Properties of Eigenfunctions An eigenfunction, associated with a specific eigenvalue of a physical system, provides a detailed description of how the system...
An eigenfunction, associated with a specific eigenvalue of a physical system, provides a detailed description of how the system behaves when it is subjected to an external potential. It can be used to calculate various physical properties of the system, such as the ground state energy, excitation energy, and other eigenvalues and eigenfunctions of the system.
Key properties of eigenfunctions include:
Normalization: Eigenfunctions must be normalized, meaning their integral over the entire physical system must equal 1. This ensures that the probability of finding the system in a specific energy state is equal to the probability of finding it in any other state.
Orthogonality: Eigenfunctions are orthogonal, meaning their inner product (or dot product) is equal to 0 when they are not related. This property allows us to use the principle of superposition to understand the behavior of the system when multiple eigenfunctions are involved.
Energy conservation: An eigenfunction associated with a specific eigenvalue represents a stationary state of the system. The energy of this eigenfunction remains constant over time, regardless of the external potential.
Physical interpretation: Eigenfunctions provide crucial information about the physical properties of the system. For instance, the ground state energy of an electron in a hydrogen atom is directly related to the energy of the electron when it is bound to the nucleus.
Examples of eigenfunctions:
Time independent harmonic oscillator eigenfunction: This function describes the energy levels and eigenfunctions of a simple harmonic oscillator.
Free particle eigenfunction: This function describes the energy levels and eigenfunctions of a free particle moving in a one-dimensional potential.
Quantum harmonic oscillator eigenfunction: This function describes the energy levels and eigenfunctions of a quantum harmonic oscillator.
These examples illustrate the various properties of eigenfunctions and their significance in understanding the behavior of physical systems