What is an eigenform?

ChatGPT said:

An eigenform (or eigenstate in the context of quantum mechanics) refers to a special solution to an equation where the system remains unchanged except for a scaling factor (an eigenvalue). In the context of quantum mechanics, this idea is closely related to eigenfunctions and eigenvalues of an operator, particularly the Hamiltonian operator, which describes the energy of a system.

Let me explain it more clearly:

1. Eigenfunction (or Eigenstate)

In quantum mechanics, the wave function $\psi(\mathbf{r})$ describes the state of a system. When an operator (such as the Hamiltonian $\hat{H}$) acts on a wave function, it often changes the wave function. However, in certain cases, the wave function doesn't change its form, it only gets multiplied by a constant factor. In these cases, the wave function is called an eigenfunction (or eigenstate).

• An eigenfunction $\psi(\mathbf{r})$ satisfies the equation:

$$\hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r})$$

Where:

• $\hat{H}$ is the Hamiltonian operator,
• $\psi(\mathbf{r})$ is the wave function (or eigenfunction),
• $E$ is a constant called the eigenvalue, which typically represents energy in quantum systems.

2. Eigenvalue

The eigenvalue $E$ is the constant that the wave function is multiplied by when the operator (in this case, the Hamiltonian $\hat{H}$) acts on it. The eigenvalue often represents a physical quantity of the system. For example, in the case of the Hamiltonian, the eigenvalue corresponds to the energy of the system.

In simpler terms:

• The wave function $\psi(\mathbf{r})$ is said to be an eigenfunction of the Hamiltonian if or when the Hamiltonian acts on it, the result is just the wave function multiplied by a constant $E$ (the energy).
• The constant $E$ is the eigenvalue, which represents the energy associated with the eigenstate (wave function).

Example in Quantum Mechanics:

For a particle in a potential well, the time-independent Schrödinger equation:

$$\hat{H} \psi(\mathbf{r}) = E \psi(\mathbf{r})$$

describes the stationary states of the system. Each eigenfunction $\psi(\mathbf{r})$ corresponds to a possible state of the particle, and the eigenvalue $E$ gives the energy of the particle in that state.

Thus, an eigenform in quantum mechanics usually refers to these eigenfunctions (or eigenstates) that describe the possible states of a system, each with a corresponding eigenvalue representing some physical quantity (e.g., energy, momentum).