Conditional Values in Quantum Mechanics - Leon Cohen (Entropy, 2024)
We consider the local value of an operator for a given position or momentum and, more generally on the value of another arbitrary observable. We develop a general approach that is based on breaking up <semantics>Aψ(x)</semantics> as <semantics><mstyle scriptlevel="0" displaystyle="true">Aψ(x)ψ(x)</mstyle>=<mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true">Aψ(x)ψ(x)</mstyle></mfenced>R<mspace width="4pt"></mspace>+<mspace width="4pt"></mspace>i<mfenced separators="" open="(" close=")"><mstyle scriptlevel="0" displaystyle="true">Aψ(x)ψ(x)</mstyle></mfenced>I</semantics> where <semantics>A</semantics> is the operator whose local value we seek and <semantics>ψ(x)</semantics> is the position wave function. We show that the real part is related to the conditional value for a given position and the imaginary part is related to the standard deviation of the conditional value. We show that the uncertainty of an operator can be expressed in two parts that depend on the real and imaginary parts. In the case of the position representation, the expression for the uncertainty of an operator shows that there are two fundamental contributions, one due to the amplitude of the wave function and the other due to the phase. We obtain the equation of motion for the conditional values, and in particular, we generalize the Ehrenfest theorem by deriving a local version of the theorem. We give a number of examples, including the local value of momentum, kinetic energy, and Hamiltonian. We also discuss other approaches for obtaining a conditional value in quantum mechanics including using quasi-probability distributions and the characteristic function approach, among others.