Time-energy uncertainty and derivative of an operator

  • #1
880
0

Homework Statement


I would appreciate feedback on the following two problems:
(1) For a given operator A with no explicit time dependence I am asked to show that d/dt(eAt)=A(eAt)
(2) A free wave packet of width Δx is traveling at a constant velocity v0=p0/m. I am asked to estimate the time-energy uncertainty relation.

Homework Equations




The Attempt at a Solution


(1) Is this as straightforward as it appears to be? Probably isn't, right? Is this to be solved using Ehrenfest's theorem? Or how?
(2) I can easily show that ΔEΔt ~ ħ but is this what I am plainly asked, or am I rather asked to arrive at a more exact result (i.e. find the proportionality coefficient)?
 

Answers and Replies

  • #2
jfizzix
Science Advisor
Insights Author
Gold Member
757
355
To take the time derivative of [itex]e^{At}[/itex] with respect to [itex]t[/itex], you can take the power series expansion of [itex]e^{At}[/itex], differentiate term by term, and re-exponentiate, since what you get left should be [itex]A[/itex] times another exponential.

For the second problem, it looks like they're asking you to calculate the uncertainty product [itex]\Delta E\Delta t[/itex] from the given information. Then you can see for yourself how much larger than [itex]\frac{\hbar}{2}[/itex] it is.
 
  • #3
880
0
For the second problem, how may I go about it then? Is it necessary to write the general wave function for the wave packet and then find the variances?
 
  • #4
jfizzix
Science Advisor
Insights Author
Gold Member
757
355
I would think so, if you know what the wavefunction of the wave packet is. I can't say off the top of my head what that would be, though.
 
  • #5
880
0
The question, as formulated, does not provide any additional information other than that in #1. Any ideas?
 
  • #6
jfizzix
Science Advisor
Insights Author
Gold Member
757
355
I'm guessing that there is a standard formula for a free wave packet of given [itex]\Delta x[/itex] and [itex]p_{0}[/itex], to which the problem is referring.

Failing that, the best I can think of is to find a limit to the momentum uncertainty with [itex]\Delta p\geq \frac{\hbar}{2 \Delta x}[/itex].

Then, knowing [itex]p_{0}[/itex] and a bound for [itex]\Delta p[/itex], you can use that [itex]E=\frac{p^{2}}{2m}[/itex] and the propagation of uncertainty

[itex](\Delta E)^{2}\approx (\frac{\partial E}{\partial p})^{2}(\Delta p)^{2}[/itex]

Getting an estimate for the time uncertainty is a bit trickier (since time is not an observable), but you can say that

[itex]\Delta t = \frac{\Delta x}{(\frac{d <x>}{dt})}[/itex]

That's about all I can say on the matter, though.
 

Related Threads on Time-energy uncertainty and derivative of an operator

  • Last Post
Replies
10
Views
5K
  • Last Post
Replies
1
Views
790
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
3K
Replies
1
Views
2K
Replies
1
Views
1K
  • Last Post
Replies
3
Views
654
  • Last Post
Replies
2
Views
10K
Replies
2
Views
3K
  • Last Post
Replies
2
Views
2K
Top