1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

What does this mathematical expression mean physically?

  1. Mar 10, 2015 #1
    This is from from Sakurai and Napolitano HW Prob #2.6. I've done the HW problem and can fit all the pieces together with the mathematical formalism, but even afterwards I'm still scratching my head wondering what I'm looking at.

    I start with the commutator [[H,x],x]. If I take <ai|[[H,x],x]|ai>, and manipulate with the formalism, I end up with

    ##\underset{k}{\sum}\left(E_{k}-E_{j}\right)\left|<a_{j}|x|a_{k}>\right|^{2}=\frac{\hbar^{2}}{2m}##

    Which is what the book is asking me to prove. What does this expression mean physically?

    My best guess looking at this, is that it is related to a transition amplitude, since I see we are taking the probability that the x-operator yields a different energy state, and then multiplying that probability by the energy gap between them.

    This is how I read it: "If we sum the energy gaps between the specific energy state j weighted with the probability of obtaining the energy gap upon operating with the position operator, then we end up with ##\frac{\hbar^{2}}{2m}##."

    Perhaps it would be easier for me to understand if there is something classical I can relate this to?

    Thanks.
     
  2. jcsd
  3. Mar 12, 2015 #2
    To be honest, it just looks like an alternative statement of the Heisenberg Uncertainty principle.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: What does this mathematical expression mean physically?
Loading...