It is a characteristic of quantum mechanics that conjugate variables are Fourier transform pairs of variables.
Conjugate variable measurement limitations affect how we perceive quantum level events as those can only be perceived by instrumented measurements at that level.
As x and k form a Fourier transform pair in quantum mechanics, the Nyquist-Shannon Sampling theorem must also apply to this pair of conjugate variables. Similar relations can be derived for the E and v pair of conjugate variables.
Equations (32) and (35) are recognized as measurement relationships for quantummechanical conjugate variables. Currently, Quantum Mechanics only considers the Uncertainty Theorem but not the Sampling Theorem.
The reason is that one expression involves the widths of conjugate variables as determined by (1) to (3), while the other involves sampling a variable and truncating its conjugate, or vice versa as determined by (32) and (35).
Quantum mechanical conjugate variables are Fourier Transform pairs of variables.
We have shown from Fourier Transform theory that the Nyquist-Shannon Sampling Theorem affects the nature of measurements of quantum mechanical conjugate variables. We have shown that Brillouin zones in Solid State Physics are a manifestation of the Nyquist-Shannon Sampling Theorem at the quantum level.
Sampling a variable x at a rate [delta]x will result in the measurement of its conjugate variable [??] being limited to its maximum Nyquist range value [[??].sub.N] as given by the Nyquist-Shannon Sampling theorem:
Conversely, truncating a variable x at a maximum value [x.sub.N] (x [less than or equal to] [x.sub.N]) will result in its conjugate variable [??] being sampled at a rate [delta][??] given by the Nyquist-Shannon Sampling theorem [delta][??] = 1/(2[x.sub.N]) resulting in the relation