Heisenberg's Uncertainty Principle's inequality [closed]








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In the uncertainty principle , whats the significance of the inequality sign ?  

Why is it not equal to over there ?  

Why do we say greater than equal to instead of just equal to ? 




As andynitrox said in the comments : 



> 
> The HUP just gives a lower bound in the uncertainty of Δx and Δpx, it is of course possible to measure both quantities with a higher uncertainty than given by the HUP, but not less.
> 
> 
> 


The inequality exists to show that, the uncertainty is atleast this much and in real-life cases, its much larger than this . ([source](https://physics.stackexchange.com/a/170686))(provided by Mithoron)


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In addition to my comment further above I might add that the Heisenberg Uncertainty Principle has nothing to do with an error in a measurement. The HUP does not state that we cannot measure something more accurately than this lower bound, but instead that it is inherently not determined to a better degree.


To illustrate, the two properties of a one-dimensional wave packet , frequency and location, are similarily interrelated.


Imagine a sharp wave packet, a peak, something like this:
![enter image description here](https://i.stack.imgur.com/CMT3S.png)


You'll have no trouble locating 'the wave packet', simply because it is fairly sharp.
Conversely, for a monochromatic (and therefore technically infinitely long wave 'packet'), it is very easy to figure out what frequency or wavelength it has:


![enter image description here](https://i.stack.imgur.com/fnEB3.png)


However, trying to find out *where* this wave is, cannot really be answered. Keeping in mind that a wave packet is essentially made up of different monochromatic waves with different frequencies and amplitudes, you'll see that the sharper a packet is, the more frequencies it contains, so it will become nearly impossible to say what frequency it has.


![enter image description here](https://i.stack.imgur.com/uL85p.png)


We can give a lower bound for the accuracy of both measurements: the certainty in the measurement of the frequency and of the location of such a wave packet. We cannot obviously measure both at the same time with arbitrary accuracy. Note that we may very well see exactly what such a wave packet looks like, but we can still not pin-point one single frequency *and* one single location that desribes such a packet. That is basically the essence of the Heisenberg Uncertainty Principle.


Using the wave number $k$ and as location $x$, we can describe this relation in the case of a wave packet as $$\Delta k \cdot \Delta x \geq 2\pi$$


The case of the wave-particle duality is analog to this purely mathematical example (the factor $2\pi$ comes from the coefficient in the Fourier transform that is used to switch between the $k$-space and the $x$-space).


Images: <http://www.newtonphysics.on.ca/heisenberg/image615.gif>, <https://skullsinthestars.files.wordpress.com/2007/12/sinesnapshot.gif> and <http://hyperphysics.phy-astr.gsu.edu/hbase/waves/imgwav/wpac5.gif>


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