Eigen functions & eigen vectors

[Amith2006]

In Quantum mechanics, we frequently deal with eigen value equations. When we speak of eigen value equations, we come across terms like eigen values,eigen vectors,eigen functions etc. When an operator is operated on certain quantities we get the same quantity multiplied by a constant. These quantities are interchangeably referred as eigen vectors and eigen functions. But do they mean the same? Is it something like, we call it as eigenfunctions in Schrodinger formalism and eigenvectors or eigenkets in Heisenberg formalism or is there a ma thematical difference between the 2?

 

[Fredrik]

They are the same. It would be weird to use "eigenket" when you're not using bra-ket notation, and it would be weird to use "eigenfunction" if you're talking about a vector that isn't actually a function, but other than that they're the same. The term "eigenstate" is also used interchangeably with the others. That's actually a little bit weird since a "state" is represented by an equivalence class of vectors. (Two vectors are equivalent if one of them is a complex number times the other).

 

[Entropia]

Eigen functions and eigen vectors are closely related concepts in quantum mechanics. They both refer to the properties of an operator acting on a particular state or system. However, there is a slight difference between the two terms.

Eigen functions are mathematical functions that represent the state of a system in quantum mechanics. They are solutions to the eigenvalue equation, which is a mathematical expression that describes the relationship between the operator, the state, and the eigenvalue. In other words, eigen functions are the possible states of a system that satisfy the eigenvalue equation.

On the other hand, eigen vectors are mathematical objects that represent the state of a system in a particular basis. They are solutions to the eigenvalue equation, just like eigen functions, but they are written in terms of a basis set. In quantum mechanics, the basis set is often referred to as eigenkets.

So, while eigen functions can be seen as the general form of a state, eigen vectors are specific representations of that state in a particular basis. In other words, eigen functions are more abstract and general, while eigen vectors are more concrete and specific.

However, there is a mathematical equivalence between eigen functions and eigen vectors. In fact, the eigen functions can be expanded in terms of eigen vectors, and vice versa. This means that the two terms are essentially interchangeable, and the choice of which one to use may depend on the context or the formalism being used.

In summary, eigen functions and eigen vectors are closely related concepts in quantum mechanics, but they have slight differences in their mathematical representations. Both terms refer to the properties of an operator acting on a state, and they can be used interchangeably in most cases.