I Heisenberg's Uncertainty Principle common misconceptions

[Jimmy87]

Let me give you part of the answer. We are talking here about momentum in the transverse direction, not total momentum. For an EM wave, diffraction is about a change in direction, not about a change in wavelength or energy.

So total momentum is conserved then, just the x component changes? And to change the wavelength the total momentum has to change? Is it not misleading when textbooks say "lasers produce photons in a momentum eigenstate because they have a precise wavelength" - surely when it diffracts and its momentum spreads out it is no longer in a momentum eigenstate/definite wavelength but you said it will be because total momentum/wavelength is conserved?

 

[PeroK]

So total momentum is conserved then, just the x component changes? And to change the wavelength the total momentum has to change? Is it not misleading when textbooks say "lasers produce photons in a momentum eigenstate because they have a precise wavelength" - surely when it diffracts and its momentum spreads out it is no longer in a momentum eigenstate/definite wavelength but you said it will be because total momentum/wavelength is conserved?

In this case you have light with a well defined wavelength, hence a well defined energy and momentum, in the sense of magnitude of momentum. If the light reflects off a surface, or diffracts, then the direction of its momentum changes, but not the magnitude.

 

[David Neves]

Noether's Theorem and quantum mechanics are related by the notion from Hamiltonian mechanics that every dynamical variable can be interpreted as an infinitesimal generator of some canonical transformation, or the quantum mechnical notion that every Hermitian operator generates a unitary transformation.

The Heisenberg principle is true of any variable with a continuous spectrum and the infinitesimal generator of translations in that variable, just because these variables always have a nonzero commutator in every possible state. Position and momentum, angle and angular momentum, charge and phase, these are all conjugates in classical mechanics. The charge operator generates infinitesimal rotations in the phase of charged-particle wavefunctions, not changes in potential.

Noether's Theorem states that when translations of a certain variable are a symmetry, the infinitesimal generator of those translations is conserved. So translations in x, translations in angle, and translations in phase give conservation of momentum, angular momentum, and charge. But these generators obey the HUP with their conjugate variables.

It is important to understand that conjugate pairs like position and momentum relate as generators of translations. If these translations leave the system unchanged, i.e. you have a symmetry generated by the conjugate momentum, then the conjugate momentum must be conserved.

In quantum theory the conjugate pairs are not independent. This is because on a Hilbert space the generators of translations and the coordinate they act on relate like the derivative and the coordinate, which don't commute.

So it really comes down to the geometry of the phase space of a system. In classical physics the phase space is just a normal manifold with the usual geometric structure. But in quantum theory the construction using translations on the Hilbert space result in a non-commutative geometry.
There are other conjugate variables, other than position and momentum. Here you can read about the Heisenberg Uncertainty Principle.

https://brilliant.org/wiki/heisenberg-uncertainty-principle