Free Particle: Prove Constant in Time

[renegade05]

Homework Statement

A free particle moving in one dimension is in the initial state ψ(x,0). Prove that
<p> is constant in time by direct calculation (i.e., without recourse to the
commutator theorem regarding constants of the motion).

Homework Equations

<p> = m*(d<x>/dt) ?
S.E.
ψ(x,0)= 1/(√(2∏))*∫Θ(k)e^(ikx)dk

The Attempt at a Solution

I am not really sure where to start... Should I find <x> ? is that the easiest way? And how do you this for a free particle ? stuff is confusing me. Can someone give me a start here...

 

[vela]

Homework Statement

A free particle moving in one dimension is in the initial state ψ(x,0). Prove that
<p> is constant in time by direct calculation (i.e., without recourse to the
commutator theorem regarding constants of the motion).

Homework Equations

<p> = m*(d<x>/dt) ?

No, don't use this. Use the definition of the expectation value of an operator.

S.E.
ψ(x,0)= 1/(√(2∏))*∫Θ(k)e^(ikx)dk

The Attempt at a Solution

I am not really sure where to start... Should I find <x> ? is that the easiest way? And how do you this for a free particle ? stuff is confusing me. Can someone give me a start here...

What is the Hamiltonian for a free particle?