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A cooling method that reduces an applied magnetic field while suppressing heat transfer to or from the material. For a suitable material exhibiting paramagnetism, an approximately reversible field reduction can lower its temperature at approximately constant entropy.

The preparation matters as much as the field reduction. First establish thermal equilibrium with a cold heat bath, then raise the magnetic field while maintaining the bath temperature: this is magnetization in an isothermal process. The bath accepts heat and the material loses magnetic entropy. Next open a heat switch, thermally isolating the working material, and lower the magnetic field. If the sweep maintains internal thermal equilibrium with negligible entropy production, this second leg is an isentropic process. An adiabatic process alone does not guarantee constant entropy.

The attraction is a controllable energy scale: changing the magnetic field changes the separation of magnetic energy levels, even across a thermally insulating gap. This lets an already cold solid become colder without needing a colder contacting heat bath. A practical refrigerator also needs a load, a period of absorbing its heat, and a reset that rejects accumulated entropy to the bath. A single field sweep has finite cooling capacity.

Alexander Prokhorov and Nikolay Basov developed a theory of the molecular maser. Their analysis combines a prepared molecular beam, stimulated emission and feedback from a resonant cavity. Oscillation begins only when the molecular radiation supplies more power than the cavity and load lose. Including saturation determines a finite steady amplitude. The calculation connects molecular energy levels with practical requirements for a stable microwave oscillator.^[ref]

A physicist.^[ref]

In Ammonia, the nitrogen chemical atom can lie on either side of the plane of the three hydrogen chemical atoms. Quantum tunnelling couples these configurations; the resulting pair of energy levels permits an electric dipole transition. This geometrical inversion is distinct from population inversion, which concerns how many molecules occupy each level.^[ref]

For an idealized pair of energy levels with equal quantum degeneracy, the Boltzmann distribution gives $N_u/N_l=\exp [-h{\nu }_0/(k_{\mathrm{B}}T)]<1$ at positive temperature $T$. A warm source therefore supplies upper-state molecules together with an unwanted lower-state population. Selecting the former can produce population inversion in the transmitted beam without exciting each transmitted molecule during selection. The energy per net downward transition isHere $h$ is the Planck constant. Taking the reported ${\nu }_0=23.87\mathrm{GHz}$ as a rounded input gives $ϵ\approx 1.582\times 10^{-23}\mathrm{J}$. This calculation is a conversion of the reported frequency, not a new measurement of the ammonia inversion transition.

Atomic spectroscopy studies the emission, absorption and scattering associated with the energy levels of chemical atoms and ions. The positions of spectral lines identify differences of energy, while their strengths and polarization also depend on populations and selection rules.

A dilute-gas discharge can excite chemical atoms; a spectrometer then separates the emitted light by wavelength. Comparing the resulting lines with an independently calibrated reference can identify an element or reveal an isotope shift. Changing a magnetic field tests Zeeman splitting, and improving energy resolution can reveal fine structure. Different experimental changes therefore isolate different pieces of the atomic model.

An atom contains an atomic nucleus and bound electrons; a neutral atom has equal numbers of protons and electrons. The atomic number specifies the number of protons, while different isotopes of the same element have different numbers of neutrons. Losing or gaining electrons produces an ion, rather than a different atomic number.

The hydrogen atom gives the simplest electronic model: one electron bound through a Coulomb interaction to one proton. Its spectral lines reveal differences between discrete energy levels. Multi-electron chemical atoms add the Pauli exclusion principle and electron correlation, so their structure is not obtained by placing every electron in the same one-electron ground state.

An oven supplies a collimated caesium beam. State-selecting magnets precede and follow two coherent microwave interaction regions; a weak static magnetic field separates sublevels. The clock transition connects $F=3,m_F=0$ and $F=4,m_F=0$. Transferred atoms reach a heated ionizing collector. A detector scan identifies the resonance before locking.^[ref]

Here $F$ labels the total angular momentum and $m_F$ its projection. The selected pair is a two-level system inside a larger set of atomic energy levels; treating every oven atom as a prepared qubit would overstate the preparation. A measured detector signal may be modeled as $S(\nu )=S_{\mathrm{bg}}+A\overline{P}(2\pi ({\nu }_0-\nu ))$, with background $S_{\mathrm{bg}}$ and gain $A>0$. Background subtraction and stable transmission are necessary before interpreting the signal as a transition probability. The velocity-averaged Ramsey line provides one explicit ideal example; its chosen dimensions are not historical specifications.

