Rick88 said:because if you use <p> = m d<x>/dt, you would determine the expectation value of the momentum via that of x.
But <p> is given only by ∫u p u* dx - it's a definition.
Expectation values represent the average value of a physical quantity, such as position or momentum, in a quantum state. They are essential in predicting the behavior of particles in quantum systems.
To calculate the expectation value of a physical quantity, such as position or momentum, for a wavefunction, the wavefunction must first be squared to find the probability distribution. Then, the integral of the product of the physical quantity and the squared wavefunction is taken over all possible values of the variable.
The expectation value of position represents the average position of a particle in space, while the expectation value of momentum represents the average momentum of a particle. They are related through the Heisenberg uncertainty principle, where the product of their uncertainties cannot be less than a certain value.
In quantum mechanics, the expectation values of observables, such as position and momentum, can change over time due to the time evolution of the wavefunction. This change is governed by the Schrödinger equation.
The uncertainty principle states that the product of the uncertainties in position and momentum must be greater than or equal to a certain value. This is directly related to the concept of expectation values, as the uncertainty in a physical quantity is inversely proportional to its expectation value. This means that the more precisely we know the expectation value of a quantity, the less certain we are about its exact value.