Why is angular momentum quantized?

[TURK]

Hello Everybody.

I wanted to find a good explanation for "Why is angular momentum quantized?" but I found a lot of explanations most of them includes math.
Could you help me about the question Why is angular momentum quantized? philosophically.

thank you form now.

 

[Meir Achuz]

Welcome to physics. A lot of explanations need math.

 

[Manchot]

Hmm...angular momentum quantization without math. Well, from the perspective of a wavefunction, angular momentum is the "generator" of rotations. However, if you rotate an object by 2*pi, it should be as if you did nothing to it at all. Quantization ensures this. (Actually, it's more general than this, but you said not to use math.)

 

[malawi_glenn]

The language / tool in physics is math, not philosophy. Sorry to dissapoint you.

Also, angular momentum is a physical concept defined by math..

 

[TURK]

thank you for your answers. but I meant why it is quantized what is the reason for that. of course basic math must be in there. philosophically in my question is by words, like you are telling it to your father. There is an expressin from a big physicsists who I don't remember the name, If you are not able to express a phenomena in physics to your grand father that means you dodnt understand it. it was something like that. :)

thank you.

 

[malawi_glenn]

Try to explain lattice gauge theories to a 80year old man! ;)

If we ask you this: what sources have you considered?

 

[TURK]

is not there a good explenation for this consept? OK: math can be in this explanation.

:(

 

[ZapperZ]

Hello Everybody.

I wanted to find a good explanation for "Why is angular momentum quantized?" but I found a lot of explanations most of them includes math.
Could you help me about the question Why is angular momentum quantized? philosophically.

thank you form now.

Is there a reason why you are asking about the "angular momentum" in particular? Does this mean that you already have a satisfactory explanation on why other parameters such as energy, spin, magnetic flux, etc. are quantized and that they no longer are puzzling to you?

Zz.

 

[nrqed]

Try to explain lattice gauge theories to a 80year old man! ;)

If we ask you this: what sources have you considered?

I had no problem talking exlaining some work I was doing related to lattice gauge theory to Hans Bethe when he was still alive and way passed 80 years old

 

[setAI]

because it is fundamentally generated by a digital algorithm- as are all features of the cosmos- continuous mathematical models are only approximations of what are in actuality logical operations on the discrete quantized states of a bit register

 

[OOO]

I think, at least part of the reason why angular momentum is quantized can be explained without math.

If you imagine angular momentum as something that runs on a circle, and then add quantum mechanics as causing some "spirals" around this circle, then you are only able to close the spiral if it winds an integer times around the circle.

Closing the spiral corresponds to a stationary quantum state. Of course this doesn't directly explain half integer spin, but I'm sure someone more clever than me can extend this explanation to spinors (however I can hardly believe that this can be done without math...).

 

[malawi_glenn]

I had no problem talking exlaining some work I was doing related to lattice gauge theory to Hans Bethe when he was still alive and way passed 80 years old

=) Of course not.. but i meant the average 80y old..

 

[Count Iblis]

Given some finite amount of energy you can only store a finite amount of information in a finite volume. Clearly, if angular momentum or any other local degree of freedom were not quantized, this principle would be violated and an infinite amount of information could be stored in a finite volume. The entropy of any system would be infinite...

 

[setAI]

Given some finite amount of energy you can only store a finite amount of information in a finite volume. Clearly, if angular momentum or any other local degree of freedom were not quantized, this principle would be violated and an infinite amount of information could be stored in a finite volume. The entropy of any system would be infinite...

you have struck upon the fundamental reasons why continuous/classical systems are unphysical- any truly continuous element of a system would lead to infinite instability- it would have infinite degrees of freedom and be expressed as pure entropy- it could never interact with the rest of the universe-

the only kinds of things that can have the quality of existence are the discrete quantized states produced by algorithmic processes

 

[OOO]

Given some finite amount of energy you can only store a finite amount of information in a finite volume. Clearly, if angular momentum or any other local degree of freedom were not quantized, this principle would be violated and an infinite amount of information could be stored in a finite volume. The entropy of any system would be infinite...

You're saying this as if it was a mathematical fact. But then classical electrodynamics would be impossible mathematically. The classical field does in fact have an infinite number of degrees of freedom in a finite volume and yet the classical field has finite energy density. So I feel the information thing doesn't explain quantization.

