In Linear Algebra and Analysis the term spectrum refers to the set of eigenvalues of an operator. A spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces.
An example of spectral theorem is the statement that every symmetric matrix is equivalent to a diagonal matrix via a orthogonal transformation:
$$ S=ODO^t $$Another one is that every Hermitian matrix is unitarily equivalent to a diagonal one (that is, $H=UDU^{\dagger}$, with $U$ an unitary matrix and the dagger is the conjugate transpose).
The first example is a particular case of the latter.
The generalization of these statements to operators in Hilbert space is called spectral theory.
It is related to the emission spectrum of atoms. In quantum mechanics, the emission and absorption frequencies of an atom are indeed related to eigenvalues of observables. Specifically, the energy levels of an atom's electrons can be represented by the eigenvalues of the Hamiltonian operator, which describes the total energy of the system.
When an electron makes a transition from a higher energy level to a lower energy level, it emits a photon with an energy equal to the difference in the energy levels. This energy is related to the frequency of the emitted light by the formula E = hf, where h is Planck's constant and f is the frequency of the light.
Thus, the emission frequencies of an atom can be represented as the differences between the eigenvalues of the Hamiltonian operator, and the eigenvalues themselves can be thought of as the energies of the energy levels. This connection between the emission frequencies of an atom and the eigenvalues of an observable in quantum mechanics is a key aspect of the connection between spectroscopy and quantum mechanics.
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Author of the notes: Antonio J. Pan-Collantes
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