wavefunction in Quantum Mechanics

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wavefunction in Quantum Mechanics

Postby krum on May 4th, 2016, 8:53 pm 

My knowledge of Quantum Mechanics is really poor, based mainly on some popularizations that I never understood very well.

I want to gain some knowledge of the mathematical side of Quantum Mechanics. I hear all that talk about the wavefunction. What I want to ask is: Can a wavefunction have volume? I mean, can it be like a sort of "cloud" in space? Can it even be a "cloud" with infinite volume, spread throughout space?
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Re: 3D waves

Postby Faradave on May 5th, 2016, 2:33 am 

krum wrote:Can a wavefunction have volume?

Yes. Atomic orbitals can be interpreted as standing waves about the nucleus in 3D. For hydrogen these are given by solutions to the Schrödinger equation. The solutions can have indefinite size but only locations with reasonable probability of finding an electron are used mapping atomic orbitals. They give an atom its typically estimated "size".
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Re: wavefunction in Quantum Mechanics

Postby bangstrom on May 8th, 2016, 4:02 am 

I agree with Faradave that atomic orbitals of electrons about a nucleus are best thought of as wavefunction "clouds in space" rather than "orbits" like the orbits of planets about a star.

In theory and in math, the ultimate extent of a Schroedinger wavefunction can span the distance of visible space and beyond but it is debatable whether or not any extension beyond a short distance is of physical consequence.
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Re: wavefunction in Quantum Mechanics

Postby Natural ChemE on May 8th, 2016, 5:16 am 

krum,

Short answer
In theory, no, wave functions tend to have non-zero values just about everywhere (with the exception of nodes), so they basically have the volume of the universe.

Practically, almost all wave functions we discuss have almost all of their probability in a very localized area. It's hard to ascribe an exact volume because there's no known fundamental cut-off.

Long answer
Wave functions have a value at all points in the universe, so you can say that wave functions have infinite volume. However wave functions tend to have near-zero values throughout most of the universe except in very confined locations.

Conceptually, I'd suggest thinking of the Copenhagen interpretation of quantum mechanics, which asserts that the square of a wave function, , describes the probability that a particle is in any particle location. And since the total probability that a particle exists somewhere in the universe must be exactly 100% (by definition), then
    .
Since the universe is huge, it's mathematically impossible for a wave function to have any significant value throughout a large portion of it.

Okay, so that's the Quantum Physics 101 answer. For a more theoretical discussion, it's worth considering if quantum mechanics is valid at the most extreme edges - which has not been proven, nor is particularly likely to be true.

So one weird prediction of quantum mechanics is quantum tunneling. We say that a particle is tunnelling when it's in a location that should require more energy than it has. Tunnelling was first predicted by the wave equations used in quantum mechanics, because they're still non-zero even when they don't have enough energy, , to move into a region of higher potential, :
    Image.
If you suspect that tunnelling is just a mathematical artifact of the wave equations that might not represent an actual, real-life phenomena, you'd be in good company. It was pretty weird.

Just about two weeks back, researchers made headline news with the report that they discovered water molecules tunnelling. In short, the researchers were looking at water in rocks, and they discovered that single water molecules could simeltaneously exist in multiple gaps in the rock.
Quantum Tunneling of Water in Beryl: A New State of the Water Molecule, Kolesnikov et al. (2016) wrote:ABSTRACT
Using neutron scattering and ab initio simulations, we document the discovery of a new “quantum tunneling state” of the water molecule confined in 5 Å channels in the mineral beryl, characterized by extended proton and electron delocalization. We observed a number of peaks in the inelastic neutron scattering spectra that were uniquely assigned to water quantum tunneling. In addition, the water proton momentum distribution was measured with deep inelastic neutron scattering, which directly revealed coherent delocalization of the protons in the ground state.

Anyway, the point here is that we've now confirmed that wave functions can cause particles to exist in ways that don't make sense through the lens of classical physics, and we can see that wave functions continue to work even when their values are near-zero, .

So is it technically possible that a particle here on Earth might somehow interact with a distant star where its wave function is technically non-zero? As far as I know (stressing that I'm not a theoretical physicist), that's an open question.

