First Axiom: States as Hilbert Space Vectors

To conclude, we formulate our first axiom of quantum mechanics:

Axiom 1: A quantum mechanical system is described by a vector $\vert\Psi(t)\rangle\equiv \Psi(t)$ of a Hilbert space ${\cal H}$. The time evolution of $\Psi(t)$ is determined by the Schrödinger equation

$$i\hbar\frac{\partial}{\partial t} \Psi(t) = \hat{H} \Psi(t)$$ (174)

The Hamilton operator $\hat{H}$ is an operator corresponding to the total energy of the system. In the case of a single particle with mass $m$ moving in the configuration space $R^d$ under a potential $V({\bf x})$, the wave function $\Psi({\bf x},t)\in {\cal H} = L^2(R^d)$ (square integrable functions) obeys

$$i\hbar\frac{\partial}{\partial t} \Psi({\bf x},t) =\left[-\frac{\hbar^2\Delta}{2m} + V({\bf x}) \right]\Psi({\bf x},t).$$ (175)

$\vert\Psi({\bf x},t)\vert^2d^dx$ is the probability for the particle to be in the (infinitesimal small) volume $d^dx$ around ${\bf x}$ at time t. The solutions of the stationary Schrödinger equation at fixed energy,

$$\hat{H}\phi=E\phi$$ (176)

are called stationary states, the possible energies $E$ eigenenergies.

Note that the form of the Hamilton operator not necessarily has to be as in (3.10), the Hamiltonian for a single particle defined over the space $R^d$. We will later encounter, for example, Hamiltonians that describe the sites of a finite lattice and have the form of a $n\times n$ matrix.