Basis Functions in Coordinate Space

Now imagine that we have a problem where the wavefunction can be expanded as a sum of only two basis functions (admittedly unlikely, but perhaps useful for a single electron spin problem):

$$\Psi({\bf r}, t) = c_1(t) \Phi_1({\bf r}) + c_2(t) \Phi_2({\bf r}).$$ (32)

This leads to the time-dependent Schrödinger equation (where we will suppress variables ${\bf r}$ and t for convenience):

$$i \hbar \left( {\dot c}_1 \Phi_1 + {\dot c}_2 \Phi_2 \right) = {\hat H} \left( c_1 \Phi_1 + c_2 \Phi_2 \right).$$ (33)

How do we solve this equation? It's a coupled differential equation, similar to eq. 1 except that it's first-order instead of second order. Just as in the classical example, it's the coupling that makes it hard to solve! In the classical case, the answer to coupling was to get the eigenfunctions. What happens if we assume $\Phi_1$ and $\Phi_2$ to be eigenfunctions of ${\hat H}$? In that case,

$${\hat H} \Phi_1 E_1 \Phi_1$$ = (34)

$${\hat H} \Phi_2 E_2 \Phi_2$$ = (35)

and the time-dependent equation becomes

$$i \hbar \left( {\dot c}_1 \Phi_1 + {\dot c}_2 \Phi_2 \right) = c_1 E_1 \Phi_1 + c_2 E_2 \Phi_2.$$ (36)

Furthermore, since the eigenvectors of a Hermitian operator are or can be made orthogonal, we can multiply by $\Phi_1^*$ and $\Phi_2^*$ and integrate over dr to obtain

$$i \hbar {\dot c}_1$$ = c₁ E₁ (37)

$$i \hbar {\dot c}_2$$ = c₂ E₂ (38)

which are simple first-order differential equations solved by

 

$$c_1(0) e^{- i E_1 t / \hbar}$$ c₁(t) = (39)

$$c_2(0) e^{- i E_2 t / \hbar}$$ c₂(t) = (40)

as you can verify by substituting and differentiating.

But what if our original wavefunction $\Psi({\bf r}, t)$ is not given as a linear combination of eigenfunctions? A good strategy is to re-write it so that it is! In the coordinate representation (i.e., ${\bf r}$ space), we can get the coefficients c_i(0) in an expansion over orthogonal eigenfunctions $\Phi_i({\bf r})$ simply as

$$c_i(0) = \int \Phi_i^*({\bf r}) \Psi({\bf r}, 0) d{\bf r}.$$ (41)

The other strategy would be to try to re-write the propagator in the original basis set. In the problems we do, we will usually use the first approach.