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A roton is an excitation near a local minimum of a superfluid's dispersion relation at nonzero momentum $p_0$. The displayed local approximation defines a positive energy gap $\Delta$ and positive curvature effective mass $m_r$; it is not a free helium atom or a macroscopic quantum vortex.^[ref] In the stipulated plot, these correspond to $p_0=2p_{\ast }$, $\Delta =0.6E_{\ast }$ and $m_r=p_{\ast }^2/(4E_{\ast })$. The plot illustrates how a low energy at appreciable momentum can lower the Landau criterion below the speed of sound, without assigning any measured parameter to real helium.

Landau's 1947 revision joined the low-momentum linear part to a spectrum with a finite-momentum minimum, retaining the macroscopic superfluid theory while changing its excitation model.^[ref]

For $V(x)=\alpha {\sum }_j\delta (x-ja)$ with $\alpha >0$, put $P=m\alpha a/{\hslash }^2$ and $x=ka$. The Floquet discriminant of the delta-comb Kronig-Penney model is $f(x)=\cos x+P\sin x/x$. Extended Bloch states require $|f(x)|\le 1$. Since $f(0)=1+P>1$, there is a positive lower band edge; a repulsive potential does not support negative-energy states in this energy convention.

The $j$th positive-energy band ends at $x=j\pi$. Its lower edge lies in $((j-1)\pi ,j\pi )$ and solves $f(x)=(-1{)}^{j-1}$. A finite periodic ring selects only the discrete allowed values of crystal momentum within these bands. Nonzero band gaps do not by themselves imply an insulator: the band filling must also place the Fermi energy in a gap.

With inverse-time convention $e^{i\omega t}$, the half-line Klein-Gordon equation ${\sigma }_{tt}-{\sigma }_{rr}+{\mu }^2\sigma =0$ has spatial normal modes $A{e}^{ikr}+B{e}^{-ikr}$, where $k=\sqrt{{\omega }^2-{\mu }^2}>0$ above threshold and $k=i\sqrt{{\mu }^2-{\omega }^2}$ below it. For the branch even in $\omega$, reality above threshold requires $A(\omega )=B(-\omega {)}^{\ast }$. Below threshold boundedness eliminates $B$, and reality requires $A(\omega )=A(-\omega {)}^{\ast }$. At threshold the spatial differential equation instead has the basis $1,r$; boundedness retains only the constant mode. A spectral integral with ordinary integrable functions as amplitudes ignores the isolated threshold points, while distribution amplitudes can represent the surviving monochromatic constant-spatial modes.

Let the Klein four-group have generators $X,Y$ and $Z=XY$, with $X^2=Y^2=I$ and $XY=YX$. For every real $p$, the matricesdefine a group representation. Indeed, $D(Y{)}^2=I_2$, and $D(X)$ commutes with every matrix, so the group relations hold.

Set $S=\left(\begin{array}{cc}1& -p/2\\ 0& 1\end{array}\right)$. A direct change of basis givesConsequently all these representations are isomorphic representations, with character values $(2,-2,0,0)$ on $(I,X,Y,Z)$. Each is a direct sum of the two one-dimensional representations whose values on $X$ are $-1$ and whose values on $Y$ are $+1$ and $-1$. A group representation need not be orthogonal; nonzero $p$ changes the basis rather than the represented group.

On the nonzero real numbers, sign reversal $a$ and reciprocation $b$ are commuting involutions. Their function composition generates exactly $e(x)=x$, $a(x)=-x$, $b(x)=1/x$ and $ab(x)=-1/x$. The relations $a^2=b^2=e$ and $ab=ba$ identify the group with $C_2\times C_2$, the Klein four-group. Its subgroups are the trivial group, its three order-two cyclic subgroups and itself. Excluding zero from the domain is necessary for all four transformations to be defined and invertible.

A flat circular contact of radius $a$ carrying total normal force $N$ has uniform pressure $p=N/(\pi a^2)$. In steady relative rotation, a ring of radius $s$ and thickness $ds$ experiences tangential kinetic friction $dF=\mu p2\pi sds$. Its resisting torque is $d\tau =sdF$. Integration from zero to $a$ gives ${\tau }_f=2\mu Na/3$. A different pressure distribution changes the result; for an axisymmetric distribution use ${\tau }_f=2\pi \mu {\int }_0^{a}p(s)s^{2}ds$.

A cylindrical plug of radius $a$, length $l$ and density $\rho$ is pushed outward by gauge pressure $P_g$. If its remaining contact length is $l-x$ and its constant radial contact pressure is $P_c$, the resisting kinetic friction is $2\pi a(l-x)\mu P_c$. At the marginal initial condition $\pi a^2{P}_g=2\pi al\mu P_c$, the net outward force becomes $\pi a^2{P}_{g}x/l$. With mass $\rho \pi a^{2}l$,Thus the exit speed at $x=l$ tends to $\sqrt{P_g/\rho }$ as $v_0\to 0^{+}$. Exactly $v_0=0$ leaves the marginal configuration stationary, and the exit time $b^{-1}\mathrm{arsinh}(bl/v_0)$ diverges in that limit. An attached extra mass changes the acceleration but not the pressure-minus-friction work, if the contact geometry is unchanged.

