Energy Quantization

Energy quantization is the occurrence of discrete allowed values of energy in a physical system. In quantum mechanics, these values constitute the discrete part of the system’s energy spectrum and are commonly called energy levels. Quantization is especially evident in confined particles and bound states, including the internal states of an atom. It does not mean that every energy in nature is discrete: quantum systems can also possess continuous spectra, or both discrete and continuous portions. The allowed values depend on the system’s interactions and boundary conditions. (ocw.mit.edu)

Historical development

In 1900, Max Planck introduced energy elements proportional to frequency while developing his explanation of blackbody radiation. The relation $\varepsilon=h\nu$ connected an energy element $\varepsilon$ with frequency $\nu$ through what became known as the Planck constant, $h$. Planck’s proposal concerned the statistical treatment of material oscillators; it was not yet the modern theory of light particles. In 1905, Albert Einstein proposed that light itself could behave as localized energy quanta, using this idea to explain the photoelectric effect. These quanta are now called photons. (nobelprize.org)

In 1913, Niels Bohr incorporated discrete stationary states into his atomic model. Radiation was emitted or absorbed when an atom changed between states, with its frequency determined by the energy difference. This connected atomic spectral lines with quantized energies. Modern quantum mechanics replaced prescribed electron orbits with states described by wave functions and derived energy levels from dynamical equations. (nobelprize.org)

Mathematical description

A system’s energy is represented by its Hamiltonian operator, $\hat H$. For a time-independent Hamiltonian, a state with definite energy satisfies the stationary Schrödinger equation:

Here $\psi_n$ is an energy eigenstate and $E_n$ its energy eigenvalue. This is an instance of the mathematical relationship described by eigenvalues and eigenvectors. For a nonrelativistic particle of mass $m$ moving in one dimension in a potential $V(x)$,

Physical requirements on the wave function, including boundary conditions and normalizability for bound states, restrict which solutions are admissible. In standard confining potentials, these restrictions produce discrete energies. Quantization therefore follows from the eigenvalue problem rather than from an arbitrary rule that energy must occur in identical packets. (ocw.mit.edu)

A general state need not have one definite energy. It can be a superposition of energy eigenstates:

For normalized states with a discrete spectrum, an ideal energy measurement returns $E_n$ with probability $|c_n|^2$. The mean energy is $\langle E\rangle=\sum_n|c_n|^2E_n$. Because the coefficients can vary continuously, this mean need not equal an allowed individual measurement outcome. (ocw.mit.edu)

Representative systems

The particle in a box illustrates how confinement generates quantization. For a one-dimensional infinite well of width $L$, the wave function vanishes at both walls. Only standing-wave solutions satisfying those boundaries are allowed, giving

The levels are not equally spaced: the separation between neighboring levels increases with $n$. Increasing the box width reduces the energy scale and makes the levels more closely spaced. (ocw.mit.edu)

For the quantum harmonic oscillator, with potential $V(x)=m\omega^2x^2/2$,

These levels are equally spaced by $\hbar\omega$. Its lowest state has zero-point energy $\hbar\omega/2$, measured relative to the potential minimum. This is a property of the oscillator’s quantum ground state, not thermal excitation. The oscillator also provides a framework for describing photon number states of a radiation mode. (ocw.mit.edu)

In the simplest nonrelativistic description of hydrogen, the bound-state energies are approximately

taking the ionization threshold as zero. The levels approach that threshold as $n$ increases; above it lies a continuum of unbound states. More detailed descriptions include additional interactions that split or shift the levels. (ocw.mit.edu)

Transitions and experimental evidence

When a system undergoes a single-photon transition between levels $E_i$ and $E_f$, energy conservation gives

Absorption raises the system’s energy; emission lowers it. The resulting characteristic frequencies underpin spectroscopy, which uses radiation to investigate atomic and molecular structure. Molecular spectra reveal rotational, vibrational, and electronic energy differences. Not every pair of levels produces an equally accessible transition: selection rules and transition matrix elements determine which processes occur and their strengths. (nobelprize.org)

Measured spectral lines provide evidence for discrete energy differences. Their wavelengths and intensities allow researchers to identify levels and test theoretical descriptions. The National Institute of Standards and Technology’s Atomic Spectra Database compiles critically evaluated atomic energy levels, wavelengths, and transition probabilities. (nist.gov)

Scope and classical behavior

Energy quantization does not imply a universal smallest unit of energy. A photon’s energy depends on its frequency, and different systems have different level spacings. Even within one system, spacings may vary substantially, as the box and hydrogen examples demonstrate. An unconfined free particle has a continuous energy spectrum, although its behavior remains quantum mechanical. (nobelprize.org)

The distinction from classical mechanics can become difficult to observe when adjacent quantum levels are much closer than experimental resolution or the relevant energy scale. For the harmonic oscillator, the relative size of one level spacing decreases at high excitation; for a particle in a large box, the confinement energy scale falls as $L^{-2}$. Such limits help explain why apparently continuous energies are compatible with an underlying quantum description. (ocw.mit.edu)