Eigenfunctions and Vector Space

Wavefunctions are analogous to vectors in 3D space. The unit vectors of our vector space are eigenstates.

In normal 3D space, we represent a vector by its components.

$$\bgroup\color{black} \vec{r}= x\hat{x}+y\hat{y}+z\hat{z} = \sum\limits_{i=1}^3r_i\hat{u}_i \egroup$$

The unit vectors $\hat{u}_i$ are orthonormal,

$$\bgroup\color{black} \hat{u}_i\cdot\hat{u}_j=\delta_{ij} \egroup$$

where $\delta_{ij}$ is the usual Kroneker delta, equal to 1 if $i=j$ and otherwise equal to zero.

Eigenfunctions - the unit vectors of our space - are orthonormal.

$$\bgroup\color{black}\langle\psi_i\vert\psi_j\rangle=\delta_{ij}\egroup$$

We represent our wavefunctions - the vectors in our space - as linear combinations of the eigenstates (unit vectors).

$$\bgroup\color{black}\psi=\sum\limits_{i=1}^\infty \alpha_i\psi_i\egroup$$

$$\bgroup\color{black}\phi=\sum\limits_{j=1}^\infty \beta_j\psi_j\egroup$$

In normal 3D space, we can compute the dot product between two vectors using the components.

$$\bgroup\color{black}\vec{r}_1\cdot\vec{r}_2=x_1x_2+y_1y_2+z_1z_2 \egroup$$

In our vector space, we define the dot product to be

$$\begin{eqnarray*} \langle\psi\vert\phi\rangle &=&\langle\sum\limits_{i=1}^\inft... ...eta_j\delta_{ij} =\sum\limits_{i=1}^\infty\alpha^*_i\beta_i \\ \end{eqnarray*}$$

We also can compute the dot product from the components of the vectors. Our vector space is a little bit different because of the complex conjugate involved in the definition of our dot product.

From a more mathematical point of view, the square integrable functions form a (vector) Hilbert Space. The scalar product is defined as above.

$$\bgroup\color{black} \langle\phi\vert\psi\rangle=\int\limits_{-\infty}^\infty d^3r \phi^*\psi \egroup$$

The properties of the scalar product are easy to derive from the integral.

$$\bgroup\color{black} \langle\phi\vert\psi\rangle=\langle\psi\vert\phi\rangle^* \egroup$$

$$\bgroup\color{black} \langle\phi\vert\lambda_1\psi_1+\lambda_... ...ert\psi_1\rangle+\lambda_2\langle\phi\vert\psi_2\rangle \egroup$$

$$\bgroup\color{black} \langle\lambda_1\phi_1+\lambda_2\phi_2\v... ...ert\psi\rangle+\lambda_2^*\langle\phi_2\vert\psi\rangle \egroup$$

$\langle\psi\vert\psi\rangle$ is real and greater than 0. It equals zero iff $\psi=0$. We may also derive the Schwartz inequality.

$$\bgroup\color{black} \langle\psi_1\vert\psi_2\rangle\leq\sqrt... ...psi_1\vert\psi_1\rangle\langle\psi_2\vert\psi_2\rangle} \egroup$$

Linear operators take vectors in the space into other vectors.

$$\bgroup\color{black} \psi'=A\psi \egroup$$