Eigenfunktionsdarstellung

Eigenfunctions are a fundamental concept in quantum mechanics, describing the possible states of a system in terms of wave functions. In mathematical terms, an eigenfunction is a function that satisfies a linear homogeneous equation, known as the eigenvalue equation. This equation is typically written as AX = λX, where A is a matrix, λ is a scalar (the eigenvalue), and X is the eigenfunction.

In the context of quantum mechanics, the eigenfunctions of a system are the possible wave functions that can describe the system's state. These wave functions are wave-like solutions to the Schrödinger equation. The eigenvalues of a system, which are the scalar values obtained by solving for λ, have a physical meaning, such as the energy levels of the system.

The eigenfunctions of a quantum mechanical system are orthogonal to each other, which means their integral over all space is zero. This property is crucial for the application of the spectral theorem, a mathematical tool for diagonalizing matrices. The resulting eigenfunctions form a complete set of orthonormal vectors, which means they can be used as a basis for the expansion of arbitrary wave functions.

The spectral theorem also implies that any wave function in a quantum system can be expressed as a linear combination of the eigenfunctions of the system. This is known as the superposition principle, a fundamental concept in quantum mechanics. The eigenfunctions of a system, together with their corresponding eigenvalues, provide a complete description of the system's properties and behavior.