Formula of Heisenberg Uncertainty

[songoku]

I came across two different forms of Heisenberg uncertainty:

\Delta x . \Delta p \geq \frac{h}{2 \pi}

and

\Delta x . \Delta p \geq \frac{h}{4 \pi}

and I read here in PF both forms are correct. I want to know how we apply both forms in solving problem.

Here is the example:
1. An electron moves with a constant velocity of 1.50 x 106 ms^-1. If its momentum is measured to a precision of 0.20 percent, find the minimum uncertainty in its position

2. An electron has kinetic energy 5.00 eV and its momentum is accurate to within 0.0100%. Find the minimum uncertainty in determining the position of this electron.

Both are example questions from two different books. Question 1 is solved using first form and question 2 is solved using second form.

How to determine when to use which form?

Thanks

 

[Nugatory]

and I read here in PF both forms are correct.

Where in PF? Give is a link.

 

[songoku]

Where in PF? Give is a link.

https://www.physicsforums.com/threads/uncertainty-in-heisenbergs-uncertainty-principle.60089/

post #3, unless I misinterpret the meaning

 

[jtbell]

Post #3 states only a single version, in two different ways. Note carefully the difference between $h$ and $\hbar$.

 

[moriheru]

planckconstant (bar) is h/2pi !

 

[songoku]

Post #3 states only a single version, in two different ways. Note carefully the difference between $h$ and $\hbar$.

planckconstant (bar) is h/2pi !

I know the difference between h and h bar. What I mean by post#3 is " The smallest possible value is found in the case of the the Gaussian Function only. For any other function it is more."

So Δx.Δp ≥ h / 4π is for the case of Gaussian Function only and there is possibility that for other case Δx.Δp ≥ h / 2π is used. Am I correct?

I am still confused about two example questions I wrote. As I said, the two books use different form to solve the problems. Is the book using Δx.Δp ≥ h / 2π wrong?

I also read this in College Physics (book) by Serway:
"If a measurement of the position of a particle is made with precision Δx and a simultaneous measurement of linear momentum is made with precision Δpx , then the product of the two uncertainties can never be smaller than h/4: Δx . Δpx ≥ h / 4π"

But after one or two paragraphs explaining photon hitting electron, there is this statement:
" Further, because the photon also has wave properties, we expect to be able to determine the electron’s position to within one wavelength of the light being used to view it, so Δx =lambda. Multiplying these two uncertainties gives Δx . Δpx = h, where Δpx = h / lambda. The value h represents the minimum in the product of the uncertainties. Because the uncertainty can always be greater than this minimum, we have Δx.Δp ≥ h. Apart from the numerical factor 1/4π introduced by Heisenberg’s more precise analysis, this inequality agrees with previous equation (Δx . Δpx ≥ h / 4π)"

Can we use Δx.Δp ≥ h / 2π (or even use Δx.Δp ≥ h) to solve problems regarding HUP? If yes, how can we differentiate between using those and Δx.Δp ≥ h / 4π? If no, why?

Thanks

 

[jtbell]

So Δx.Δp ≥ h / 4π is for the case of Gaussian Function only and there is possibility that for other case Δx.Δp ≥ h / 2π is used. Am I correct?

Keep in mind that the HUP is a statistical statement. For a Gaussian wave packet, and assuming that Δx and Δp are defined using the standard deviation of a large number of measurements (and not the deviation in any single measurement), your first equation becomes an equality. For other shape wave packets, you get an equality with a different constant on the right side. The Gaussian has the smallest value for the constant. Therefore, if you don't know what shape your wave packet is, only that you have a wave packet of some kind, your first equation (inequality) holds.

In introductory-level problems like the ones you see in Serway and similar books, people are normally interested only in order-of-magnitude (nearest power of ten) estimates of uncertainty. In that case the difference between $\hbar/2$, $\hbar$ and $h$ isn't significant. I suppose one can argue that if you want a realistic estimate of uncertainty, you should use something bigger than the minimum of $\hbar/2$ on the right side of the HUP.

At this level, it's not worth stressing out over the differences. Simply take your cue from the author of whatever book you're using.

 

[songoku]

Ok

Thanks a lot for the explanation