Heisenberg Uncertainty Principle

In the previous tutorial https://acsharmablog.com/2018/07/02/quantum-computing-for-newbies/ ,we saw the evolution of quantum computing and find out that every particle at subatomic level behaves like a wave ,so if a single particle can carries all the information which is encoded in entire spectrum of wave ,then it’s amazing ,then it seriously looks like that quantum computing can give us exponential speed up on traditional computing.

Now the next set of questions arises –

  • How to measure the wave function of a particle
  • How to control the wave function of a particle
  • How to measure the superimposed wave function of two particles

To understand all this let’s revisit the double slit experiment once more .

As we have see before ,that when researchers send one electron in double slit experiment ,it creates a diffraction pattern on the other wall .But that sound’s weird because light  creates diffraction pattern on the other wall ,because some light waves are blocked by the wall ,but some light waves passes through the two holes simultaneously and after passing from the these two holes interact with each other and these two light waves either cancel out each other(destructive interference) ,where the phase of the light waves will be opposite or creates a more stronger intensity light wave (constructive interference)where phase of these light waves will be same ,because of this reason we show this bright and dark spot or striped patterns on the other wall.

So this diffraction pattern was produced by interference of two light waves ,but in double slit experiment of electron we are sending only one electron at a time ,so if it is still creating the diffraction pattern ,it means it is passing from both the slits simultaneously that’s weird huh.

Anyway to measure this weird phenomenon ,scientist put the detector on both the slits ,so if one electron passes through both the slits simultaneously ,then detectors should be able to detect it

Now boom!!!!! ,that didn’t happen ,as soon as scientist put the detectors on both the slits ,electron started behaving like other particles like(tennis ball) ,diffraction or striped pattern disappeared ,instead now electron created brightest points on the portion of wall which was just behind the slit or in other words ,same phenomenon which was experienced with tennis balls .Read this  https://acsharmablog.com/2018/07/02/quantum-computing-for-newbies/ first  to understand it better.

 

Very good video on this – Quantum Mechanics: Animation explaining quantum physics

https://www.youtube.com/watch?v=iVpXrbZ4bnU&index=1&list=PLkyBCj4JhHt-elH-mR1d1NfTZ-W0_DCRl

Wow ,so now when we try to measure the wave function of electron ,its wave function collapsed ,that’s really spooky

But doesn’t it makes this all the more complicated

why?

Because to know the information carried by a particle ,we need to know its wave function ,but we can’t do that ,because electron exhibits wave like properties only when we don’t measure it

Then how we will utilize this wave like behavior of electron for exponential speed up on traditional computers.

As mentioned in previous blog ,Wave function of a matter wave is defined by below equation

where V=V(r,t)=r and t are the position vector and time respectively,

Ђ=h/2π where h is momentum

So basically to define the wave function of a matter wave we need momentum and position at different time steps

And how do we measure the position of an object ,we check it time to time and if object is displaced from position x to position y in time t ,then we know its velocity ,and once we know its velocity we can calculate its momentum as well .

But in case of sub atomic level ,its not that easy ,position and momentum of an electron or photon or any particle at sub atomic level are conjugate with each other ,means we can’t measure both position and momentum of a particle with complete precision .How much inaccuracy will be there in this measurement is denoted by heisgenberg uncertainty principal.

As per that principal ,the product of variance of  measurement in position ϭx and variance in measurement of momentum ϭy  should be greater than planck’s constant ђ/2

Ϭxϭy>=ђ/2

But this looks weird ,why we can’t measure the position and momentum both precisely at subatomic level ?We can do it very precisely for bigger objects like trains,cars etc.

To understand this ,first let’s see what’s the very basic step in  calculating position and momentum of bigger object’s –

To measure position and momentum of these object ,we need to see these objects and how we will see them ,we will throw some light on it ,that’s where the catch is 😊

A light ray contains multitude of photons ,so if we will throw light on a single electron ,to see it clearly ,it will displace it right ? ,its like that – to measure the position of a small stone ,we are throwing a very big stone ,this big stone will displace the smaller stone ,so if we want to correctly measure the position of smaller stone ,we need to throw a smaller stone right ?

In case of light ,smaller stone means ,we need to throw a light ,which comprises only one photon ,now light with one photon means very dim light ,in this light you can’t see the position of electron very clearly ,you will get to know that electron present in this region ,but you won’t know ,that where exactly it  present , and if you want to measure the position correctly ,you have a throw a light with more number of photons ,but this light will displace the electron from its position a little bit ,so because of this displacement ,we won’t be able to calculate the momentum correctly .

That’s why this uncertainty constraint ,so basically we can measure the position and momentum of a particle at sub atomic level ,but with this uncertainty threshold ,and because position and momentum is error prone ,then the wave function which is dependent on this ,will also be error prone .

I hope all of you now must have some understanding of heisenberg uncertainty principle.To understand the concept visually ,Please refer this- https://www.youtube.com/watch?v=qwt6wUUD2QI

I hope till now you all have understood the wave function of an electron ,uncertainty associated with measurement of position and momentum of subatomic particle ,and the biggest thing ,if we will try to measure an electron its wave function collapses ,it starts behaving like a normal particle .

So if we want to use electron wave function ,to store spectrum of information simultaneously ,what do we need ?

  • We should be able to measure the wave function without collapsing it
  • We should know that how much error prone this measurement is (because of heisenberg uncertanity) and what are the counter measures to correct this error
  • And the most important question ,how we are going to make a qubit

In next tutorial ,we will explore possible answers of these striking questions.

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