Quantization

In physics, the term quantization refers to the energy of physical systems being discontinuous—quantized. The size of discrete energy steps is determined by Planck's constant h. This natural constant is so small (h ≈ 6.626 ×10⁻³⁴ Js) that on a macroscopic scale (energies on the order of joules, time intervals on the order of seconds), quantization is a minute effect that can hardly be observed. For all intents and purposes, macroscopic energies are continuous. However, for microscopic systems, such as electrons orbiting a nucleus, quantization is important.

Energy quantization was first introduced in 1900 by Max Planck in his theory of black-body radiation, when he assumed that the walls of a black-body consist of harmonic oscillators and that the energies of these oscillators are discrete, i.e., quantized. He was forced to introduce this assumption in his explanation of the experimentally observed deviations from Wien's distribution law. Planck did not quantize the black-body radiation—a form of electromagnetic radiation—this was done by Albert Einstein five years later, who postulated that the electromagnetic field consists of light quanta (energy parcels, that were later called photons). Einstein's energy parcels are of size hν, where ν is the frequency of the electromagnetic waves.

In 1923 Louis de Broglie announced that the relativistic kinetic energy of material particles is also quantized and he derived as the consequence that the motion of material particles is wave-like. The very small value of h explains why the wavelike nature of matter is very difficult to demonstrate on a macroscopic scale. The work of de Broglie inspired Erwin Schrödinger to postulate a wave equation that describes the motion of very light material particles, such as electrons. When Schrödinger's wave equation is solved with appropriate boundary conditions, energy quantization follows automatically for bound systems (not for unbound systems where particles come and go to and from infinity). In Schrödinger's theory certain operators play a role that correspond to classical—electromagnetic or mechanical—properties.

Quantization rules

In a second meaning of the word, quantization refers to the replacement of classical quantities by quantum mechanical operators. In general, two kinds of operators are distinguished:

1. Those corresponding to dynamic variables; they are Hermitian and referred to as observables.
2. Those corresponding to transformations of the system (rotations and translations); they are unitary.

The process of quantization only applies to operators of the first kind, i.e., to observables. A classical dynamic quantity ${\mathcal {A}}$ (defined in classical mechanics or electromagnetism) has a quantum mechanical counterpart, an observable (Hermitian operator) A. To construct the latter, we consider a single particle without spin subject to a scalar potential. Classically the quantity is a function of its momentum p, its position vector r, and the time t, ${\mathcal {A}}(\mathbf {r} ,\mathbf {p} ,t)$. With r = (x, y, z) is associated the observable R and with p = (p_x, p_y, p_z) the observable P. The two operators satisfy the commutation relations

${\begin{aligned}\left[R_{i},R_{j}\right]&\equiv R_{i}\,R_{j}-R_{j}\,R_{i}=0\\\left[P_{i},P_{j}\right]&=0\\\left[R_{i},P_{j}\right]&=i\hbar \delta _{ij},\end{aligned}}$

where ℏ is Planck's constant and δ_ij is the Kronecker delta. To obtain the observable A one could replace in the expression for ${\mathcal {A}}(\mathbf {r} ,\mathbf {p} ,t)$ the variables r and p by the observables R and P,

${\mathcal {A}}(\mathbf {r} ,\mathbf {p} ,t)\Longrightarrow A(t)={\mathcal {A}}(\mathbf {R} ,\mathbf {P} ,t)$

However, this mode of action would be, in general, ambiguous. For instance, in classical mechanics the inner product p⋅r is equal to r⋅p, but simple replacement in the two forms gives two operator forms that are not equal, because

$\mathbf {P} \cdot \mathbf {R} \neq \mathbf {R} \cdot \mathbf {P} .$

Moreover, neither of these two operator expressions is Hermitian. Hence to the replacement rule must be added a symmetrization rule, which classically is allowed and usually trivial. For example, the observable associated with p⋅r is obtained by first symmetrizing the classical expression, and then making the replacement

$\mathbf {p} \cdot \mathbf {r} ={\frac {1}{2}}(\mathbf {p} \cdot \mathbf {r} +\mathbf {r} \cdot \mathbf {p} )\Longrightarrow {\frac {1}{2}}(\mathbf {P} \cdot \mathbf {R} +\mathbf {R} \cdot \mathbf {P} )$

which is indeed Hermitian. In systems consisting of more than one particle, the coordinates and momenta are replaced one particle at the time and this is done for all particles consecutively.

Remarks:

1. There exist quantum physical quantities which have no classical equivalent and which are defined directly as an observable (this is the case for particle spin).
2. The procedure just sketched applies only to quantities in Cartesian coordinates. For other coordinate systems, such as spherical polar coordinates the procedure must be generalized.

