The Schrödinger Equation

\[ i \hbar \frac{\partial}{\partial t} \Psi = - \frac{\hbar^2}{2m} \Psi + V \Psi\]

The Schrödinger equation governs quantum mechanics. It is analogous to Newton's \(F=ma\), by virtue of its ability to predict how the system evolves in time. The function \(\Psi(x,t)\) is complex valued, and called the wave function.

The wave function is interpreted in the following manner:

• \(|\Psi(x,t)|^2\) gives the probability of observing the particle at position, x ,and time, t (Born's statistical interpretation).
• \(\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1\), the total probability of finding the particle anywhere must be 1 (Normalized).
• As a result of the wave function being normalized, the possible types of solutions are reduced to the square integrable functions.

One important property of the wave function, is that once normalized, it will stay normalized for any future time (proof in Griffith's). The wave function can be thought of as a vector in a vector space, over the complex numbers. In quantum mechanics the inner product is always Hermitian.

\[ <v,w> = \overline{<w,v>}\]

this ensures that the inner product is a real number, because the only way \(z = \bar{z}\) is iff \(z \in \mathbb R\).