Question

Consider the one-dimensional motion of a particle of mass ? in a space where uniform gravitational acceleration ? exists. Take the vertical axis ?. Using Heisenberg's equation of motion, find the position-dependent operators ? ? and momentum arithmetic operator ?? in Heisenberg display that depend on time. In addition, calculate ??, ?0, ??, ?0.

Answer 1

the Heisenberg equation of motion is

here the Schrodinger hamiltonian is

here it has no implicit time-dependence so the second term in the Heisenberg equation vanishes. to change from Schrodinger to Heisenberg hamiltonian we just need to replace the operators

now using Heisenberg equation

  

since [x, f(x)]=0

so our first equation is

     

now using the operator of momentum P

so the second equation is

  

differentiating (A) again wrt time and substituting (B) in it

integrating both sides two times

which is similar to our kinematic equation S= Ut + (1/2) a t^2

so at t=0 is the initial position

now we can put the value of x(t) in the equation (A) in order to obtain P(t)

so at t=0   

is the intial momentum