structural:scholar_edition, stylistic:style_transfer
Schrödinger equation

An example of an eigenvalue equation where the transformation T {\displaystyle T} is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:

where H {\displaystyle H}, the Hamiltonian, is a second-order differential operator and ψ E {\displaystyle \psi _{E}}, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue E {\displaystyle E}, interpreted as its energy.

However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for ψ E {\displaystyle \psi _{E}} within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which ψ E {\displaystyle \psi _{E}} and H {\displaystyle H} can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.

The bra–ket notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by | Ψ E ⟩ {\displaystyle |\Psi _{E}\rangle }. In this notation, the Schrödinger equation is:

where | Ψ E ⟩ {\displaystyle |\Psi _{E}\rangle } is an eigenstate of H {\displaystyle H} and E {\displaystyle E} represents the eigenvalue. H {\displaystyle H} is an observable self-adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above H | Ψ E ⟩ {\displaystyle H|\Psi _{E}\rangle } is understood to be the vector obtained by application of the transformation H {\displaystyle H} to | Ψ E ⟩ {\displaystyle |\Psi _{E}\rangle }.