Charles H. Townes, James Power Gordon and Herbert Jack Zeiger developed the molecular maser, using stimulated emission to amplify or generate coherent microwaves. Their report treats the same device as an amplifier, oscillator and spectrometer. Preparing the molecular excitation and providing resonant feedback turns microscopic transitions between energy levels into a macroscopic signal. The measured frequency stability and narrow spectroscopic resolution showed why the maser could serve as both an instrument for studying molecules and a frequency reference.^[ref]

A physicist.^[ref]

A description of an atomic nucleus in terms of coordinated motion of many nucleons, especially changes in shape and orientation. The nuclear shell model organizes individual nucleon states in a nuclear mean field; the collective nuclear model organizes low-lying energy levels by nuclear deformation, nuclear rotational bands and nuclear quadrupole vibrations. These are complementary organizations of the same nuclear many-body problem. A deformed nuclear mean field can support individual-particle states while its orientation supplies a slow collective coordinate.

In his 1975 lecture, Ben Roy Mottelson described coupling individual-particle motion to nuclear shape coordinates, and distinguished observed rotational spectra from vibrations about spherical equilibrium. This connects the two descriptions without treating the collective coordinates as extra constituent particles.^[ref]

James Rainwater's 1975 account identifies unexpectedly large electric quadrupole moments as a motivation for a deformable nuclear mean field. The challenge concerned their magnitude, even where spherical-shell trends had the correct sign.^[ref]

A simple original potential energy example makes the coupling tangible. Let $q$ be a dimensionless signed shape coordinate for a chosen intrinsic configuration, $C>0$ its restoring stiffness, and $f$ a configuration-dependent coupling, both with units of energy. SupposeThe minimum is $q_{\ast }=f/C$, with energy gain $f^2/(2C)$. For chosen values $C=20u$ and $f=4u$, where $u$ is an arbitrary positive energy unit, $q_{\ast }=0.20$ and the gain is $0.40u$. This illustrates how a configuration can favor deformation when the core can respond. It is not a fitted nuclear surface or a claim that every configuration chooses the same shape; competing configurations and nuclear pairing can favor a spherical minimum.

The predictive attraction is economy: one collective scale relates several energy levels, while electric quadrupole transitions independently test their common structure. Agreement in both is much more restrictive than assigning a separate free parameter to every level. Nuclear magnetic moments test magnetic structure and angular momentum assignments; they are distinct observables from the electric quadrupole moments used to infer charge deformation.

Condensed matter comprises physical systems such as solids and liquids in which interactions among many constituents determine the observed properties. The term refers to the material system; condensed matter physics studies its structure, dynamics and phases.

A crystal illustrates the change of viewpoint. Its chemical atoms can have collective vibrational modes described as phonons, while its electronic response can involve energy bands rather than isolated-atom energy levels. Collective excitations and phase transitions show why a many-constituent system can display useful organizing principles that are not apparent from one atom alone.

The correspondence principle requires quantum mechanics to reproduce established classical predictions in the regimes where those predictions work. It guides the relation between quantum observables and classical variables, but it does not say that every large quantum number makes every feature of a state classical.

For a quantum harmonic oscillator, neighboring energy levels differ by $\hslash \omega$, so the associated transition angular frequency equals the classical oscillator frequency. At large excitation, the relative level spacing becomes small compared with the total energy, although an individual energy eigenstate still lacks the definite phase of a classical oscillation. A suitable localized wave packet can supply that phase for an appropriate time interval. The classical limit must therefore specify both the state and the quantities being compared.

Two allowed optical transitions sharing a lower quantum state can produce an extra saturation spectroscopy feature even where there is no additional energy level difference. Approximate their wavenumbers by the same $k$, and let their angular frequencies be ${\omega }_a,{\omega }_b$. A pump resonant with one transition and an opposed probe resonant with the other satisfysoExchanging the two transitions selects the opposite velocity. Both beams can then modify the same shared population, producing a crossover near the midpoint. This requires the relevant velocities to be occupied and the transitions to be connected by the population dynamics; not every pair must produce a visible feature. Strength and sign depend on optical pumping and the multilevel structure. Assigning every narrow feature to a separate atomic transition would therefore give false energy levels.

A Josephson junction can make a collective quantum phase behave as a measurable dynamical coordinate. In current-biased Josephson phase dynamics, the Gauge-invariant Josephson phase $\varphi$ moves in a tilted Josephson washboard potential; the junction's capacitance supplies its effective inertia. A trapped phase gives zero mean electric potential difference, while sustained phase advance produces a nonzero mean electric potential difference. Quantized phase levels in a Josephson junction and quantum escape of a current-biased Josephson junction test quantum mechanics for this collective coordinate.

