6.4 How it Feels to be Free
Solution of the Schrödinger equation for a free particle.
Let’s start by imagining a system where the Schrödinger equation is simplified because there are no forces acting on the particle, so the potential energy, V(x), is everywhere zero. The particle is therefore free in space. The equation then becomes relatively simple:
The key feature here is that the second derivative of ψ is the same as ψ itself, except that it is multiplied by a constant. What kind of functions behave like that?
If you completed this question, then congratulations once again: you’ve now not just generated the Schrödinger equation, you’ve solved it too. The wave function you have deduced has no localisation: it is just an endless sinusoid, so it is equally likely that the particle could be found anywhere in its one-dimensional universe. Most importantly, you can see that k can take any value and the wave function will still be a valid solution to the equation. Looking at the energy expression you derived in Question 6-4, you can further see that this in turn means that any energy level is possible.
This is a hugely important result: it means that the Schrödinger equation does not, of itself, lead to quantised energy levels. Cormorant and Nefertiti are momentarily horror-struck: have we been on the wrong track all along? Erwin wraps his ghostly arm reassuringly around Nefertiti: “fear not, little one: this was just one special case. Now let’s move on to something more revealing”.
