Why is the total number of quantum states = 2n^2 for some n?

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Discussion Overview

The discussion revolves around the calculation of the total number of quantum states, specifically questioning why it equals 2n² for some integer n. Participants explore the relationships between quantum numbers L and m, as well as the implications of spin directions in this context.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that the total number of states should be calculated as 2*(number of possible L)*(number of possible m), leading to confusion over the resulting expression of 4n² - 2n.
  • Another participant clarifies that each value of l corresponds to a different number of possible values of m, indicating that a simple multiplication of the counts is not valid.
  • A different participant points out that the values of l range from 0 to n-1, implying that the number of possible values of l is n but does not include n itself.
  • One participant acknowledges the clarification regarding the relationship between l and m, expressing gratitude for the insight.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the calculation method, and multiple competing views regarding the relationships between quantum numbers remain evident.

Contextual Notes

The discussion highlights potential misunderstandings in the relationships between quantum numbers and their respective counts, but does not resolve these issues or clarify all assumptions involved.

jerronimo3000
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If the number of possible values of L is n, and the number of possible values of m is 2*L-1, and there are 2 spin directions.. shouldn't the total number of states be 2*(number of possible L)*(Number of possible m)? But this gives 4n^2 - 2n. I am extremely confused. Thanks for your help!
 
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Each (different) value of l has a different number of possible of values of m, so you can't simply multiply (number of possible l)*(number of possible m).
 
Looks to me like the values of l range from 0 to n-1 so no l value equals n.
The number of possible values of l is n, but the values of l don't equal n.
 
jtbell said:
Each (different) value of l has a different number of possible of values of m, so you can't simply multiply (number of possible l)*(number of possible m).

Oh, duh..thanks, definitely clears that up! Thank you!
 

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