When can one clear the operator

In summary, the conversation discusses difficulties in understanding operator algebra in a projection case, specifically in terms of state expansion and moving operators between bras and kets. It is suggested to think of bras and kets as row and column vectors, and to use linear algebra concepts to help understand the problem. The conversation concludes with the recommendation to consult Ballentine's chapter 1 for further guidance.
  • #1
Ernesto Paas
5
0
Hi all!
I'm having problems understanding the operator algebra. Particularly in this case:
Suppose I have this projection ## \langle \Phi_{1} | \hat{A} | \Phi_{2} \rangle ## where the ##\phi's ## have an orthonormal countable basis.
If I do a state expansion on both sides then I suppose I'd get this: [tex]\sum_{n,m} \langle n | b_{n}^* \, \hat{A} \, c_{m} | m \rangle [/tex]
And what's to stop me from moving the operator to the left and getting a kronecker-delta?
 
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  • #2
I presume your expansion states are eigenstates of ##\hat A##? If so, then I'd say "yes..."
 
  • #3
strangerep said:
I presume your expansion states are eigenstates of ##\hat A##? If so, then I'd say "yes..."
Ah no... Sorry for not pointing that out. They are just of the same same arbitrary orthonormal basis.
 
  • #4
Oh, well, if your expansion states are not eigenstates of ##\hat A##, then... what are they? E.g., what is $$\hat A |m\rangle ~=~ ?$$
 
  • #5
I don't know, that's the problem. I've seen bras and kets being moved around without being eigenstates of an operator. Like in this case $$ \langle a | b \rangle \langle c1 | A | c2 \rangle \langle d | e \rangle = \langle a | b \rangle \langle d | e \rangle \langle c1 | A | c2 \rangle $$
 
  • #6
Is the operator like glued to the left ket? Can I move the bra around?
 
  • #7
Think of the bra ##\langle a|## as a row vector "##a^T##'', the ket ##|b\rangle## as a column vector "##b##", and the operator ##\hat A## as a matrix ##A##. I'll write it the overall expression as ##a^T A b##. Does that much make sense?
(I'm not sure how much ordinary linear algebra you already know.)
 
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  • #8
Ok it hadn't occurred to me to think of them in that way. It wouldn't make sense to do a commutation there if the rows didn't equal the columns. Nevertheless, moving scalars around wouldn't be a problem.

Thanks, it was actually really easy...
 
  • #9
More generally, see the second line of my signature below... :oldbiggrin:

Ballentine, ch1, can help a lot with these sort of questions.
 
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Related to When can one clear the operator

1. When can one clear the operator?

The timing for when one can clear the operator depends on the specific experiment or study being conducted. It is important to consult the protocol or methodology being used to determine the appropriate timing for clearing the operator.

2. What does it mean to clear the operator?

Clearing the operator refers to removing any potential biases or influences from the scientist or experimenter in order to ensure the validity and reliability of the results. This can involve minimizing personal opinions, emotions, and other external factors that may impact the experiment.

3. Why is it important to clear the operator?

Clearing the operator is crucial in obtaining accurate and unbiased results in scientific studies. By minimizing personal biases and influences, the results are more likely to reflect the true nature of the phenomenon being studied.

4. How does one clear the operator?

There are various techniques and strategies that can be used to clear the operator, such as blinding the experimenter to the experimental conditions, using standardized procedures, and having multiple experimenters to reduce individual biases. It is important to carefully consider and implement these techniques in order to effectively clear the operator.

5. Can the operator ever be completely cleared?

While it is impossible to completely eliminate all biases and influences, it is important for scientists to strive to minimize them as much as possible. Through careful planning, implementation of techniques, and constant self-reflection, the operator can be significantly cleared and the validity of the results can be strengthened.

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