 

[Count Iblis]

Well, you need to quantize the field. Impose some boundary conditions on the field and solve for the eigenstates. The energy eigenvalues are of the form \sum_{k, \sigma}\left(n_{k,\sigma} + 1/2) \hbar\omega_{k}, the quantum numbers n_{k,\sigma}.

In the classical limit you get the ultraviolet catastrophe, which is caused by the fact that you can let the frequencies of the occupied modes go to infinity while keeping the energy bounded.

 

[Count Iblis]

I agree with SetAi.

Note that Classical Mechanics, perhaps also classical electromagnetism, allows one to build an apparatus that can solve the Halting problem by brute force. One can make a so-called "rapidly accelerating computer". The clock cycle of this computer will accelerate so fast that an infinite number of computations can be performed in a finite time.

Classical physics is thus not reconcilable with computability.

 

[OOO]

Classical physics is thus not reconcilable with computability.

Who says that nature must conform to our limited ability to compute something ? Probably you see more clearly than me that this leads to unavoidable mathematical inconsistencies. But in the meantime I tend to think that nature is quantized because it's quantized and we have designed mathematical tools that reflect this fact.

 

[Meir Achuz]

A lot of explanations need math.

 

[TURK]

?

Is there a reason why you are asking about the "angular momentum" in particular? Does this mean that you already have a satisfactory explanation on why other parameters such as energy, spin, magnetic flux, etc. are quantized and that they no longer are puzzling to you?

Zz.

actually thers is no specific reason about angular momentum. But in quantum mechanics as known to all quantization of angular momentum is a main body I think. Of course I know some about the quantization of energy or via bohr model about the quantization of angular momentum. However I would be more satisfy if I heard more about this fact.

thank you.

 

[Lunawser]

This is the same as asking, "Why does a guitar string vibrate at a specific frequency when it is held at two points and plucked?"

A particle obeys a wave equation called the Schrodinger equation. It is like the wave in the string. When the particle is confined to an atom, it is like holding the string at two endpoints. When you pluck the string, it vibrates at different frequencies or harmonics, each one of which is a multiple of the fundamental frequency. If you draw two points on a paper and then draw a sine wave going from one to the other, you can do it in several ways. Just draw one arch going from left to right, or draw one arch and one trough from left to right, or draw one arch, one trough and one arch. These are the three fundamental modes, each of which has a different frequency and corresponding energy. See http://en.wikipedia.org/wiki/Vibrating_string -- the picture to the right.

The same thing happens when you hit a drum in two dimensions, except each radial slice of the drum is a sine wave. See http://en.wikipedia.org/wiki/Vibrations_of_a_circular_drum at the bottom where they animate the modes. The same will happen in three dimensions if you strike a sphere; the air inside will oscillate in discrete modes (spherical harmonics). This is the closest to what is happening with the electron in an atom, except the wave also has something called quantum phase, which we will not worry about.

This explains why you only see certain energy levels (quantized modes). It is harder to see why the angular momentum is quantized, but it comes from the same equation in multiple dimensions. There are different ways to distribute the mass to get the same energy, so each of these have a different angular momentum, see http://en.wikipedia.org/wiki/Atomic_orbital, the orbitals table.

To sum up, you get multiple discrete modes because it is confined to a sphere. (In reality, it is not confined to a sphere, but something spherically symmetric, but I think that's close enough.) The modes have different angular momenta because the mass distribution is different for each one and they are discrete because if you demand that the particle is confined to the atom (spherically symmetric potential), then only discrete modes are allowed.

Now go explain this to someone else.

 

[Meithan]

The language / tool in physics is math, not philosophy. Sorry to dissapoint you.

Also, angular momentum is a physical concept defined by math..

'You do not really understand something unless you can explain it to your grandmother.'
— Albert Einstein (?)

Math is a very convenient way to describe nature, but it is by no means the only way.

 

[haael]

Classical physics is thus not reconcilable with computability.

Quantization has nothing to do with computability. It's the same error, as to say that quantum mechanics requires fuzzy logic.

 

[nattydread]

From Bohr's model of the atom we see that the angular momentum L = mvr.
As the mass and velocity can be considered constant we see that L changes with the radius of the orbit.
As the radii are fixed to give whole wavelengths of the electrons, then the quantisation of the radius of the orbit results in the quantisation of the angular momentum.