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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.
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.