4.2: Schrödinger's Equation

Consider a dynamical system consisting of a single non-relativistic particle of mass $\begin{equation}m\end{equation}$ moving along the x-axis in some real potential $\begin{equation}V(x)\end{equation}$. In quantum mechanics, the instantaneous state of the system is represented by a complex wave function $\begin{equation}\psi(x, t)\end{equation}$. This wavefunction evolves in time according to Schrödinger's equation: 

$$\begin{equation}\mathrm{i} \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \psi}{\partial x^{2}}+V(x) \psi\end{equation}$$

The wavefunction is interpreted as follows: $\begin{equation}|\psi(x, t)|^{2}\end{equation}$  is the probability density of a measurement of the particle's displacement yielding the value x. Thus, the probability of a measurement of the displacement giving a result between $\begin{equation}a \text { and } b \text { (where } a<b \text { ) is }\end{equation}$ 

$$\begin{equation}P_{x \in a: b}(t)=\int_{a}^{b}|\psi(x, t)|^{2} d x\end{equation}$$

Note that this quantity is real and positive definite.

Contributors

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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