Is there an official Eigenvalue Condition in Quantum Mechanics?

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Discussion Overview

The discussion revolves around the concept of the "Eigenvalue Condition" in quantum mechanics, with participants exploring its definition and relevance within the context of quantum physics and linear operators. The scope includes mathematical interpretations, potential applications in quantum mechanics, and the search for an official condition related to eigenvalues.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on what is meant by the "Eigenvalue Condition" and asks for insights from others.
  • Another participant explains that eigenvalues in physics relate to the outputs of quantum mechanical operators, suggesting a mathematical equivalence to operators in equations.
  • A mathematical definition is provided, stating that for a linear operator A, the eigenvalue equation is Ax = [lambda]x, where non-trivial solutions indicate eigenvalues.
  • It is noted that in quantum physics, measurements such as position and momentum are represented as linear operators, and their eigenvalues correspond to possible measurement outcomes.
  • One participant speculates that the inquiry might pertain to boundary value problems in quantum mechanics and questions the existence of an official Eigenvalue Condition.
  • Another participant suggests the operator/eigenvalue postulate of quantum mechanics and considers the possibility of an eigenvalue boundary condition.
  • A later reply mentions the determinant equation det(A-lambda I) = 0 as a method to find eigenvalues but clarifies that it does not specifically relate to quantum mechanics.

Areas of Agreement / Disagreement

Participants express differing interpretations of what the "Eigenvalue Condition" refers to, with no consensus on a specific definition or official condition existing in quantum mechanics.

Contextual Notes

There is ambiguity regarding the specific context of the inquiry about the Eigenvalue Condition, and participants highlight the potential relevance of boundary conditions in quantum mechanics without reaching a definitive conclusion.

eNtRopY
I was recently asked to explain the eigenvalue condition, but I'm sure exactly which condition the inquirer was asking about.

Are any of you nerds familiar with the Eigenvalue Condition?

If so, please enlighten me.

eNtRopY
 
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eigenvalues are (in physics) values that define the output of a quantum mechanical operator in an equation.

There is a mathematical meaning, and i think it is pretty much the same thing i.e. mathematical operators in equations.

what was the context of the problem?
 
Mathematics: If A is a linear operator on a vector space, the
"eigenvalue" equation is Ax= [lambda]x. x= 0 is a "trivial" solution. If there exist non-trivial (i.e. non zero) solutions [lambda] is an eigenvalue of A.

Physics: As "jonnylane" said, in quantum physics, various possible measurements (position, momentum) are interpreted as linear operators. The only possible specific numerical results of such measurements are eigenvalues of the linear operators. That may be what your inquirer was asking about.
 
I think that the inquirer was asking about something else. I think that he meant something specifically related to the boundary value of a QM problem. I was just wondering if there was an official Eigenvalue Condition. I see now that there is not.

eNtRopY
 
Perhaps the operator/eigenvalue postulate of quantum mechanics?

Most likely an eigenvalue boundary condition, though.
 
Originally posted by eNtRopY
I think that the inquirer was asking about something else. I think that he meant something specifically related to the boundary value of a QM problem. I was just wondering if there was an official Eigenvalue Condition. I see now that there is not.

The closest I can think of (as far as an "official" condition) would be the equation
det(A-lambda I) == 0.

Which is used to determine the eigenvalues (lambda) of the linear operator A.

However, just as that, it has nothing to do with QM.
 

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