Why Is It Called Space Quantization in Quantum Mechanics?

[Roo2]

I'm not quite sure which forum I should post this in; it's related to my school work but I don't have any homework questions about it. Rather, I am just confused about what it really is and what it means. Hopefully it goes here.

I get that the projection of the angular momentum along some reference axis is quantized. My question is what does this mean physically and how does it lead to the band splitting in the Stern-Gerlach experiment? Also, why is it called space quantization rather than angular momentum quantization?

I read that the angular momentum is quantized against some reference axis, which may be arbitrary. This is precisely why I am confused about the S-G experiment. If the reference axis is arbitrary, then there is no reason to apply the same reference axis to all atoms in a collection. Therefore, why does the Stern-Gerlach experiment only show two bands, rather than a dispersion as each atom travels to a location on the plate as dictated by its own angular momentum projection?

Finally, if it is "just" the angular momentum projection that's quantized, why is it called space quantization? Is this some sort of mild misnomer, or are the implications of angular momentum quantization going over my head? Does a.m. quantization imply something about quantization of position (or something else) across all coordinates in space?

Thanks for any replies; I'm pretty confused on this topic.

 

[Bill_K]

I read that the angular momentum is quantized against some reference axis, which may be arbitrary. This is precisely why I am confused about the S-G experiment. If the reference axis is arbitrary, then there is no reason to apply the same reference axis to all atoms in a collection. Therefore, why does the Stern-Gerlach experiment only show two bands, rather than a dispersion as each atom travels to a location on the plate as dictated by its own angular momentum projection?.

When it says the axis is arbitrary, it means that for a spin 1/2 system, for example, you will always get just the two values +/- 1/2, no matter which axis you choose. This is quite different from what you would see in a classical system, and quite different from what you would expect.

An atom does not keep a history. It does not remember the axis from the last time it was quantized. The Stern-Gerlach apparatus imposes an axis and all the atoms must use it, and they will all have spin projection either +/- 1/2 along that axis.

 

[wawenspop]

Just as a circle cannot have an exact area (because pi is irrational) a perfect curve cannot exist - it must always be incremented in units and thus at some level be 'pixellated' you could even say quantized in a certain sense. This has consequences for our metric that we call space. Is it continuous?


(I thought this thread was about space quantization...)

 

[Roo2]

Just as a circle cannot have an exact area (because pi is irrational) a perfect curve cannot exist - it must always be incremented in units and thus at some level be 'pixellated' you could even say quantized in a certain sense. This has consequences for our metric that we call space. Is it continuous?(I thought this thread was about space quantization...)

Are you my P chem professor? He uses the pixellation argument too. What I don't understand is how the quantization of angular momentum or spin requires the incrementation (incrementedness?) of the curve. I get both concepts individually, but I don't see the connection between them and at least in my textbook, they're both covered under the umbrella of space quantization.

 

[khemist]

Are you my P chem professor? He uses the pixellation argument too. What I don't understand is how the quantization of angular momentum or spin requires the incrementation (incrementedness?) of the curve. I get both concepts individually, but I don't see the connection between them and at least in my textbook, they're both covered under the umbrella of space quantization.

The quantization of the curve IS the incrementation of the curve. Quantization means there are discrete values, which is the same thing as an increment.

Doesnt really describe why space itself is quantized, but maybe its more of a definition problem?

 

[wawenspop]

The quantization of the curve IS the incrementation of the curve. Quantization means there are discrete values, which is the same thing as an increment.

Doesnt really describe why space itself is quantized, but maybe its more of a definition problem?

From what you said space must be quantized - isn't it a logical following of this argument? Why not?

 

[Roo2]

The quantization of the curve IS the incrementation of the curve. Quantization means there are discrete values, which is the same thing as an increment.

Doesnt really describe why space itself is quantized, but maybe its more of a definition problem?

Agreed. But, in order to make the connection in my original question in the OP, the following statement would have to be true:

The quantization of the angular momentum IS the quantization of the curve.

Is this the case? If so, why? And what is "curve" referring to? The wavefunction by which the particle whose angular momentum is quantized is bound?

 

[wawenspop]

angular momentum formula contains pi which is an irrational number and cannot have an exact value, so neither can angular momentum - correct?
Is it a question then of what precision is being used before quantization occurs?

