Question

Problem 1 Consider the expectation value of the momentum of a particle to follow from the clas- sical relation da Using the expectation value of position of the particle dt Jo derive the expression for the momentum operator in the position representation, i.e -ih

Answer 1

The explanation value of the position operator is given by < x > = integral^infinity_-infinity dx psi^* (x, t) x psi(x, t) the rate of change of position operator is given by d/dt < x > = d/dt integral^infinity_-infinity dx psi^* (x, t) x psi(x, t) = integral^infinity_-infinity dx d/dt psi^* (x, t) x psi(x, t) + integral^infinity_-infinity dx psi^* (x, t) d/dt x psi(x, t) + integral^infinity_-infinity dx psi^* (x, t) x d/dt psi(x, t) Here in Schrodinger picture, operators do not change with time, hence dx/dt = 0 From the Schrodinger equation -h^2/2m partial differential^2 psi/partial differential x^2 = ih partial differential psi/partial differential t implies -h^2/2m partial differential^2 psi^*/partial differential x^2 = -ih partial differential psi^*/partial differential t The previous result becomes d/dt < x > = -integral^infinity_-infinity x dx ih/2m partial differential^2 psi^*/partial differential x^2 psi(x, t) + integral^infinity_-infinity x dx ih/2m partial differential^2 psi/partial differential x^2 psi^* (x, t) d/dx < x > = integral^infinity_-infinity x dx ih/2m (partial differential^2 psi/partial differential x^2