Position and Momentum are random variables in QM?

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Discussion Overview

The discussion revolves around the characterization of position and momentum as random variables in quantum mechanics (QM). Participants explore the implications of this perspective, particularly in relation to classical mechanics and the interpretation of quantum operators and probability distributions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that position and momentum should be viewed as random variables in QM, arguing that this perspective changes the understanding of derivatives in quantum mechanics.
  • Another participant agrees to some extent, noting that in the classical limit, the time derivative of the expectation value of the position operator relates to the expectation value of the momentum operator divided by mass.
  • There is a discussion about the availability of the textbook by Ballentine, with some participants asserting it is freely available while others clarify that it is not free but suggest alternatives for accessing it.
  • One participant expresses uncertainty about their previous comment regarding the textbook's availability, indicating a correction from another participant.

Areas of Agreement / Disagreement

Participants generally agree on the notion that position and momentum can be considered random variables in QM, but there is no consensus on the implications of this view or the specifics of the classical limit. The discussion about the textbook's availability also reveals differing opinions.

Contextual Notes

Some claims about the relationship between quantum and classical mechanics remain unresolved, particularly regarding the interpretation of derivatives and the nature of probability distributions in QM.

mike1000
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A paradigm shift for me occurred when, I now realize, that position and momentum are random variables in QM. As such, it does not make any sense to say things like "take the derivative of the position with respect time".

Instead QM has the position and momentum operators which operate on the probability distribution. The probability distributions are inherently multi-modal (except for the ground state?). In the classical limit, the number of modes becomes infinitely dense and they approach the well know classical curves.

Here is a picture of the probability distribution for the 100th state of the quantum harmonic oscillator. The thick line is the probability distribution for the classical harmonic oscillator.

harmonicosc2.gif


The light bulbs are beginning to turn on and I think I am ready to read a textbook on Quantum Mechanics. I have heard about the one by Ballentine and I think I will start there.
 
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mike1000 said:
position and momentum are random variables in QM
To some extent, yes. Ballentine is free and widely recommended.
mike1000 said:
it does not make any sense to say things like "take the derivative of the position with respect time".
No, but in the classical limit (##h\downarrow 0##) the time derivative of the expectation value for the position operator is the expectation value for the momentum operator divided by the mass. Somewhat comparable at least!
 
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mike1000 said:
Why do you say it is free? Is it available online somewhere? I found a fairly long preview of it at this link, but it is not complete.

I am sure he meant freely.

It is not free which is the same for all academic books, although a very few authors occasionally make it free such as Griffiths book on Consistent Histories:
http://quantum.phys.cmu.edu/CQT/index.html

If money is a problem look into second hand:
https://www.amazon.com/gp/product/9810241054/?tag=pfamazon01-20

Also look into your local library. Most university libraries have it and at least at the universitys I went to (ANU and QUT) anyone was welcome to go to the library and read - student or not.

Thanks
Bill
 
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o:) I shouldn't have posted that. Bhobba subtly puts me right.
 
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