Why the Integration Constant in the Schrödinger Equation is Set to L(L+1)?

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Discussion Overview

The discussion revolves around the integration constant in the angular wave equation of the Schrödinger equation, specifically why it is set to L(L+1). The context includes theoretical aspects of quantum mechanics, particularly in relation to angular momentum in systems like the hydrogen atom.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions the reason for setting the integration constant to L(L+1) without specifying the system in question.
  • Another participant suggests that in the context of the hydrogen atom, the total angular momentum being a constant of motion is relevant, and that the operator L^2 appears in the spectral equation.
  • A different participant recommends examining the solution details for the \theta part of the Schrödinger equation, particularly in relation to associated Legendre polynomials.
  • It is noted that the eigenvalues of the angular momentum operator L^2 are L(L+1).

Areas of Agreement / Disagreement

Participants express varying levels of understanding and focus on different aspects of the topic, indicating that multiple competing views remain without a consensus on the primary reason for the integration constant's value.

Contextual Notes

The discussion does not clarify the specific system being analyzed, which may affect the interpretation of the integration constant's significance. Additionally, the relationship between the angular momentum operator and the spectral equation is mentioned but not fully explored.

member 141513
in the Schrödinger equation the integration constant in the angular wave
equation is set to L(L+1).
may i know why this is set ,what is the reason , thx!
 
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Your question lacks naming the system for which the Schrödinger's equation is used.

If it was for the hydrogen atom, it may be worthy to know that the total angular momentum is a constant of motion and the H_{dummy particle} contains L^2 in its expression. When separating the spherical variables from the radial one, you're bumping into the spectral equation for L^2.
 
I think you need to look at the details of the solution for the [itex]\theta[/itex] part of the Schrödinger equation in terms of associated Legendre polynomials.
 
The eigenvalues of the angular momentum operator L^2 are L(L+1).
 

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