenberg Uncertainty Principle does not apply in dynamical systems whose coordinates are transformed into the symmetric plane.

Hermitian matrices have the following properties: (i.e. )

• Diagonalizable, i.e.
• The eigenvectors are real
• The eigenvalues for unique eigenvectors are orthogonal to each other
• A matrix with complex elements plays the role of a symmetric matrix - i.e. they are equal to their conjugate transpose;
• The diagonal entries are real (implying the kind of symmetry that was seen with the symmetric plane)

The principle axis theorem of mechanics applies to orthogonally diagonalizable matrices:

• If the eigenvectors define the axes, then the axes are perpendicular
• In deformable body mechanics, the eigenvectors give the directions in which there is pure compression or pure tension. In other directions, there is pure shear.

This is just the phenomena that was implied by the Mohr Circle analysis earlier.

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© 2004 WH Clark