Position and Momentum are random variables in QM?

In summary: It is not free, but it is available at most academic libraries. If money is a problem, second hand may be the way to go.I should have been more clear. It is not free, but it is available at most academic libraries.
  • #1
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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  • #2
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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FAQ: Position and Momentum are random variables in QM?

What is the significance of position and momentum in quantum mechanics?

Position and momentum are two fundamental physical quantities in quantum mechanics that describe the location and velocity of a particle. These variables play a crucial role in determining the behavior and properties of quantum systems.

How are position and momentum related in quantum mechanics?

In classical mechanics, position and momentum are two independent variables. However, in quantum mechanics, they are related through the Heisenberg uncertainty principle, which states that the more precisely we know the position of a particle, the less we know about its momentum, and vice versa.

Why are position and momentum considered random variables in quantum mechanics?

According to the principles of quantum mechanics, the state of a particle cannot be precisely determined, and its behavior is described by a probability distribution. Therefore, position and momentum are considered random variables as they can take on a range of values with varying probabilities.

How are position and momentum measured in quantum mechanics?

In quantum mechanics, the act of measuring a particle's position or momentum causes the wave function describing the particle to collapse and yield a specific value. The measurement process involves interacting with the particle, which can alter its state and introduce uncertainties in the measured values.

Can the position and momentum of a particle in quantum mechanics be known simultaneously?

No, according to the Heisenberg uncertainty principle, the position and momentum of a particle cannot be known simultaneously with absolute precision. This is a fundamental limitation of quantum mechanics and reflects the probabilistic nature of the quantum world.

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