- #1

Silversonic

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There must be some lapse in my understanding of this. I understand that you can have an eigenstate of a system with an angular momentum magnitude value and a value for one component of the angular momentum (z). Using the lowering and raising operators we can create states (or deduce that states exist) with the same angular momentum magnitude value and the value for one component of the angular momentum increased or decreased by one unit.

We can keep doing this until the value for the component reaches a maximum or minimum value that is within the bounds of its square not exceeding the square of the magnitude of the whole angular momentum.

So we have deduced that there definitely exist states with same L quantum number, but with M values that run from L to -L in steps of one. Using this fact we can deduce L is either a positive half-integer or integer.

But how does this let us know that states can ONLY have an M-value that runs in steps of one from L to -L? We only deduced that states with M from L to -L in steps of one exist, but the M values might not be exclusive to those. What if we created some other mythical operator that could act on an eigenstate which created another eigenstate with the same L value, but M increased by a half?

For orbital angular momentum the argument is clear. In order for a full rotation to a return a system to itself, the M value (and therefore L) must be limited to integer values. Hence it runs from L to -L in integer steps and in this case, those are the only M-value states that can exist.

However L values that are half-integers are mathematically consistent, this is spin and full rotation bringing the same state isn't a necessity. So we know M, in this case, can have values from L to -L in steps of one, but there is nothing telling us that M is exclusive to those values?

We can keep doing this until the value for the component reaches a maximum or minimum value that is within the bounds of its square not exceeding the square of the magnitude of the whole angular momentum.

So we have deduced that there definitely exist states with same L quantum number, but with M values that run from L to -L in steps of one. Using this fact we can deduce L is either a positive half-integer or integer.

But how does this let us know that states can ONLY have an M-value that runs in steps of one from L to -L? We only deduced that states with M from L to -L in steps of one exist, but the M values might not be exclusive to those. What if we created some other mythical operator that could act on an eigenstate which created another eigenstate with the same L value, but M increased by a half?

For orbital angular momentum the argument is clear. In order for a full rotation to a return a system to itself, the M value (and therefore L) must be limited to integer values. Hence it runs from L to -L in integer steps and in this case, those are the only M-value states that can exist.

However L values that are half-integers are mathematically consistent, this is spin and full rotation bringing the same state isn't a necessity. So we know M, in this case, can have values from L to -L in steps of one, but there is nothing telling us that M is exclusive to those values?

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