Prove some relations but going round in circles

In summary, the conversation discusses proving a relation using commutator notation. The solution involves manipulating the commutator using general algebra rules and factoring out a half to arrive at the correct solution.
  • #1
Gregg
459
0

Homework Statement

I need to prove some relations but going round in circles.

## [\hat{J}_z, \hat{J}_+] = \hbar J_+ ##

I've got this:

##\left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)\left(a_+^{\dagger }a_-\right)-\left(a_+^{\dagger }a_-\right)\left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)##

Homework Equations



##J_{\pm} = \hbar a_{\pm}^{\dagger} a_{\mp} ##

##J_z = \frac{\hbar}{2} \left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right)##

The Attempt at a Solution

I have these sort of obvious relations to work from##[a_{\pm}^{\dagger}, a_{\pm}] = 1##
##[a_{\pm}^{\dagger}, a_{\mp}] = 0##
##[a_{\pm}^{\dagger}, a_{\mp}^{\dagger}] = 0##
##[a_{\pm}^{\dagger}, a_{\mp}] = 0##
##[a_{\pm}, a_{\mp}] = 0##

I start expanding and it gets too tough, what am I missing?
 
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  • #2
I'd suggest working in commutator notation instead of expanding it out completely. I.e. start from:
[tex]
[\hat{J_z},\hat{J_+}] = [\frac{\hbar}{2} \left(a_+^{\dagger }a_+-a_-^{\dagger }a_-\right),\hbar a_{+}^{\dagger}a_{-}]
[/tex]

Then manipulate that using the general rules of algebra for commutators.
 
  • #3
Using

## [A+B,C] = [A,C] + [B,C] ##
## [AB,C] = [A,C]B + A[B,C] ##
## [A,BC] = [A,B]C + B[A,C] ##

It quickly boils down to

##\frac{\hbar^2}{2} \hat{a}_{+}^{\dagger} \hat{a}_{-} ##

I am out by a factor of half but I'm sure it works!
 
  • #4
Yes it works fine, I factored a half out of the commutator wrongly. Thanks!
 
  • #5
Sweet! I'm glad it worked out. These types of problems can and will get messy sometimes. Good thing I didn't have to puzzle through it myself. :P
 

FAQ: Prove some relations but going round in circles

1. What does it mean to prove some relations but going round in circles?

Proving some relations but going round in circles refers to a circular argument, where the conclusion is already assumed in the premises. It is a logical fallacy and does not provide any meaningful evidence to support a claim.

2. How can you avoid going round in circles when proving relations?

To avoid going round in circles, it is important to carefully examine the premises and ensure that the conclusion is not already assumed or implied. It is also helpful to seek out different perspectives and consider opposing arguments.

3. Is going round in circles a common mistake in scientific research?

Yes, going round in circles is a common mistake in scientific research, especially when researchers are biased towards a particular outcome or have a vested interest in the results. It is important for scientists to remain objective and avoid circular reasoning in their studies.

4. How can circular reasoning impact scientific findings?

Circular reasoning can greatly impact scientific findings by leading to false conclusions and hindering the progress of knowledge. It can also diminish the credibility of research and make it difficult for other scientists to replicate or build upon the findings.

5. Can circular reasoning ever be useful in scientific research?

No, circular reasoning is never useful in scientific research. It goes against the principles of the scientific method, which requires evidence-based reasoning and logical conclusions. Any findings based on circular reasoning should be reevaluated and further research should be conducted to verify the results.

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