Skip to main content
Chemistry LibreTexts

Postulate 3: Quantum Mechanics

  • Page ID
    20838
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    Every observable in quantum mechanics is represented by an operator which is used to obtain physical information about the observable from the state function. For an observable that is represented in classical physics by a function \(Q(x,p)\), the corresponding operator is \(Q(\hat{x},\hat{p})\).

    Discussion

    For every observable property of a system there is a quantum mechanical operator. The operator for position of a particle in three dimensions is just the set of coordinates x, y, and z, which is written as a vector

    \[ r = (x , y , z ) = x \vec {x} + y \vec {y} + z \vec {z} \tag {3.1}\]

    The operator for a component of momentum is

    \[ \hat {P} _x = -i \hbar \dfrac {\partial}{\partial x} \tag {3.2}\]

    and the operator for kinetic energy in one dimension is

    \[ \hat {T} _x = \left ( -\dfrac {\hbar ^2}{2m} \right ) \dfrac {\partial ^2}{\partial x^2} \tag {3.3}\]

    and in three dimensions

    \[ \hat {p} = -i \hbar \nabla \tag {3.4}\]

    and

    \[ \hat {T} = \left ( -\dfrac {\hbar ^2}{2m} \right ) \nabla ^2 \tag {3.5}\]

    The Hamiltonian operator \(\hat{H}\) is the operator for the total energy. In many cases only the kinetic energy of the particles and the electrostatic or Coulomb potential energy due to their charges are considered, but in general all terms that contribute to the energy appear in the Hamiltonian. These additional terms account for such things as external electric and magnetic fields and magnetic interactions due to magnetic moments of the particles and their motion.


    Postulate 3: Quantum Mechanics is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?