Tuesday, January 20, 2009

Stay Still!

I think I’ll go into the idea of stationary states today; what exactly is a stationary state? Well, it’s essentially a state of a system, where the energy of the state remains constant over time. Defined more rigorously, it’s a state where the expectation values of observables remain constant over time.

For instance, let us take a look at the particle-in-a-box wavefunctions (as functions of time and position):

So let’s consider the probability density function of finding the particle within the box:

Notice that the phase factors (exponential time factors) cancel out when the complex conjugation is taken and multiplied! That is, the probability density function isn’t a function of time! It’s solely a function of position and thus, doesn’t vary with time.

How about the expectation value of the position? Well let’s take a look:

Hey! It turns out that the expectation value of the position doesn’t depend on time as well! So, it turns out that for all states of the system that are eigenstates (that is, if the system exists only as an eigenstate), the state is a stationary state!

So what isn’t a stationary state? Well, let’s look at linear combinations of eigenstates; we know that any linear combination (properly normalized of course) of eigenstates will still result in an arbitrary state that is a solution of the Schrödinger equation, so let’s try a positive linear superposition of the ground state and first excited state (I’ve normalized it for all of you already):

Let us now evaluate the probability density function (click on it to enlarge it):

Notice that now the probability density function is a function of time as well! The time phase factors no longer cancel out! It turns out that for any linear combination of eigenstates, the state will no longer be a stationary state – and therefore observables like its energy, momentum and even position will not have time-constant expectation values.

If you’ve heard physics professors go, “It’s all because of the cross terms!” this is what it means. Haha.

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