Commutators and Expectation Values

In summary, the conversation revolves around a homework question that is due in the upcoming week. The question involves proving a conclusion, but the textbook and course notes only state the conclusion without providing a way to prove it. The individual shares a link to an image of the exam question and their attempts at solving it. Another person suggests using the equation ##[X,Y]+\{X,Y\}=(XY-YX)+(XY+YX)=2XY## to solve the question, which leads to the understanding of where the factor of 1/2 comes from. The conversation concludes with the individual expressing gratitude for the explanation.
  • #1
Bravus
23
0

Homework Statement



Here's a link to an image of the exam question. It appears in the exam every couple of years, and it's due in my exam this coming week. I've looked in both the textbook and the course notes, and they simply *state* the conclusion, so I don't have a way of proving it, and in particular don't know where the factors of 1/2 come from.

http://www.bravus.com/commutators1.jpg

Homework Equations



See above.

The Attempt at a Solution



Here's an image of my attempts so far:

http://www.bravus.com/commutator2.jpg
 
Last edited by a moderator:
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  • #2
You have ##[X,Y]+\{X,Y\}=(XY-YX)+(XY+YX)=2XY##, right? See where the 1/2 comes from now?
 
  • #3
Yep, that works, thanks! (As ever, it looks easy once someone explains!)
 

Related to Commutators and Expectation Values

1. What is a commutator?

A commutator is a mathematical operation used in quantum mechanics to determine the order in which operators are applied. It is defined as [A, B] = AB - BA, where A and B are operators. The commutator measures how much two operators do not commute, or do not give the same result when applied in different orders.

2. How is the commutator related to uncertainty?

The commutator is related to uncertainty through the Heisenberg uncertainty principle, which states that the product of the uncertainties in two incompatible observables (represented by operators) must be greater than or equal to the magnitude of their commutator. This means that the more two operators do not commute, the more uncertain their corresponding observables are.

3. What is an expectation value?

An expectation value is the average value of an observable in quantum mechanics. It is calculated by taking the inner product of the state vector and the corresponding operator, and then applying that operator to the state vector. The result is a single numerical value that represents the expected measurement of that observable in the given state.

4. How is the expectation value related to the commutator?

The expectation value of a commutator is related to the average value of the corresponding observables. If the expectation value of the commutator is zero, then the observables commute and their values are not affected by the order in which they are applied. However, if the expectation value is non-zero, then the observables do not commute and their values are affected by the order of application.

5. Can the expectation value of a commutator be negative?

Yes, the expectation value of a commutator can be negative. This indicates that the corresponding observables do not commute and their values are affected by the order of application. A negative expectation value does not necessarily mean that the corresponding observables are incompatible, as the magnitude of the expectation value is also important in determining the level of uncertainty.

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