Begin with a stipulated parallel electrical circuit: an ideal sinusoidal Josephson junction, capacitance $C>0$, and effective electrical resistance $R>0$. The Kirchhoff node law and Josephson effect giveHere $I_b$ is the constant bias electric current, $I_{\mu }$ is a deliberately applied microwave electric current, and $I_N$ represents unwanted fluctuating electric current. Multiplication by $\hslash /(2e)$ puts the undriven equation into the mechanical formThe effective inertia is a coefficient for a dimensionless quantum phase, not a literal material mass moving through the insulating layer. The capacitor's energy, $C{V}^2/2=M_{\varphi }{\dot{\varphi }}^2/2$, provides a direct check of that identification. The effective $R$ includes loading by the external electrical circuit; a constant parallel $R$ is an approximation over the relevant frequency range.

The useful design freedom is that $I_b$ changes the confinement while $C$ changes its inertia. The same manufactured device can therefore connect a classical oscillation, discrete energy levels, and escape to an electrical readout. The Martinis-Devoret-Clarke circuit experiments separate these questions experimentally. John Clarke, Michel H. Devoret and John M. Martinis shared the 2025 Nobel Prize in Physics for circuit experiments establishing macroscopic quantum tunnelling and discrete energy levels.^[ref]

C. V. Raman and K. S. Krishnan observed additional spectral lines in light scattered by liquids. Filtering the illumination made it possible to distinguish these new components from lines already present in the incident spectrum. In the Raman effect, exchanged energy changes the scattered photon's frequency, providing information about molecular energy levels. The experiment thus turned a faint part of scattering of light into a probe of molecular structure.^[ref]

A physicist.^[ref]

A degeneracy factor counts distinct microstates sharing the same specified energy or other grouped quantum labels. If an energy level $E_i$ has multiplicity $g_i$, its contribution to the canonical partition function is $g_i{e}^{-\beta E_i}$. The probability of occupying that entire level is consequently proportional to both its Boltzmann factor and its state degeneracy.

For a lower nondegenerate level at zero energy and an upper level at $\Delta$ with degeneracy factor $g$, the upper-level probability is $g{e}^{-\beta \Delta }/(1+g{e}^{-\beta \Delta })$. Equal level probabilities occur at $\Delta =k_{B}T\log g$, rather than necessarily at equal energies. This makes state counting a source of observable entropic shifts. The multiplicity must match what has already been counted explicitly; summing separately over spin states and then multiplying by the same spin multiplicity would double-count microstates.

An energy level is degenerate when its eigenspace has dimension greater than one: several linearly independent energy eigenstates share the same energy eigenvalue. Any quantum superposition within that eigenspace is also an energy eigenstate with the same energy.

For a particle in a ring without applied flux, clockwise and anticlockwise momentum eigenstates have equal kinetic energy, except for the single zero-momentum state. A field or other perturbation may split such a pair if it distinguishes the two states. Degenerate perturbation theory identifies the combinations selected by a weak perturbation. The energy-level degeneracy counts independent states, rather than the number of different basis descriptions one can write for them.

Degenerate perturbation theory treats a perturbation of an energy eigenvalue shared by several independent quantum states. Ordinary nondegenerate formulas contain vanishing energy denominators in this situation. Instead, for the orthogonal projection $P$ onto that eigenspace, diagonalize $PVP$ within the eigenspace before treating coupling to other levels.

Its eigenvalues are the first-order energy shifts, and its eigenvectors choose combinations with a well-defined leading response. For a two-state block $V=\left(\begin{array}{cc}0& v\\ v^{\ast }& 0\end{array}\right)$, the shifts are $\pm |v|$: zero diagonal entries in the original basis do not imply zero shifts. Coupling to states outside the block enters at higher order when their energy separations remain large compared with the perturbation. This method explains how a weak field or broken symmetry splits a degenerate energy level.

A double-well potential has two local minima separated by a potential energy barrier. A symmetric confining example isIts minima are at $x=\pm a$ and the central barrier has height $\lambda a^4$. Near either minimum, writing $x=\pm a+\xi$ gives $V\simeq 4\lambda a^2{\xi }^2$, a harmonic potential with local angular frequency $\sqrt{8\lambda a^2/m}$ for mass $m$.

