Required math: calculus
Required physics: Schrödinger equation
References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 3.40.
Since the Hamiltonian for a free particle is , the Schrodinger equation in momentum space is
so the solution can be found by simply integrating with respect to :
We looked at the travelling Gaussian wave packet in free space earlier. Its initial state in position space is
To find we use the conversion to momentum space we found earlier:
From the analysis of the travelling Gaussian packet we see that the integral is the same as that done when calculating if we replace with . Therefore
Using 2, we have the full solution for :
which is independent of time. (As a check, we can integrate this over all and verify that this integral is 1.)
We can calculate the means for momentum in the usual way:
Both results agree with those in the analysis of the travelling Gaussian packet.
For the mean energy, we have
Referring back to the stationary Gaussian wave packet in free space, we see that , so the energy is the sum of that for a stationary Gaussian wave packet and the term . For the travelling packet, there is a net non-zero average momentum, so is non-zero. Thus the energy arises from the inherent energy of the wave packet, plus the kinetic energy of motion of the packet.