In quantum mechanics, determinate operations, also known as projective measurements, are a type of measurement that yields definite and unambiguous results. These measurements are associated with observables that have a complete set of eigenstates, meaning that the observable's values are determined by the eigenstates.
A common example of a determinate operation in quantum mechanics is the measurement of spin. Consider a spin-1/2 particle, such as an electron, which has two possible spin states: "spin up" and "spin down." The observable associated with spin in the z-direction is the z-component of spin, denoted as Sz.
Performing a measurement of the spin in the z-direction corresponds to a determinate operation. If the measurement yields the outcome "spin up," then the particle is found to have a spin projection of +ħ/2 in the z-direction. If the outcome is "spin down," the spin projection is -ħ/2.
Upon measurement, the quantum state of the system "collapses" into the corresponding eigenstate of the observable being measured. This collapse is a key feature of determinate operations and distinguishes them from general measurements that can result in a superposition of states.
It's important to note that not all measurements in quantum mechanics are determinate operations. Other types of measurements, such as measurements of non-commuting observables like position and momentum, give rise to probabilistic outcomes and do not yield definite results.