I'd suppose that, if particles can't interact with distant objects via near-zero wave functions, then it might be appropriate to revise wave mechanics such that wave functions do have finite volumes.
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Re: wavefunction in Quantum Mechanics

Postby hyksos on May 8th, 2016, 3:24 pm 

I want to gain some knowledge of the mathematical side of Quantum Mechanics.

I would advise temporarily dispensing with trying to visualize what the equations are saying. At least in the first few chapters, you will need to 'trust the equations' even when no helpful visual is in your mind from day to day. If you are going to read quantum mechanics textbooks, get more than one of them, and cross-check them. Different books will use different notation for the same ideas. Whenever you make a connection between two different books, you will begin learning.

The internet is a giant library of resources. Some links :



And finally, my strongest suggestion. Do not trust anything that you read on this forum about physics. Not from anyone. Not from NaturalChemE. Not from me included.

The only exception is Lincoln (...but he doesn't post here anymore).
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Re: wavefunction in Quantum Mechanics

Postby krum on May 9th, 2016, 12:31 am 

What's the difference between a wavefunction and a probability wave?
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Re: wavefunction in Quantum Mechanics

Postby bangstrom on May 9th, 2016, 12:32 am 

hyksos » May 8th, 2016, 2:24 pm wrote:
The only exception is Lincoln (...but he doesn't post here anymore).

"Those who know don't speak. Those who speak don't know." Lao tse, 500 BC.
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Re: Sharring a Shared State

Postby Faradave on May 9th, 2016, 11:55 am 

hyksos wrote:Do not trust anything that you read on this forum about physics. ...Not from me included.

You may not have intended to make this point but it's a nice analogy of quantum indeterminism associated with entanglement.

Your statement reduces to the form: "This statement is not true."

For which the logical truth value is purely ambiguous. Though it appears to oscillate between True and False with each successive reading, that is attributable to the nonzero reading time we impose upon it. Its actual value however, is inherently neither True nor False . It's a verbal "shared state" analogous to total-spin-zero entanglement, where neither particle has definable spin of its own.

That the statement is self referential compares to what is achieved via instantaneous quantum "spinhole" (spatial-interconnecting wormhole), now established as connecting entangled particles.[1, 2, 3] This maintains the relationship of their relative values (e.g. their "oppositness"), without requiring their exact values to have been determined. They, in fact, remain "indeterminate".
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Re: What's Waving?

Postby Faradave on May 9th, 2016, 12:20 pm 

krum wrote:What's the difference between a wavefunction and a probability wave?

A wave function is a mathematical expression which may have classical (e.g. describing sound waves) or quantum (e.g. probability) embodiments. The later is often given as an answer to the question "What's waving?" when students are first introduced to wave-particle duality, including the concept of a particle, such as an electron, having a wave aspect. The probability, in that case, refers to the probability of finding the electron at a given location within its described state.
(If you only have time for one of these very brief videos, the second is more important.)


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Re: wavefunction in Quantum Mechanics

Postby Natural ChemE on May 10th, 2016, 3:11 pm 

krum » May 8th, 2016, 11:31 pm wrote:What's the difference between a wavefunction and a probability wave?

The wave function, , is the basic description of quantum mechanical entities. It's what actually exists, as far as quantum mechanics is concerned.

The phrase "probability wave" is non-standard, but I'd guess that the speaker means .
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Re: wavefunction in Quantum Mechanics

Postby Infinite_Observer on September 12th, 2017, 10:46 am 

I was staring up at a fan today and I was wondering...Since when I look at the fan out of the corner of my eye its spinning so fast its a ringed blur but when I focus on one fan blade I can clearly see its position, would that be an example of the collapse of wave function on the visible level or just optical and how our eyes and brain see it?
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Re: Probably Right

Postby Faradave on September 12th, 2017, 11:44 am 

I think that's a great model! You can think of a fan blade as being distributed in its "field" of rotation. If it could rotate infinitely fast, it would be in a "superposition" state, existing probabilistically at all radial locations at once. Only by interaction with the field will an observer find out if the blade is or is not at a particular location. Otherwise the field represents a potential for interaction.
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