Heisenberg uncertainty principle (simple stuff)

[Kenny Lee]

I'm having a little trouble with my textbook's explanation. This is regarding the energy - time variation of the uncertainty principle; very basic stuff, so I hope helping me out won't take too much. Let me quote exactly:

"... the energy conservation can appear to be violated by an amount delta E, as long as it is only for a short time interval, delta t, consistent with the equation (delta E) * (delta t) >= (h / 4 pi )"

I understand what is being said - and it makes sense. Otherwise we'd all be noticing these 'violations' of energy conservation.
But I don't see how it is 'consistent' with the equation. Why must delta t be small? To me, it seems more 'consistent' with the equation that delta t is large, so that the equation becomes greater than h/4pi.

Someone please help clarify. And thanks for your time.

 

[barob1n]

(deltaE)*(delta t) is always greater than " ". Think about it this way: If you take more measurements (greater accuracy-much data) over a longer time, say for a single body whose energy is conserved, then the uncertainty in energy tends toward zero. So, if you make a quick (inacurate-few data points) measurement the uncertainty in energy is high. This could lead you to believe that the energy was not conserved. Atleast, I think that was the point the book was trying to make.

 

[kyle8921]

The Heisenberg uncertainty principle is a fundamental principle in quantum mechanics that states that it is impossible to simultaneously know the exact position and momentum of a particle. This also applies to other pairs of complementary variables, such as energy and time. The equation (delta E) * (delta t) >= (h / 4 pi) represents the uncertainty in energy and time, where h is Planck's constant and 4 pi is a mathematical constant.

The reason why delta t must be small in this equation is because of the nature of quantum mechanics. In classical physics, we can measure the position and momentum of a particle with absolute precision, but in quantum mechanics, the act of measurement itself affects the state of the particle. This means that the more precisely we try to measure the energy of a particle, the less we know about its time evolution, and vice versa.

So, if we want to decrease the uncertainty in energy, we must decrease the uncertainty in time. This is why the equation requires delta t to be small. If delta t were large, it would mean that we have a very precise measurement of time, but a large uncertainty in energy, which goes against the principle.

In summary, the Heisenberg uncertainty principle tells us that there will always be a trade-off between the precision of our measurements of complementary variables. We cannot have both variables known with absolute certainty at the same time. The equation (delta E) * (delta t) >= (h / 4 pi) quantifies this trade-off and shows us that in order to decrease the uncertainty in one variable, we must increase the uncertainty in the other. I hope this helps clarify the concept for you.