"An eigenvector of a linear transformation is a non-zero vector that is either left unaffected or simply multiplied by a scale factor after the transformation (the former corresponds to a scale factor of 1).

The eigenvalue of a non-zero eigenvector is the scale factor by which it has been multiplied.

A number λ is an eigenvalue of a linear transformation T : V → V if there is a non-zero vector x such that T(x) = λx.

The eigenspace corresponding to a given eigenvalue of a linear transformation is the vector space of all eigenvectors with that eigenvalue.

The geometric multiplicity of an eigenvalue is the dimension of the associated eigenspace.

The spectrum of a transformation on a finite dimensional vector space is the set of all its eigenvalues. (In the infinite-dimensional case, the concept of spectrum is more subtle and depends on the topology on the vector space).

For instance, an eigenvector of a rotation in three dimensions is a vector located within the axis about which the rotation is performed. The corresponding eigenvalue is 1 and the corresponding eigenspace contains all the vectors along the axis. As this is a one-dimensional space, its geometric multiplicity is one. This is the only eigenvalue of the spectrum (of this rotation) that is a real number.

In order to find the energy levels and the corresponding energy eigenstates, we must solve the time-independent Schrödinger equation.

We can solve the differential equation in the coordinate basis, using a power series method. It turns out that there is a family of solutions."

The above are excerpted from Rapaire's Introduction to Quantum Mathematics, Rapaire Publications, Rapaire, ed., 2007, Rapaire, Idaho.

Further elucidation can be found in this Rapaire article.



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