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At fixed positive bias ratio $s=I_b/I_c$ with $0<s<1$, the Josephson coupling energy and work of the ideal bias source combine intoThis is the effective potential energy for an unwrapped Gauge-invariant Josephson phase; it includes the bias source and decreases by $2\pi s{E}_J$ on advancing one period. A local minimum and the adjacent downhill barrier areTo verify the barrier, solve $U^{\prime }=E_J(\sin \varphi -s)=0$, select $U^{\prime \prime }({\varphi }_m)>0$ and $U^{\prime \prime }({\varphi }_b)<0$, and subtract their potential energies. The small-oscillation angular frequency is fixed by the curvature:Thus the barrier and oscillation frequency both vanish as $s\to 1^{-}$, but with different powers. For $ϵ=1-s\ll 1$, expanding $\arccos (1-ϵ)$ and $\sqrt{2ϵ-{ϵ}^2}$ givesThe first approximation is for a nearly critical bias. At $s=0$ the exact downhill barrier is $2E_J$; at $s\ge 1$ there is no confining local minimum in this model.

For those stipulated parameters, direct evaluation gives $E_J/h=496.684\mathrm{GHz}$, $\Delta U/h=29.7670\mathrm{GHz}$, and ${\omega }_p/(2\pi )=2.89608\mathrm{GHz}$. Hence $\Delta U/(\hslash {\omega }_p)=10.2784$. The nearly critical barrier approximation gives $29.6165\mathrm{GHz}$, about $0.51\%$ below the exact result. These are original circuit calculations, not parameters assigned to the historical samples.

In the Heisenberg picture, an observable without explicit time dependence hasHere $\hslash$ is the reduced Planck constant. An emitted photon on a transition $n\to m$ with $E_n>E_m$ has positive angular frequency $(E_n-E_m)/\hslash =|{\omega }_{mn}|$. The sign of the array's quantum phase follows its index convention; it is not a negative measured photon frequency. The frequencies involve differences of energy levels, rather than the absolute values $E_n/\hslash$. Adding a constant to $H$ leaves these frequencies unchanged.

Insert the complete orthonormal basis between two linear operators:The sum runs over intermediate states. The second identity follows by cancellation of $E_k$; it makes every term in the product acquire the same time factor $e^{i{\omega }_{mn}t}$. This is a useful reason for matrix multiplication: it composes the two indexed arrays consistently with the addition of transition frequencies. It does not multiply matching entries independently. Swapping $A$ and $B$ generally changes the intermediate-state sum and hence the result.

The transition frequency between two stationary energy levels is the positive frequency satisfying $h\nu =|E_f-E_i|$. In absorption, the system gains this energy from the radiation; in emission, it loses it. The corresponding angular frequency is $\omega =2\pi \nu$.

An energy difference specifies where a spectral line can occur, but does not by itself guarantee that the transition is strong or allowed. The interaction and its selection rules determine the transition amplitude. External fields, motion and finite lifetimes can shift or broaden the measured line.

A vibrational partition function sums over molecular vibrational energy levels. A single quantum harmonic oscillator, with energies measured relative to its ground level and vibrational temperature ${\theta }_v=h\nu /k$, has $q_{\mathrm{vib}}=(1-e^{-{\theta }_v/T}{)}^{-1}$. Retaining its absolute zero-point energy multiplies this by $e^{-{\theta }_v/(2T)}$. That factor shifts the internal energy but contributes no entropy.

Vladimir Bulović's organic-optoelectronics course connects molecular excited states with practical light-emitting and photovoltaic devices. Its treatment follows excitons from their creation through diffusion and energy transfer, then asks whether that excitation produces light or collected electric charge. This distinction explains why efficient absorption alone does not guarantee an efficient solar cell or organic light-emitting diode. Comparing molecular energy levels, interfaces and transport makes device behavior understandable from the processes that compete inside the material.^[ref]

MIT lists the instructor of Organic Optoelectronics lecture notes by Vladimir Bulović as Vladimir Bulovic.

A theoretical atomic physicist who coauthored calculations of the Lamb shift in hydrogen and light hydrogen-like ions. The work reviews theoretical contributions and uncertainties needed for precision spectroscopy. Such predictions let a small measured difference between energy levels test relativistic quantum interactions rather than just the leading Coulomb potential.^[ref]

Wave mechanics represents a quantum state by a wavefunction and observables by operators acting on it. For a nonrelativistic spinless particle in a potential energy $V$, the Schrödinger equation readsThe spatial shape and boundary conditions determine possible energy eigenfunctions; the Born rule gives their experimental interpretation as probability amplitudes.

For the infinite square well, vanishing at both walls selects standing sine waves and hence discrete energy levels. This explains quantization through a boundary-value problem rather than by imposing a separate allowed-orbit rule. Erwin Schrödinger's first 1926 paper developed quantization as an eigenvalue problem for the hydrogen atom. Matrix and wave mechanics equivalence explains how the same physical predictions arise from matrix mechanics in a suitable orthonormal basis.^[ref]

An atomic physicist who coauthored a NIST compendium of atomic spectroscopy with Wolfgang Wiese. It connects the labeling of energy levels with spectral lines, transition strengths and selection rules. These conventions allow a measured spectrum to be compared with tabulated atomic data without confusing a level designation, a wavelength and an intensity.^[ref]

Willis Lamb and Robert Curtis Retherford measured a small separation between hydrogen energy levels that the simplest relativistic quantum mechanics placed at equal energy. Their 1947 experiment used microwave transitions involving metastable atoms, rather than trying to resolve the separation in an ordinary optical line. The Lamb shift required improved quantum electrodynamics, including the interaction of an electron with the quantized electromagnetic field. A small spectral discrepancy thereby became a quantitative test of the theory.^[ref]

A physicist.^[ref][ref]

An atomic physicist who coauthored a NIST reference on atomic spectroscopy with William C. Martin. Its treatment of transition probabilities distinguishes the probability of an atomic transition from the population of its initial energy level. This matters because the strength of an observed spectral line depends on both atomic properties and the conditions of the emitting sample.^[ref]

The change of an energy level in a magnetic field. The displayed weak-field form applies to a hyperfine level with magnetic projection $m_F$, when $g_F$ is the corresponding Landé g-factor. A transition frequency shift is the difference of the two level shifts divided by Planck constant. A clock transition can have zero first-order transition shift even when its two individual level slopes are nonzero. For $m_F=0$ hyperfine levels, the linear level shift vanishes and quadratic terms remain.