Water Cherenkov neutrino reconstruction infers a vertex, time and charged particle direction from photomultiplier pulses. The 1987 analysis recorded charge and timing, calibrated with decaying muons and source-produced gamma rays, and rejected outer-detector activity and events following muons.^[ref]

In an ideal point-flash approximation, a light source at ${\mathbf{r}}_0$ and time $t_0$ gives the displayed travel-time relation at sensor ${\mathbf{r}}_i$ in a medium of refractive index $n$. Subtracting two times eliminates $t_0$. With a chosen constant $n=1.33$ and path lengths $3\mathrm{m}$ and $6\mathrm{m}$, the arrival-time difference is $n(3\mathrm{m})/c\simeq 13.31\mathrm{ns}$. These are synthetic inputs, not detector measurements. Several sensors constrain the vertex; an extended moving track requires the light-emission position and time along that track as well.

For a charged particle moving at $v=\beta c$, the Cherenkov cone angle satisfies $\cos {\theta }_C=1/(n\beta )$ when $n\beta >1$. With $\beta \simeq 1$ and the illustrative $n=1.33$, ${\theta }_C\simeq 41.25^{\circ }$. The geometrical cone carries track-direction information; electron deflections and the spread of emission positions broaden the observed pattern. The neutrino itself does not draw the light cone.

A Cherenkov radiation threshold and an analysis threshold are different constraints. For a fixed $n$, an electron must exceed $T_C=m_e{c}^2[(1-n^{-2}{)}^{-1/2}-1]$. With $m_e{c}^2=0.511\mathrm{MeV}$ and the chosen $n=1.33$, this gives about $0.264\mathrm{MeV}$. Recording and selecting a useful event requires much more light: the 1987 detector's trigger accepted $8.5\mathrm{MeV}$ electrons with 50% efficiency.^[ref]

For $d{s}_5^2={\eta }_{ab}d{x}^{a}d{x}^b+(dy+A_{a}d{x}^a{)}^2$, with $A$ independent of $y$, an affine parameter $\lambda$ makes $\xi =dy/d\lambda +A_{a}d{x}^a/d\lambda$ and the tangent norm $\mathcal{N}=g_{AB}{\dot{x}}^A{\dot{x}}^B$ constant. The four-dimensional interval satisfiesConsequently the formula $d{\lambda }^2=d{u}^2/(1-{\xi }^2)$ assumes the additional normalization $\mathcal{N}=1$; affine parametrization alone does not set this value. With base metric signature $+---$ and $u=c\tau$, the projected curve is timelike only for $\mathcal{N}>{\xi }^2$. The normalized case gives $d\tau /d\lambda =\sqrt{1-{\xi }^2}/c$. The positive fibre term and the $+---$ base together have two positive directions; the same sign convention must be retained when interpreting the reduction.

Take $L=[{\eta }_{ab}{\dot{x}}^a{\dot{x}}^b+W^2]/2$ with $W=\dot{y}+A_a{\dot{x}}^a=\xi$ constant. The Euler-Lagrange equation for $x^a$ isThus, with $F_{bc}={\partial }_b{A}_c-{\partial }_c{A}_b$,Reversing the explicit derivative order while retaining this metric and the same definition of $\xi$ changes the equation rather than merely changing a field-tensor naming convention. More generally, a fibre term $\sigma W^2$, $\sigma =\pm 1$, gives $\sigma \xi F^a{}_b{\dot{x}}^b$; its conserved canonical fibre momentum is $p_y=\sigma \xi$.

In the normalized positive-fibre case with $|\xi |<1$, four-dimensional proper time gives $d{u}^a/d\tau =c\xi F^a{}_b{u}^b/\sqrt{1-{\xi }^2}$. This has the Lorentz force form. The scale relating the geometrical $A$ to a physical electromagnetic four-potential fixes the conversion to an SI charge-to-mass ratio.

A mass unit based on Jupiter, conventionally written $M_J$; numerical calculations must state the adopted value or gravitational parameter.

For constant nonzero voltage $V$, direct integration givesA constant electric potential difference therefore produces an oscillating pair electric current. The average of this ideal sinusoid over a complete cycle is zero; a measured nonzero steady electric current can include single-electron superconducting tunneling. Josephson's lecture explicitly separates the ordinary quasiparticle term from the phase-dependent term. ^[ref]

For an original numerical example, use the exact SI values $e=1.602176634\times 10^{-19}\mathrm{C}$ and $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$. ThenThe corresponding period is about $0.2067834\mathrm{ns}$. These values describe an assumed exact applied voltage, not a historical measurement. The factor $2e$ identifies coherent pair transfer; replacing it by $e$ would halve the predicted frequency.

