![]() ![]() However, the effect of an infinite V ( x ) is very easy to deal with. You may think that we’re no longer dealing with a free particle now that we’re subjecting it to an infinite potential energy. We get very similar physics by looking at a particle trapped in an infinite potential well of width L. The collection of all possible energies is known as the spectrum of the Hamiltonian. At some point, this will be experimentally indistinguishable from momentum p taking arbitrary real values. ![]() As the circle gets larger, the gap between the allowed momentum states becomes smaller. The possible values of the momentum are split by ℏ / R. For the particle on a circle, we see that the momentum is quantised This word refers to the fact that certain quantities, which are always continuous variables in the classical world, can only take certain discrete values in this new framework. This is our first sign of the quantum in quantum mechanics. We must haveĮ i k ( x + 2 π R ) = e i k x ⇒ e 2 π i k R = 1 Not all solutions ( 2.12) obey this requirement. The Schrödinger equation is still ( 2.10) and its solutions are still ( 2.12), but now there is the additional requirement that the wavefunction should be periodic, so that One simple fix is to consider a particle moving on a circle of radius R. ![]()
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