eigenfunctions

(noun)

A class of mathematical function that is non-zero and returns the same function except for a multiplicative scaling factor when acted on by a linear operator (such as a Hamiltonian).

Related Terms

  • Hamiltonian
  • quantum number
  • degenerate

Examples of eigenfunctions in the following topics:

  • A single electron in a central field

    • The functions $Y_{lm}$ are eigenfunctions of the angular momentum operator.
    • This means quantum-mechanically that the Hamiltonian commutes with the angular momentum operator, and that the wavefunctions that satisfy the Hamiltonian also are eigenfunction of the angular momentum operator (${\bf L}={\bf r}\times {\bf p}$).We have
    • The angular eigenfunctions take this form regardless of the form of the central potential.
    • They are simply the eigenfunctions of the angular momentum operator.
    • Because the radial eigenfunctions for different values of $l$ satisfy different equations, there is no orthogonality relation for the radial wavefunctions with different $l$ values.
  • Problems

    • You can find the eigenfunctions of the Morse potential on Wikipedia.
  • Atomic Structure

    • We can imagine the operator $H$ as a matrix that multiplies the state vector $\psi$, so this equation is an eigenvalue equation with $E$ as the eigenvalue and $\psi$ as an eigenvector (or eigenfunction) of the matrix (or operator) $H$.The Hamiltonian classically is the sum of the kinetic energy and the potential energy of the particles.
  • Description of the Hydrogen Atom

    • Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely, generally from this isotropy of the underlying potential.
  • Wave Equation for the Hydrogen Atom

    • Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows generally from this isotropy of the underlying potential.
  • B.10 Chapter 10

    • You can find the eigenfunctions of the Morse potential on Wikipedia.
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