At zero electric potential difference, $\dot{\varphi }=0$ permits a nonzero constant quantum phase and hence a steady pair electric current. Under an imposed electric current $I_b$ with $|I_b|<I_c$, a static solution satisfies $\sin \varphi =I_b/I_c$. The low-energy branch has $\cos \varphi >0$. For example, a stipulated $I_b=I_c/2$ gives $\varphi =\pi /6$ modulo $2\pi$; it does not require a voltage drop. In the ideal current-biased energy $-E_J\cos \varphi -(\hslash I_b/2e)\varphi$, the positive second derivative on that branch explains its local stability.

The superconducting critical current $I_c$ is a maximum available static pair electric current, not the inevitable electric current under a voltage bias. At $V=0$ the initial state and circuit select a phase and a current. Above $I_c$ this sinusoidal model has no static phase solution, but its two relations alone do not predict the complete resistive current-voltage characteristic, switching dynamics or hysteresis. Those require the circuit and dissipative channels.

The ideal Josephson junction has a phase-dependent energyDifferentiating and using the voltage–phase relation yieldsThus the ideal pair element stores and returns energy. At $V=0$ it carries a static electric current without electrical power loss. Under a nonzero periodic drive its instantaneous power need not vanish, but its energy returns after a full phase cycle. Real junctions also have a capacitance, quasiparticle transport and an external circuit; their total losses cannot be inferred from this conservative pair term alone.

Write the superconducting order parameters as ${\psi }_j=|{\psi }_j|e^{i{\theta }_j}$, using a field of pair electric charge $q=-2e$. For a specified path across the superconducting weak link, defineHere $\mathbf{A}$ is the magnetic vector potential. To check electromagnetic gauge invariance, use a scalar function $\chi$ and transformThe change of ${\theta }_2-{\theta }_1$ is $q({\chi }_2-{\chi }_1)/\hslash$, exactly cancelled by the change of the line integral. Thus $\varphi$ is unchanged although each local quantum phase changes. This is the same charge/sign bookkeeping discussed in gauge-coupling sign in Ginzburg-Landau superconductivity. Different physical paths can enclose magnetic flux, so fixing the path matters.

For stationary reservoirs with equal internal pair chemical potentials and a time-independent vector potential, the local Schrödinger equation gives ${\dot{\theta }}_j=-q{U}_j/\hslash$ after dropping the common internal term. With electric potentials $U_j$ and $V=U_2-U_1$, subtraction gives $\dot{\varphi }=-qV/\hslash =2eV/\hslash$. For a general circuit, $V$ denotes the corresponding measured electrochemical voltage and the electromagnetic contribution must be treated consistently. A bare difference of phases without the vector potential term is insufficient in an arbitrary gauge.

A SQUID uses Josephson junctions in a loop made from a superconductor to detect changes in magnetic flux.

With occupied fermion sites representing up-spins, use $s_1^{-}=f_1$, $s_2^{-}=P_1{f}_2$ and $P_1=(-1{)}^{n_1}=1-2n_1$. The canonical anticommutation relations imply $P_1^2=I$, $P_1{f}_1=-f_1{P}_1$ and $[P_1,f_2]=0$. Consequently the two sites commute as spin operators, while $[s_j^{-},s_j^{+}]=1-2n_j=-{\sigma }_j^z$. This normalization is essential: the unscaled combinations ${\sigma }_j^x\pm i{\sigma }_j^y$ are twice $s_j^{\pm }$.

The two-spin XY Hamiltonian becomesIn particular, $f_1^{†}{P}_1=f_1^{†}$ and $f_1{P}_1=-f_1$ fix the hopping signs. The modes $b=(f_1+f_2)/\sqrt{2}$ and $c=(f_1-f_2)/\sqrt{2}$ preserve the canonical anticommutation relations and give $H=2h+(-2h-2J)b^{†}b+(-2h+2J)c^{†}c$.

For a uniform ideal gas sphere, set $\rho =mn$, $p=n{k}_{B}T$ and $M=4\pi \rho R^3/3$. Building the sphere by concentric shells gives gravitational potential energy $W=-3G{M}^2/(5R)$. During a small homologous contraction, released gravitational energy balances the work against gas pressure when $3G{M}^2|dR|/(5R^2)=4\pi R^{2}p|dR|$, or $GMm/R=5k_{B}T$. Eliminating $R$ gives the displayed mass. At the same initial mass density and temperature, larger masses have stronger gravitational contraction than pressure support in this estimate. Its numerical coefficient belongs to the uniform-sphere pressure-work model; the infinite-medium Jeans instability and a pressure-confined isothermal sphere use different equilibrium and boundary assumptions.