Probability fluid velocity and quantum entanglement question

[JPaquim]

Hey everyone, I'm a second year undergraduate student in aerospace engineering and I've been learning a bit of quantum mechanics for the past few weeks, for recreational purposes.
I've been following this textbook: http://www-thphys.physics.ox.ac.uk/people/JamesBinney/qb.pdf

On pages 40-41, the author makes an analogy between the probability density function associated with the wave function, and regular incompressible fluid flow/flow of electrons in a conductive wire. In the process, he associates a probability current density [itex]J[/itex] and a probability fluid velocity [itex]v[/itex]. Then he mentions that in a state of well-defined momentum the probability fluid velocity reduces to classical particle velocity. First of all, how exactly does the computation [itex]\nabla(\vec{x}\cdot\vec{p}) = \vec{p}[/itex] work? What sort of product rule should I use and how will it all work out? And does this way of interpreting the wave function as being somehow represented by a fluid have any real utility? Is it something I should pursue, or is it useful only for this particular situation? What interpretation should I have of the fact that in a situation of definite momentum, the fluid velocity coincides with the particle velocity?

Another perhaps more general question regarding functional analysis. The author defines the product of two operators as simply function composition. But when he differentiates a product of operators, he uses the regular product rule, instead of the chain rule. How come?

And a very simple question, just to see if I've got the basics of quantum entanglement down. In the EPR experiment, the fact that one of the electron's z spin is measured to be +1/2 and subsequently the other one's x spin is measured to be +1/2 places the first one back in a state of indefinite z spin, right? So subsequent measurements of the z spin of the first could yield either +1/2 or -1/2, right? And supposedly that information can travel faster than light, being a so called "spooky action at a distance".

Cheers

 

[eljose79]

Hello,

I'm glad to hear that you are exploring quantum mechanics for recreational purposes! It's a fascinating subject with many real-world applications.

Regarding the analogy between the probability density function and fluid flow, it is indeed a useful way to think about the behavior of particles in quantum mechanics. The computation \nabla(\vec{x}\cdot\vec{p}) = \vec{p} is simply using the product rule for differentiation. In this case, it is the dot product rule, which states that \nabla(\vec{a}\cdot\vec{b}) = \vec{a}\cdot\nabla\vec{b} + \vec{b}\cdot\nabla\vec{a}. In the context of quantum mechanics, this means that the probability current density (J) is equal to the product of the probability density (|\Psi|^2) and the particle's momentum (p).

The interpretation of the fact that in a state of definite momentum the fluid velocity coincides with the particle velocity is simply a way of understanding the behavior of particles in quantum mechanics. In classical mechanics, particles have a well-defined position and momentum, but in quantum mechanics, these quantities are described by probability distributions. The analogy with fluid flow helps us understand how these probabilities can change and how they relate to the behavior of particles.

Regarding the use of the regular product rule instead of the chain rule in functional analysis, it is simply a matter of convenience and convention. Different authors may choose to use different notations, but the end result should be the same.

As for your question about quantum entanglement, you are correct in your understanding. When one particle's spin is measured, the other particle's spin becomes indefinite. This is known as the "collapse of the wave function" and is a fundamental aspect of quantum mechanics. As for the apparent "spooky action at a distance," it is still a topic of debate and research in the field. Some theories suggest that there may be some underlying mechanism that allows for this apparent instantaneous communication between entangled particles, while others argue that it is simply a result of our limited understanding of the quantum world.

I hope this helps answer your questions and encourages you to continue exploring the fascinating world of quantum mechanics. Best of luck in your studies!