Sunday, January 11, 2009

Separate Them!

I would like to highlight in this post, a nice mathematical technique known as separation of variables, which reduces a more complicated partial differential equation into a less complex ordinary differential equation. To illustrate this technique, I shall be using the time-dependent Schrodinger wave equation in 1 dimension, which is as follows:

This equation tells us that the time evolution of the wavefunction depends on how the Hamiltonian operator acts on the wavefunction. That is, the time evolution depends in some manner, on the energy of the system.

With that, we shall now consider the Hamiltonian operator acting on the wavefunction:

However, in most cases, the potential is independent of time, say like how an electron always orbits around a fixed centre of charge; therefore, we indicate this by re-writing:

Notice now that the potential function only depends on the position coordinate, and not time. Now, we can write out the full time-dependent wave equation as:

In quantum mechanics, it turns out that the overall time and position dependent wavefunction can be factored out into a product of separate time wavefunctions and position wavefunctions. That is, we can now write:

This assumption (generally valid when the potential is not a function of time) leads to the technique known as separation of variables, where you effectively factor out one variable from a function. And if we make the substitution, we see that:

Dividing throughout by φ(x)τ(t):

Alright! Now on the left we have a function depending solely on time, and on the right a function depending solely on position! Notice that we have separated the variables! So what’s so good about this you say? Consider this:

‘If I vary x only, then only the right hand side of this equation should change, since the left hand side doesn’t depend on time. However, if the left hand side doesn’t change and remains constant, so should the right hand side! This means that both sides are actually equal to a constant!’

With this revelation, can you fashion a guess as to what this constant might be?

If you said energy, you are absolutely right! We can now equate both sides to the energy of the system:

Also notice that the equations are no longer partial differential equations, but ordinary differential equations! We’ve made life simpler! Hurrah!

Let us look at the time-dependent wavefunction:

Voila! The time-evolution depends on the energy of the system as shown! Now, you might be wondering about the position-dependent wavefunction, so let me rearrange it as shown:

Hark! Isn’t this the usual time-independent Schrodinger wave equation that we always see? Haha.

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