Example

Consider the Hamiltonian (energy) of a spinless particle of charge q and mass m placed in an electric field derived from a scalar potential U(r). The potential energy of the particle is therefore V(r) = qU(r). The kinetic energy is ½m v² = p²/(2m), because p≡ m v. The classical Hamiltonian is time-independent,

${\mathcal {H}}(\mathbf {r} ,\mathbf {p} )={\frac {\mathbf {p} ^{2}}{2m}}+V(\mathbf {r} )\quad {\hbox{with}}\quad \mathbf {p} ^{2}\equiv \mathbf {p} \cdot \mathbf {p} .$

This example is simple, no symmetrization is necessary, since neither P² nor V(R) involves products of non-commuting operators. Therefore:

${\mathcal {H}}(\mathbf {r} ,\mathbf {p} )\Longrightarrow H={\frac {\mathbf {P} ^{2}}{2m}}+V(\mathbf {R} ).$

X-representation

Quantum mechanics is often formulated in the so-called X-representation in which wave functions Ψ of N-particle systems are functions of the position vectors of the N particles,

$\Psi (\mathbf {R} _{1},\mathbf {R} _{2},\ldots ,\mathbf {R} _{N}).$

In the X-representation the observables R_k act pointwise (are multiplicative operators) and the momentum operators P_k are given by differential operators

$\mathbf {P} _{k}=-i\hbar \left({\frac {\partial }{\partial X_{k}}},\;{\frac {\partial }{\partial Y_{k}}},\;{\frac {\partial }{\partial Z_{k}}}\right)\equiv -i\hbar {\boldsymbol {\nabla }}_{k},\quad k=1,2,\ldots ,N.$

The one-particle kinetic energy operator is

${\frac {\mathbf {P} ^{2}}{2m}}=-{\frac {\hbar ^{2}}{2m}}\left({\frac {\partial ^{2}}{\partial X^{2}}}+{\frac {\partial ^{2}}{\partial Y^{2}}}+{\frac {\partial ^{2}}{\partial Z^{2}}}\right)\equiv -{\frac {\hbar ^{2}}{2m}}\nabla ^{2},\quad {\hbox{where}}\quad \nabla ^{2}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }}.$

The N-particle kinetic energy is

$\sum _{k=1}^{N}\left[-{\frac {\hbar ^{2}}{2m_{k}}}\nabla _{k}^{2}\right].$

Other coordinate systems

When an expression is required for the kinetic energy in other coordinate systems than Cartesian, one must transform the gradient operator ∇ to the new coordinates and remember that an inner product involves, in general, a metric tensor g. Because the metric tensor is represented by the unit (identity) matrix in Cartesian coordinates, the latter fact is easily overlooked. One could go through the transformation for every coordinate system (cylinder, spherical, ellipsoidal, etc.) separately, but then one repeats the same calculation over and over again, with the only difference being the form of the metric tensor. In the 1860s this fact was recognized by the Italian mathematician Eugenio Beltrami, who gave once and for all the following expression for what is now known as the Laplace-Beltrami operator,

$\nabla ^{2}={\frac {1}{\sqrt {|\mathbf {g} |}}}\sum _{jk}{\frac {\partial }{\partial x^{j}}}{\sqrt {|\mathbf {g} |}}\;g^{jk}\;{\frac {\partial }{\partial x^{k}}}.$

Here the coordinates are written as x^j. For example, for spherical polar coordinates, x¹ ≡ r, x² ≡ θ, x³ ≡ φ. The expression is valid for higher and lower dimensional systems as well. The quantity |g| is the determinant of the metric tensor g, and g^jk stands for the (j, k) element of the inverse g⁻¹.

In 1928 Podolsky saw that the Beltrami form is the correct quantum mechanical operator for the kinetic energy in arbitrary curvilinear coordinates, provided the operator is multiplied by

$-{\frac {\hbar ^{2}}{2m}}.$

In more general systems the mass m is often included in the metric tensor, then it must not be included in this factor.

Example: Sperical polar coordinates

The kinetic energy operator T of a single particle follows easily from the metric tensor derived in the article spherical polar coordinates:

$\mathbf {g} ={\begin{pmatrix}1&0&0\\0&r^{2}&0\\0&0&r^{2}\sin ^{2}\theta \\\end{pmatrix}}\Longrightarrow \mathbf {g} ^{-1}={\begin{pmatrix}1&0&0\\0&r^{-2}&0\\0&0&{\frac {1}{r^{2}\sin ^{2}\theta }}\\\end{pmatrix}}$

The determinant is |g| = r⁴sin² θ, so that

${\begin{aligned}T&=-{\frac {\hbar ^{2}}{2mr^{2}\sin \theta }}\left({\frac {\partial }{\partial r}}r^{2}\sin \theta {\frac {\partial }{\partial r}}+{\frac {\partial }{\partial \theta }}(r^{2}\sin \theta )(r^{-2}){\frac {\partial }{\partial \theta }}+{\frac {\partial }{\partial \phi }}{\frac {r^{2}\sin \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial }{\partial \phi }}\right)\\&=-{\frac {\hbar ^{2}}{2mr^{2}}}\left({\frac {\partial }{\partial r}}r^{2}{\frac {\partial }{\partial r}}+{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\sin \theta {\frac {\partial }{\partial \theta }}+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}\right).\end{aligned}}$