What the differ between eigenstate and eigenfunction ?

[eman2009]

what the differ between eigenstate and eigenfunction ?

 

[StatusX]

They're both types of eigenvectors. Recall that an eigenvector is an element of a vector space V, which is associated to a linear operator A on that vector space (ie, we say it is an eigenvector of that operator), with the property that Av=av, where a is a constant.

An eigenstate is a vector in the Hilbert space of a system, things we usually write like |$\psi$>. An eigenfunction is an element of the space of functions on some space, which forms a vector space since you can add functions (pointwise) and multiply them by constants. Specifically, you're probably talking about wavefunctions, and operators like x and -ih d/dx.

In the case when the system has no spin degrees of freedom, this wavefunction is just a particular representation of the state, and so eigenfunctions and eigenstates are basically the same thing. So, in most cases you'd be fine not to distinguish them. If there is spin, the full state consists of a wavefunction together with the spin state. Thus it is possible the state is an eigenstate of some operator without the underlying wavefunction being an eigenfunction (indeed, the operator might not even be something that can act on the space of functions, like a spin matrix).

 

[Entropia]

Eigenstate and eigenfunction are two concepts that are closely related and often used interchangeably, but there are some key differences between them.

An eigenstate is a state of a physical system that is described by a specific set of quantum numbers, such as energy or spin. It is a state in which the system will remain if left undisturbed, and it represents a stable, stationary state of the system. Eigenstates are also known as stationary states or energy eigenstates.

On the other hand, an eigenfunction is a mathematical function that describes the behavior of a system in an eigenstate. It is the mathematical representation of the physical state of the system. Eigenfunctions are often used to solve the Schrödinger equation, which describes the behavior of quantum systems.

In summary, eigenstate refers to the physical state of a system, while eigenfunction refers to the mathematical description of that state. Eigenstates are represented by eigenfunctions, but not all eigenfunctions represent eigenstates. Eigenfunctions can also represent non-stationary states, which are not eigenstates of the system. Therefore, it is important to distinguish between these two concepts when discussing quantum systems.