Uncertainty Principle mechanics

[terp.asessed]

Can one use uncertainty principle for Classical mechanic wave and still get the same equation for Quantum mechanics, as in (root-mean square uncertainty of position) (" of momentum) > hbar/2? It's just that V(x) [Potential equation] is same for both Classical and Quantum mechanics so I wonder if the principle applies too.

 

[Leaph]

No, you can't becouse the uncertainty principle comes from the mathematical tools used in q.m. which aren't the same of classical mechanic, so position and momentum are defined differently (they are operators and not simply vectors). This seems a simple "mathematical trick", but is proven by several experiments.

 

[moriheru]

HUP can be derived by the dirac notation in combination with cauchy schwarz inequality. And as Diracnoation is QM representation and not a classical rhere is no uncertainty principle in classical mechanics. Classical mechanics is a deterministic theory.

 

[terp.asessed]

Thanks for the info, but could you expand on what you mean by:

the dirac notation in combination with cauchy schwarz inequality

 

[jtbell]

A superposition (sum) of waves has the same uncertainty principle regardless of whether they are classical waves or QM waves, when described in terms of wavenumber versus position ($\Delta k \Delta x \ge \frac{1}{2}$) or frequency versus time ($\Delta \omega \Delta t \ge \frac{1}{2}$).

The frequency versus time uncertainty is well known in signal processing: a pulse with width $\Delta t$ contains a range of frequencies $\Delta \omega$ at least big enough to satisfy the uncertainty relation.

What makes QM waves different is that for them, wavenumber (or wavelength) and frequency are associated with momentum and energy: $p = \hbar k = \frac{p}{\lambda}$ and $E = \hbar \omega = hf$. This is not true for classical waves.

 

[moriheru]

I can give you the deriviation, if you ask specifically, but to elaborate what I said before without going into detail of the deriviation:
You substitute ket and bra terms into the cauchy schwarz inequaltiy which gives you a new expression from which you can derive the HUP, which gives you the general UP and by substituting [X,P]=-ih(bar) into the general form you get the HUP.