Required math: calculus
Required physics: Schrödinger equation
References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 3.19.
As another example of the energy-time uncertainty relation, we can look again at the example of a travelling free particle with a Gaussian wave packet. We have already worked out most of what we need to test the uncertainty relation:
From this we get
We still need . From our previous calculations, we have the wave function:
We can calculate by direct integration, using Maple:
As a check on this result, we can work out the units (always a good test to make sure you haven’t dropped a factor somewhere). From the original wave function, since exponents must be dimensionless, we know that has dimensions and has . Planck’s constant has dimensions of , so the expression above has overall units of . (Recall kinetic energy is .) It’s also worth noting that is independent of time.
From here, we can get :
We also have, from above
We’re trying to show that from above. So we want
The minimum of the LHS occurs at so if the inequality is true there, it is true always. In this case, it reduces to
This final condition is certainly true (it is required for the Gaussian wave form to converge at large ), so the uncertainty condition is verified.