Position and Momentum are random variables in QM?

[mike1000]

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.

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.

 

[BvU]

position and momentum are random variables in QM

To some extent, yes. Ballentine is free and widely recommended.

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!

 

[mike1000]

To some extent, yes. Ballentine is free and widely recommended.

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.

https://books.google.com/books?id=sHJRFHz1rYsC&pg=PA7&source=gbs_toc_r&cad=3#v=onepage&q&f=false

 

[bhobba]

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

 

[BvU]

I shouldn't have posted that. Bhobba subtly puts me right.