Final answer:
The Hamiltonian operator can be applied to any wave function; it does not need to initially be an eigenfunction of the Hamiltonian. The act of measurement is what causes the system to 'collapse' into an eigenstate of the Hamiltonian associated with the observed energy.
Step-by-step explanation:
The Hamiltonian operator is integral to the field of quantum mechanics, serving a vital role analogous to Newton's second law of motion in classical mechanics. When we discuss a wave function, we are referring to a mathematical entity that encapsulates the quantum state of a system, including all information about the system's energy and spatial distribution of probabilities. Particularly, when the Hamiltonian operator is applied to a wave function, we are effectively working towards extracting the total energy of the system. This relationship is conventionally written as Ĥψ = Eψ, where Ĥ stands for the Hamiltonian operator, E for the energy eigenvalue, and ψ for the wave function.
Addressing the student's query, the wave function to which the Hamiltonian operator is applied does not necessarily have to be an eigenfunction of the Hamiltonian. Indeed, it can be any wave function that represents the system in question. Should the wave function happen to be an eigenfunction of the operator, the result after application will be that same eigenfunction multiplied by a constant, which represents the eigenvalue or the measurable quantity, in this case, the energy. However, any wave function can be expressed as a linear combination of eigenfunctions due to the principle of superposition. When an operator acts on this linear combination, the result is the linear combination of the eigenfunctions multiplied by their corresponding eigenvalues, which can subsequently be measured.
In quantum mechanics, once a measurement is performed, the system 'collapses' into a state described by an eigenfunction associated with the observed eigenvalue, in this case, the total energy. Thus, a wave function needs not to be an eigenfunction of the Hamiltonian initially; it is the act of measurement that forces it into one. Solving the Schrödinger equation — a core component of quantum mechanics — yields these eigenfunctions and associated eigenvalues for a given system, entirely analogous to solving differential equations in classical mechanics to find the motion of objects.