Final answer:
It is possible to measure the energy of a particle even if the wave function is not an eigenfunction of the Hamiltonian, by calculating the expectation value, which gives a probability distribution of possible energy values.
Step-by-step explanation:
It is possible to measure the energy of a particle even if the wave function psi is not an eigenfunction of the Hamiltonian. However, rather than receiving a single energy value, what you get is an expectation value of the energy by performing a quantum mechanical calculation involving the wave function psi.
The expectation value of an observable, like energy, in quantum mechanics can be calculated by integrating the product of the conjugate transpose of the wave function, the operator corresponding to the observable (in this case, the Hamiltonian operator for energy), and the wave function itself over all space. This process is outlined in the Schrödinger's time-independent equation and requires the wave function to be normalized.
As per the Copenhagen interpretation, the position of a particle described by a wave function psi only becomes definite upon measurement, which causes the wave function to collapse, producing a probability distribution for the particle's location. Similarly, measurements of energy yield a distribution of possible values with certain probabilities.
The Heisenberg uncertainty principle also plays a vital role in our understanding of these measurements, as it implies that certain pairs of variables, such as position and momentum, cannot both be known to arbitrary precision simultaneously. At the quantum level, the act of measuring can significantly affect the system, altering its subsequent evolution.