Before a measurement is made, a quantum particle is in a superposition of two energy eigenstates. According to the postulates, what determines the possible outcomes when energy is measured?
AThe average (expectation) value of the energy operator.
BThe eigenvalues of the Hamiltonian operator corresponding to those eigenstates.
CThe amplitude of the highest-energy eigenstate in the superposition.
DThe energy of the particle at the moment just before measurement.
Postulate 3 states that the possible outcomes of measuring an observable are the eigenvalues of the corresponding Hermitian operator. For energy, that operator is the Hamiltonian H. The actual outcome of any single measurement will be one of these eigenvalues; the probabilities of each are given by Born's rule (the squared moduli of the expansion coefficients). The expectation value is the average over many measurements, not a possible single outcome.
Question 2 True / False
Before measurement, a quantum particle in superposition is 'really' in one definite eigenstate — we just don't know which one. The measurement merely reveals a pre-existing value.
TTrue
FFalse
Answer: False
This is the hidden-variable interpretation, which contradicts the standard (Copenhagen) reading of the postulates. According to the postulates, the state vector IS the complete description of the system — there is no hidden underlying state. Bell's theorem and experiments (Aspect et al.) rule out local hidden-variable theories. The superposition is physically real, not merely a statement of ignorance. Measurement causes a genuine change (collapse) to an eigenstate, not a revelation of a pre-existing value.
Question 3 Short Answer
Why must quantum mechanical observables correspond to Hermitian operators rather than arbitrary linear operators?
Think about your answer, then reveal below.
Model answer: Hermitian operators have real eigenvalues, which is required because measurement outcomes must be real numbers (we can't measure an imaginary energy). Hermitian operators also have orthogonal eigenstates, which allows any state to be expanded uniquely in terms of them. Both properties are essential for the postulates to be physically consistent.
A non-Hermitian operator could have complex eigenvalues, which would correspond to unmeasurable complex-valued physical quantities. The mathematical requirement of Hermiticity is exactly what ensures the physical requirement of real-valued measurement outcomes. Hermitian operators also guarantee a complete orthonormal basis of eigenstates, enabling Born rule calculations.