Consider a quantum particle of mass \(m\) moving in an \(n\)-dimensional space in the presence of an external (scalar) potential \(V(\boldsymbol{x})\). The Hamiltonian governing the motion is \[H = \frac{\boldsymbol{P} \cdot \boldsymbol{P}}{2 m} + V(\boldsymbol{X}) \,,\] where \(\boldsymbol{X}\) and \(\boldsymbol{P}\) are the position and momentum operators, respectively. The time-dependent state of the particle \(| \Psi_t \rangle\) satisfies the Schrödinger equation \[i \hbar \frac{\partial | \Psi_t \rangle}{\partial t} = H | \Psi_t \rangle \,.\] In position representation, we have \[\boldsymbol{X} = \boldsymbol{x} \,, \qquad \boldsymbol{P} = -i \hbar \frac{\partial}{\partial \boldsymbol{x}} \,,\] and so the Schrödinger equation has the following familiar form: \[i \hbar \frac{\partial \psi(\boldsymbol{x},t)}{\partial t} = -\frac{\hbar^2}{2 m} \frac{\partial}{\partial \boldsymbol{x}} \cdot \frac{\partial \psi(\boldsymbol{x},t)}{\partial \boldsymbol{x}} + V(\boldsymbol{x}) \

Nuts and bolts of quantum mechanics