Dealing with unmeasurable observables in quantum mechanics is a challenging aspect of the theory. Quantum mechanics describes physical systems in terms of mathematical objects called operators, which represent observables such as position, momentum, energy, and spin. However, not all observables in quantum mechanics can be measured directly or precisely.
In quantum mechanics, the act of measurement is described by the process of applying a measurement operator to the quantum state of a system. The measurement operator corresponds to the observable being measured. When a measurement is performed, the system's state "collapses" into one of the eigenstates of the observable, and the measured value corresponds to the eigenvalue associated with that eigenstate.
However, not all observables have a complete set of eigenstates. Some observables, known as unmeasurable observables or incompatible observables, do not commute with each other. This means that their corresponding operators do not commute, and as a result, there is no common set of eigenstates. This implies that it is not possible to measure these observables simultaneously with arbitrary precision.
For example, position and momentum are incompatible observables in quantum mechanics. The corresponding position and momentum operators, denoted as X and P, do not commute: [X, P] = iħ, where ħ is the reduced Planck constant. As a consequence, there is no well-defined simultaneous measurement of position and momentum.
In dealing with unmeasurable observables, one can focus on measuring compatible observables that commute with each other. For example, in the case of spin, one can measure the spin along different axes (e.g., the x, y, and z directions) separately. However, it is important to note that measuring one observable can affect the outcome of subsequent measurements of incompatible observables due to the collapse of the quantum state.
Another approach to dealing with unmeasurable observables is through the framework of quantum information theory. This field explores the manipulation and transmission of information encoded in quantum systems. In quantum information, one can employ techniques such as quantum tomography and quantum state estimation to indirectly infer properties of unmeasurable observables by making measurements on related observables.
Overall, dealing with unmeasurable observables in quantum mechanics requires careful consideration and often involves indirect methods, compatible measurements, or the use of quantum information techniques to gain insights into the properties of these observables.