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4.2: Quantum Operators Represent Classical Variables

An observable is a dynamic variable of a system that can be experimentally measure (e.g., position, momentum and kinetic energy). In systems governed by classical mechanics, it is a real-valued function (never complex), however, in quantum physics, very observable in quantum mechanics is represented by an independent operator which is used to obtain physical information about the observable from the wavefunction. It is a general principle of quantum mechanics that there is an operator for every physical observable. For an observable that is represented in classical physics by a function \(Q(x,p)\), the corresponding operator is \(Q(\hat{x},\hat{p})\).

Postulate 2

For every observable property of a system there is a quantum mechanical operator.

Classical dynamical variables, such as \(x\) and \(p\), are represented in quantum mechanics by linear operators which act on the wavefunction. 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, \(r\):

\[ r = (x ,  y , z ) = x \vec {i} + y \vec {j} + z \vec {k} \label {4.2.1}\]

The operator for a component of momentum is

\[ \hat {P} _x = -i \hbar \dfrac {\partial}{\partial x} \label {4.2.2}\]

and the operator for kinetic energy in one dimension is

\[ \hat {T} _x = \left ( -\dfrac {\hbar ^2}{2m} \right ) \dfrac {\partial ^2}{\partial x^2} \label {4.2.3}\]

and in three dimensions

\[ \hat {p} = -i \hbar \nabla \label {4.2.4}\]


\[ \hat {T} =  \left ( -\dfrac {\hbar ^2}{2m} \right ) \nabla ^2 \label {4.2.5}\]

The Laplacian operator

The three second derivatives in parentheses together are called the Laplacian operator, or del-squared,

\[ \nabla^2 = \left ( \frac {\partial ^2}{\partial x^2} +  \dfrac {\partial ^2}{\partial y^2} +  \dfrac {\partial ^2}{\partial z^2} \right ) \label {3-20}\]

with the del operator,

\[\nabla = \left ( \vec {x} \frac {\partial}{\partial x} + \vec {y} \frac {\partial}{\partial y} + \vec {z} \frac {\partial }{\partial z} \right ) \label{3-21}\]

also is used in Quantum Mechanics. Remember, symbols with arrows over them are unit vectors.

The total energy operator is called the Hamiltonian operator, \(\hat{H}\) and consists of the kinetic energy operator plus the potential energy operator.

\[\hat {H} = - \frac {\hbar ^2}{2m} \nabla ^2 + \hat {V} (x, y , z ) \label{3-22}\]

The Hamiltonian

The term Hamiltonian, named after the Irish mathematician Hamilton, comes from the formulation of Classical Mechanics that is based on the total energy,

\[H = T + V \nonumber\]

rather than Newton's second law,

\[\vec{F} = m\vec{a} \nonumber\]

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.

Name Observable Symbol Operator Symbol Operation
Table \(\PageIndex{1}\): Some common Operators in Quantum Mechanics
Position \(x\) \(\hat{X}\) Multiply by \(x\)
  \(r\) \(\hat{R}\) Multiply by \(r\)
Momentum \(p_{x}\) \(\hat{P_{x}}\) -\(\imath\)\(\hbar\)\(\dfrac{d}{dx}\)
  \(p\) \(\hat{P}\)
Kinetic Energy \(K_{x}\) \(\hat{K_{x}}\) \(\dfrac{-(\hbar^{2})}{2m}\)\(\dfrac{d^{2}}{dx^{2}}\)
  \(K\) \(\hat{K}\)


Which can be simplified to


Potential Energy \(V(x)\) \(\hat{V(x)}\) Multiply by \(V(x)\)
  \(V(x,y,z)\) \(\hat{V(x,y,z)}\) Multiply by \(V(x,y,z)\)
Total Energy \(E\) \(\hat{E}\)

\(\dfrac{-(\hbar^{2})}{2m}\)\(\bigtriangledown^{2}\) + \(V(x,y,z)\)

Angular Momentum \(L_{x}\) \(\hat{L_{x}}\)

-\(\imath\)\(\hbar\)[\(\text{y}\)\(\dfrac{d}{dz}\) - \(\text{z}\)\(\dfrac{d}{dy}\)]

  \(L_{y}\) \(\hat{L_{y}}\)

-\(\imath\)\(\hbar\)[\(\text{z}\)\(\dfrac{d}{dx}\) - \(\text{x}\)\(\dfrac{d}{dz}\)]

  \(L_{z}\) \(\hat{L_{z}}\) -\(\imath\)\(\hbar\)[\(\text{x}\)\(\dfrac{d}{dy}\) - \(\text{y}\)\(\dfrac{d}{dx}\)]