Operators and commutation of operators

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    Commutation Operators
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Discussion Overview

The discussion revolves around the quantization of angular momentum components in quantum mechanics, focusing on why only one component is often considered quantized and the implications of non-commutation among the components. The scope includes theoretical aspects and mathematical reasoning related to quantum mechanics.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions why only one component of angular momentum is quantized and what determines this choice.
  • Another participant asserts that all components of angular momentum are quantized, but the z component is preferred for mathematical simplicity due to isotropy in space.
  • It is noted that different components of angular momentum do not commute, which implies they cannot be quantized simultaneously.
  • A later reply clarifies that while components do not commute, all three can be quantized simultaneously, suggesting a distinction between simultaneous quantization and diagonalization.

Areas of Agreement / Disagreement

Participants express differing views on the quantization of angular momentum components, particularly regarding the implications of non-commutation and the choice of the z component. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

The discussion highlights the complexity of quantization in quantum mechanics, particularly the dependence on the mathematical treatment of angular momentum components and the implications of their commutation relations.

solas99
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why is only one component of angular momentum is quantised, and what determines which component is quantised?
 
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All components of Angular momentum are quantized but it is much easier to work with only the z component. And since space is isotropic, it does not really matter which direction you use. However, z component is simpler to work with mathematically.
 
Different components of angular momentum do not commute, so they cannot be quantized at the same time. Of all the components, z component is the easiest to deal with.
 
karlzr said:
Different components of angular momentum do not commute, so they cannot be quantized at the same time. Of all the components, z component is the easiest to deal with.
Do you mean can not be simultaneously diagonalized? All three are quantized simultaneously.
 

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