Why Does Electron Angular Momentum Not Align with External Field?

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Discussion Overview

The discussion centers on the behavior of an electron's angular momentum in the presence of an external magnetic field, specifically why the total angular momentum vector does not align with the field. It explores concepts related to intrinsic and orbital angular momentum, quantum mechanics, and the implications of angular momentum quantization.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that the total angular momentum vector, denoted as pj, precesses around the magnetic field rather than aligning with it, questioning the reasons behind this behavior.
  • Another participant explains that the eigenvalues of angular momentum operators prevent simultaneous alignment of angular momentum in the direction of the external field.
  • A contribution highlights that in quantum mechanics, only the magnitude and projection of angular momentum can be determined along one axis, due to the non-commuting nature of angular momentum operators.
  • A participant expresses understanding that the projection of the total angular momentum vector can only take specific magnitudes, which cannot equal the total angular momentum itself.
  • There is a query about whether this principle applies solely to orbital angular momentum or to total angular momentum, including both spin and orbital components.
  • Another participant confirms that the principle applies to spin, orbital, and total angular momentum for one or many electrons.

Areas of Agreement / Disagreement

Participants generally agree on the quantum mechanical principles governing angular momentum projections, but there remains some uncertainty regarding the specifics of how these principles apply to different types of angular momentum.

Contextual Notes

Some participants express a lack of familiarity with the mathematics involved, which may limit their understanding of the discussion. The conversation also reflects a dependence on quantum mechanical definitions and the implications of angular momentum quantization.

tasnim rahman
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An electron has intrinsic angular momentum(spin) and orbital angular momentum, which gives rise to the total angular momentum of the electron, let's call it pj. When the electron is placed in an external magnetic field, the pj vector precesses around the magnetic field in one of two states(with or against the field). My question is why doesn't the pj vector align itself parallel to the direction of the external field? Is it because the pj vector can only have particular components in the the direction of the axis of the field, and none of the allowed components are equal to the magnitude of pj?
 
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The eigenvalues of L2 are l(l+1), and the eigenvalues of Lz are -m...+m. So you can't simultaneously satisfy those conditions and have all of the angular momentum pointing in the z-direction.
 


In QM, you can only ever determine the magnitude and the projection of the angular momentum vector on one axis (such as an external magnetic field). These correspond to the quantum numbers L and m as Vanadium pointed out.

This is because the 3 operators L_x, L_y and L_z do not commute.
 


Thanks M Quack and Vanadium 50. But I am sorry, I am not really familiar with the mathematics involved. But what I believe I understood, is that the projection of the total angular momentum vector on one axis(z-direction:here assumed to be the direction of the magnetic field) can have only particular magnitudes, as set by quantum mechanics. And that the magnitude of the projections, can not be equal to the magnitude of the total angular momentum vector. Right?
 
Last edited:


Quick help anyone?:bugeye:
 


What you say sounds correct to me.
 


M Quack. Thank you very much for the verification.:biggrin:
 


tasnim rahman said:
Thanks M Quack and Vanadium 50. But I am sorry, I am not really familiar with the mathematics involved. But what I believe I understood, is that the projection of the total angular momentum vector on one axis(z-direction:here assumed to be the direction of the magnetic field) can have only particular magnitudes, as set by quantum mechanics. And that the magnitude of the projections, can not be equal to the magnitude of the total angular momentum vector. Right?
Does this work only for the orbital angular momentum, or the total angular momentum (as in the angular momentum by vector addition of the orbital and spin momentum)?
 


All of them.

Spin, orbital, total. One electron or many.
 
  • #10


Thanks a lot M Quack.
 

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