 

[Hurkyl]

Just as a circle cannot have an exact area (because pi is irrational)

Yes it does: pi. :tongue: Don't confuse the ideas "value of something" with "a decimal numeral with finitely many digits".

a perfect curve cannot exist

Only for certain meanings of "exist".

But these are issues for a different thread -- they are a discussion of philosophical biases rather than anything else. There are honest, scientific reasons to explore theories involving the quantization of space-time. (but I'm not entirely sure any of this has to do with the question in the opening post!)


Back to the OP: could "space quantization" just be an abbreviation of "phase-space quantization"? Wiki's page on the latter was the second hit when I googled the phrase.

 

[jtbell]

Just as a circle cannot have an exact area (because pi is irrational) a perfect curve cannot exist - it must always be incremented in units and thus at some level be 'pixellated' you could even say quantized in a certain sense. This has consequences for our metric that we call space. Is it continuous?

(I thought this thread was about space quantization...)

I think you and the posts following yours are confusing two meanings of "space quantization."

The "space quantization" that Roo2 originally asked about is simply a short name for "quantization of the component of angular momentum along a specified direction." I don't know how that name came to be, but it goes back all the way to the days of the Stern-Gerlach experiment in the 1920s, perhaps even earlier.

[I now see that Hurkyl has suggested that it's short for "phase-space quantization" which sounds plausible to me.]

The notion of space (and time) quantization in terms of Planck units or whatever, that people often ask about here on PF, is something different. It's a speculative concept and is not related to the "space quantization" of the preceding paragraph, as far as I know.

 

[khemist]

Thanks for clearing that up jtbell.

 

[Roo2]

Thanks jtbell, that clears things up for me!

 

[wawenspop]

Yes it does: pi. :tongue: Don't confuse the ideas "value of something" with "a decimal numeral with finitely many digits".


.

There is no exact area of a circle in binary, hex, oct, qubits decinal or any other base. By inspection a unit of area is a square. And no matter how hard one tries one cannot make a curve into a square. There can be no discrete number of squares in a circle. Thats why pi is irrational. There is no such beast in the cosmos as a perfect circle.

Its a question of precision. The precision of universal quantities must be limited. Thats also why we cannot get to zero separation and so get infinite forces.

Suppose there will be no sensible reply to this.

 

[alxm]

Wavenspop, everything you said is a load of nonsense. You seem to be into computers, and trying to shoehorn both physics and mathematics into that kind of thinking.

The universe isn't a computer. As far as anyone knows, it doesn't work with discrete math. It doesn't have a special preference for integers. Space is not made up of 'pixels' or voxels, AFAIK not even in the more speculative theories jtbell mentions. In fact I believe that would violate a number of quite fundamental assumptions about how everything does work. (e.g. Noether's theorem). There is no 'question of precision'. The universe isn't a computer that needs to store digits somewhere, making exact representations of irrational numbers impossible. Besides which, every single integer can trivially be represented by an any number of infinite sums (e.g. 1 = 0.9999..) - just as pi can. So you're basically defying logic by claiming that integers are exact and things represented by an infinite series sum cannot be.

There are also basic factual errors in what you're saying. There's an infinite number of curves whose area can be squared. A trivial example is the area under the parabolic curve 3x² on the interval [0,1], which is unity. It's just that a circles and ellipses are not among those which can have an integer area with integer radii. That doesn't prove at all that they don't exist. It proves you can't draw a perfect circle on a computer screen, but as I already said, the universe doesn't work that way as far as anyone knows. Besides which, why should area specifically have to be integral? Every square with integer sides has an irrational diagonal, so you might as well argue that squares don't exist either.

And you can "go to zero separation and get infinite forces". In an atom, two electrons (with opposite spin), modeled as point charges with a 1/r coulomb potential, will still have a non-zero probability of being in the same place at the same time. Because the infinity in the potential is canceled out by the infinity in the kinetic energy. (http://www.archive.org/details/oneigenfunctions00kato" a detailed and rigorous proof of that fact as well as some important properties of those cusps).

Just because something might not be intuitive doesn't mean it's not logical. And if there's anything we know, it's that the universe obeys logic, not human intuition.