Is the Expectation Value of Uncertainty Constant in Time?

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Homework Help Overview

The discussion revolves around the concept of expectation values in quantum mechanics, specifically examining the conditions under which the uncertainty in an observable remains constant over time. The original poster attempts to demonstrate that if the Hamiltonian and an observable commute, along with the time derivative of the observable being zero, then the uncertainty in that observable is also constant.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of a specific equation related to the time derivative of expectation values and explore the implications of commutation relations. Questions arise regarding the definition of uncertainty and the application of Heisenberg's equation of motion, with some participants expressing unfamiliarity with this concept.

Discussion Status

The discussion is ongoing, with participants exploring different aspects of the problem. Some guidance has been offered regarding the differentiation of uncertainty and the use of relevant equations, but there is no clear consensus on the approach to take, particularly concerning the application of Heisenberg's equation.

Contextual Notes

There is a noted lack of familiarity with certain concepts, such as Heisenberg's equation of motion, which may be affecting the participants' ability to engage fully with the problem. Additionally, the original poster mentions constraints based on their current coursework and textbook content.

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Homework Statement


Show that, if [H,A] = 0 and dA/dt = 0, then <&Delta;A> is constant in time.


Homework Equations


d<A>/dt = <i/ℏ[H,A] + dA/dt>


The Attempt at a Solution


I am trying to use the above equation to show that d<&Delta;A>/dt is 0, and I can get to d&Delta;A/dt = 0, but I can't figure out how to compute [H,&Delta;A]. The only thing I can think of is that since &Delta; A is just a function of A and A commutes with H, then &Delta; A also commutes with H, but I can't find a theorem that says that.
 
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What's the definition of [itex]\delta A[/itex] ? Then differentiate it wrt time and use Heisenberg's equation of motion.
 
Sorry, I don't know Heisenberg's equation of motion. We haven't gone over it in class and it doesn't show up in my book until much later.
 
What is the name of the equation you placed under <Relevant equations> ?
 

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