What is a degenerate solution

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Area of ​​Expertise - Math, quantum physics

In general, in mathematics, degeneracy is understood to be a special borderline case of a different kind of mathematical object. For example, the point is both a degenerate line segment (length 0) and a degenerate square (side length 0), as well as a degenerate cube (edge ​​length 0).

In equation theory, degeneracy is understood to be the multiplicity of a solution. The number of solutions to one and the same value is then called the degree of degeneracy. For example, the equation

exactly two solutions that degenerate into one solution in the borderline case (i.e. the solution is twofold degenerate)

Analogously, the multiplicity of the solutions to an eigenvalue problem for the same eigenvalue is called degeneracy, i.e. there is an eigenvalue n linearly independent eigenvectors, then the eigenvalue n-fold degenerate. Every linear combination of the eigenvectors for this eigenvalue is then also the solution of the eigenvalue equation for this eigenvalue.

In quantum theory, for example, the eigenfunctions of the Schrödinger equation provide the same for the hydrogen atom n the same energy intrinsic value. The solutions are thus degenerate.

Learning units in which the term is dealt with

Eigenvalues ​​of a matrix45 min.

mathematicsLinear AlgebraMatrix eigenvalue problems

What are the eigenvalues ​​of a matrix and their determination are dealt with in detail (eigenvalue problem of a matrix).