For this symmetric one-dimensional bound-state problem, the lowest two energy eigenstates can be chosen as an even function and an odd function. They are delocalized across both wells, with splitting $\Delta E=E_1-E_0>0$ produced by quantum tunnelling. Define orthonormal combinations $|L⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|R⟩=(|0⟩-|1⟩)/\sqrt{2}$. For well-separated wells with a sufficiently high barrier, these approximate states localized on the left and right. An initial $|L⟩$ evolves within this doublet withThe first full transfer takes $\pi \hslash /\Delta E$. A two-mode approximation for a Bose gas uses analogous left/right orbitals; its validity also requires suppression of higher energy levels and appropriate control of interactions. Two local minima alone do not guarantee that every arbitrary nonconfining potential has two discrete bound levels.

Choose an isolated pair of energy levels, with ground state $|g⟩=(0,1{)}^T$ and excited state $|e⟩=(1,0{)}^T$. Use the Pauli matrices in the ordered basis $(|e⟩,|g⟩)$, so ${\sigma }_z=\mathrm{diag}(1,-1)$. After subtracting the mean energy, prescribe the laboratory Hamiltonian operatorThe unperturbed energy gap is $\hslash {\omega }_0$, the drive has angular frequency $\omega$, and $\varphi$ fixes its oscillation phase at the chosen time origin. All of ${\omega }_0$, $\omega$, the Rabi angular frequency $\Omega$ and the Rabi detuning $\delta$ are in radians per second; ordinary frequency is angular frequency divided by $2\pi$. In particular, $\Omega$ here is the coefficient multiplying the cosine in $H_{\mathrm{lab}}/\hslash$, not twice that coefficient.

Write $|{\psi }_{\mathrm{lab}}⟩=R(t)|{\psi }_{\mathrm{rot}}⟩$ with $R(t)=\exp (-i\omega t{\sigma }_z/2)$. Transforming the Schrödinger equation gives $H_{\mathrm{rot}}=R^{†}{H}_{\mathrm{lab}}R-i\hslash R^{†}\dot{R}$. The rotating-wave approximation drops terms at angular frequency $2\omega$, leavingThis approximation requires $\Omega ,|\delta |\ll {\omega }_0,\omega$, and an interaction duration over which omitted corrections remain small. It also assumes that other energy levels can be neglected. Resonance alone does not justify either assumption. An exactly co-rotating drive has no discarded counter-rotating component; the linearly oscillating laboratory drive above does.

The Pauli matrix multiplication law gives $K^2={\Omega }_{\mathrm{eff}}^{2}I$. Separating even and odd terms of the matrix exponential therefore proveswith the continuous limit $U=I$ when $\Omega =\delta =0$. Starting from $|g⟩$, the upper probability amplitude is $-i{e}^{-i\varphi }(\Omega /{\Omega }_{\mathrm{eff}})\sin ({\Omega }_{\mathrm{eff}}t/2)$. The Born rule givesThe rotating transformation only changes the quantum phases of these basis states, so their laboratory populations are the same. Rabi detuning tilts the rotation axis of the Bloch vector: it increases the oscillation rate to ${\Omega }_{\mathrm{eff}}$ but caps the transferred population at ${\Omega }^2/({\Omega }^2+{\delta }^2)$. A fixed drive oscillation phase cancels from this single-pulse population when the initial state is $|g⟩$. It still fixes the relative quantum phase of a resulting quantum superposition and matters when another pulse supplies a phase reference.

The electron magnetic moment couples an electron's angular momentum to a magnetic field. Its intrinsic spin contribution isHere ${\mu }_B$ is the Bohr magneton, and $g_s$ is the positive electron spin g-factor under this sign convention. The minus sign makes magnetic dipole moment antiparallel to spin angular momentum. An electron in a chemical atom can also contribute an orbital magnetic dipole moment; the measured response of a whole atomic level combines these contributions.

Measuring a tiny departure of $g_s$ from $2$ tests how well a theory predicts an electron's interaction with its surroundings. The Dirac equation gives $g_s=2$ before radiative corrections; the anomalous magnetic moment makes a small correction experimentally accessible.^[ref] A useful strategy is to compare two atomic energy levels whose orbital and spin contributions have different weights, rather than trying to calibrate an absolute magnetic field more accurately.

The fermion self-energy of an electron in quantum electrodynamics. After renormalization, its contribution in a bound chemical atom is one source of the Lamb shift; other radiative corrections and nuclear structure also affect the observed energy levels.