Verifying Eigenfunctions for Operators

Verify whether a given function is an eigenfunction for a specified operator by calculating derivatives and checking if the function returns in the form $a \psi$.

In quantum mechanics, operators play a crucial role in defining the physical properties of quantum systems, and understanding how functions interact with these operators is critical. An eigenfunction of an operator is a function that returns the same function, up to a multiplicative constant, when the operator is applied to it. This constant is known as the eigenvalue, and it represents one of the measurable values of the property associated with the operator. In this problem, you are asked to verify whether a given function is an eigenfunction of a specified operator.

To approach this, the first step is to calculate the action of the operator on the function, which often involves taking a derivative, depending on the operator. This involves applying the operator to the function and simplifying the expression. If the resulting expression can be written as the product of a constant (the eigenvalue) and the original function, it confirms that the function is indeed an eigenfunction of the operator.

Understanding this process is essential as it helps in solving the Schrödinger equation, where finding eigenfunctions and eigenvalues corresponds to finding the states a quantum system can occupy and their associated energies. This concept is foundational as it ties into many areas of quantum mechanics, including quantum states, wavefunctions, and the measurement process.