- #1
buraqenigma
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How can i prove that [tex]u(x)=exp(-x^{2}/2)[/tex] is the eigenfunction of [tex]\hat{A} = \frac{d^{2}}{dx^{2}}-x^2 [/tex].(if i don't know the eigenfunction how can i find it from expression of A operator)
An eigenfunction is a mathematical function that, when acted upon by a certain operator, returns a scaled version of itself. It is a special type of solution to an operator equation.
An operator is a mathematical function that operates on other functions, transforming them in some way. Examples of operators include differentiation, integration, and multiplication by a constant.
An eigenfunction is a function that, when acted upon by an operator, returns a scaled version of itself. This scaling factor is called the eigenvalue, and it represents the "strength" of the function's response to the operator.
Eigenfunctions are used extensively in physics to describe the behavior of physical systems. In quantum mechanics, for example, the eigenfunctions of the Hamiltonian operator represent the possible states of a particle, and the corresponding eigenvalues represent the energy of the particle in each state. In this way, eigenfunctions help us understand the behavior of complex physical systems.
Eigenfunctions and eigenvectors are closely related, but they are not the same thing. An eigenvector is a vector that, when acted upon by a linear transformation, returns a scaled version of itself. An eigenfunction is simply a special type of solution to an operator equation. In some cases, an eigenfunction can be thought of as an eigenvector in an infinite-dimensional vector space, but this is not always the case.