͟d͟ ͟\<͟p͟\>͟ = - 〈͟d͟ ͟V͟(͟X͟)͟〉
  dt           dx

  If a particle is subject to potential 〈V(x)〉

The potential is known, and in the absense of any non-conservative influences, the Hamiltonian is equal to the potential.

H(x) = V(x) 

H|E〉 = E|E〉

Ĥ = ͟p̂͟²͟ + V(x̂)
     2m

x̂ ≐ x

p̂ ≐ -ι ħ  ͟d͟
          dx

p̂² ≐ -ħ²  ͟d͟²
          dx²

Ĥ = ͟p̂͟²͟ + V(x̂)
     2m


      ∞
〈p̂〉 = ∫ dx p(x) = 
     -∞

d/dt ∫ dx p(x) = 
      


      
Probably start here:

〈V(x)〉 = ∫ dx V(x) 

〈d/dx V(x)〉 = ∫ d/dx V(x) dx = V(x)

d/dt 〈p〉 = d/dt ∫ dx p(x) 

the definition of momentum in function space is 

    d/dt p(x) = -d/dx V(x)