Is there an official Eigenvalue Condition in Quantum Mechanics?

In summary, the conversation discusses the concept of eigenvalues in both mathematics and physics. Specifically, the discussion touches on the definition and purpose of eigenvalues, as well as their role in boundary conditions in quantum mechanics. The closest thing to an official eigenvalue condition is the equation det(A-lambda I) == 0, which is used to determine the eigenvalues of a linear operator in mathematics. However, this does not directly relate to the eigenvalue condition in quantum mechanics.
  • #1
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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  • #2
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?
 
  • #3
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.
 
  • #4
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
 
  • #5
Perhaps the operator/eigenvalue postulate of quantum mechanics?

Most likely an eigenvalue boundary condition, though.
 
  • #6
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.
 

FAQ: Is there an official Eigenvalue Condition in Quantum Mechanics?

What is the Eigenvalue Condition?

The Eigenvalue Condition is a mathematical concept used in linear algebra to determine the eigenvalues of a matrix. It is a necessary condition that must be satisfied in order to find the eigenvalues of a matrix.

How is the Eigenvalue Condition used in practical applications?

The Eigenvalue Condition is used in various fields such as physics, engineering, and computer science to solve problems involving linear transformations and eigenvalues. It is also used in machine learning and data analysis to understand the behavior of data.

What are the implications of satisfying the Eigenvalue Condition?

If the Eigenvalue Condition is satisfied, it means that the matrix has a set of linearly independent eigenvectors and corresponding eigenvalues. This allows for the simplification of calculations and the understanding of the behavior of the matrix.

Can the Eigenvalue Condition be violated?

No, the Eigenvalue Condition cannot be violated. If the condition is not satisfied, it means that the matrix does not have a set of linearly independent eigenvectors and corresponding eigenvalues, making it difficult to analyze and work with.

How is the Eigenvalue Condition related to other concepts in linear algebra?

The Eigenvalue Condition is closely related to other concepts such as eigenvectors, eigenvalues, and diagonalization. It is also used in conjunction with other methods, such as the characteristic polynomial, to find the eigenvalues of a matrix.

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