phy-4600/solutions/exam1/prob2

Show that ͟d͟〈͟p͟〉͟ = -〳͟d͟V͟(͟x͟)͟〵 when a particle is subjected to a potential 〈V(x)〉.
           dt     〵dx   〳


The time derivative of the expectation value of the momentum is a known quantity, from
Time Dependence of Expectation Value of General Momentum Operator:

d〈p〉 = 1.
dt     ιħ

The problem is therefore reduced to finding whether -/dV(x)\ reduces to 1.
                                                     \dx   /            ιħ

-/dV(x)\ = -〈Ψ| d V(x) |Ψ〉. 
 \dx   /        dx        

 Viewing the expression in this form reveals a relationship between the space derivative and the operators V(x) and |Ψ〉. The chain rule allows this derivative to be computed.

-〈Ψ| d V(x) |Ψ〉 = -〈Ψ| ⎛d V(x)|Ψ> + d |Ψ> V(x)⎞. 
     dx                ⎝dx          dx        ⎠