At a separation of $1$ fm, the proton Coulomb energy is approximately $V=e^2/(4\pi {\epsilon }_{0}r)\simeq 1.44$ MeV, whereas $k_{B}T\simeq 0.862$ keV at the stated solar-core temperature. As a deliberately crude rectangular-barrier estimate, take a width of $1$ fm and use the two-proton reduced mass $\mu =m_p/2$. ThensoTreating one proton as fixed and using $m_p$ instead gives about $0.59$. Both estimates only say that this arbitrarily shortened barrier would be fairly transparent; the opaque-barrier exponential is itself uncontrolled for such a small exponent.

There is an important conditioning issue. If the protons have already reached a separation of one femtometre, most of the long-range Coulomb suppression has already been overcome. The rectangular estimate cannot represent the probability for thermal protons starting far apart. Their classical turning separation at this energy is about $1.7\times 10^3$ fm, not one femtometre.

Inside a rectangular barrier with $E<V$, the stationary Schrödinger equation givesIts independent solutions are $e^{\pm \alpha x}$ withFor an opaque barrier the amplitude is suppressed by approximately $e^{-\alpha L}$ and the probability by $e^{-2\alpha L}$. Matching contributes a prefactor, so this exponential approximation is not an exact transmission coefficient for a transparent or very thin barrier. For two moving particles the relative-coordinate mass is their reduced mass.

Quantum tunnelling is transmission through a region where the particle's energy is below the classical potential energy. The stationary wavefunction decays within that region but need not vanish at its far boundary, so matching can leave a nonzero transmitted probability current.

This does not require a particle to borrow energy temporarily. Energy is conserved for a time-independent barrier; the classically forbidden region is described by an evanescent wave, with the same stationary energy as the incident and transmitted waves.

For a single slit of width $d$, the far-field amplitude is proportional to $\mathrm{sinc}(\pi d\sin \theta /\lambda )$. The first zeros satisfy $d\sin \theta =\pm \lambda$. At small angles the angular width between those zeros is $2\lambda /d$, or $2L\lambda /d$ on a screen a distance $L$ away.The result assumes the wavelength, propagation distance and far-field conditions stay fixed. If the slit becomes comparable to the wavelength, the exact sine relation replaces the small-angle scaling.

Both first moments vanish, so the standard deviations follow directly from the second moments:The Heisenberg uncertainty relation is satisfied and is not saturated. A Gaussian wave packet can saturate the position-momentum bound; this compactly supported polynomial is not Gaussian.

The wavefunction is real and vanishes continuously at both endpoints. HenceThe momentum second moment is most safely evaluated from its quadratic form:ThusThe derivative jumps where the wavefunction joins zero. On the whole real line the strong second-derivative operator therefore produces endpoint delta distributions, so the state is not in the usual operator domain of $p^2$. The finite momentum second moment, equivalently the kinetic-energy quadratic form or the integral of $p^2$ against the momentum probability density, remains well defined and gives the value above.

The probability density is even, so its first moment vanishes. Its second moment follows by integrating the polynomial:Using the normalization from part a givesThe variance in position is consequently $a^2/7$.

Choose the overall phase so that $A$ is positive. Normalization requiresThereforeAn arbitrary constant phase multiplying this answer has no effect on any expectation value.

At room temperature, electrons require quantum mechanics, including their occupied orbitals or bands and Fermi statistics. Heavy nuclei can often be treated as distinguishable particles near classical equilibrium lattice sites, because their exchange wavelengths are short. Their coupled displacements form phonons, however, and the occupation of a vibrational mode is classical only if $k_{B}T\gg \hslash \omega$ for that mode. Even at room temperature some high-frequency modes remain quantum mechanical.

At $1$ K, quantum lattice vibrations, zero-point motion and the scarcity of thermally excited phonons are essential. Many heavy nuclei still do not have overlapping exchange wavefunctions; quantum vibration and quantum exchange are distinct criteria. For the same spacing, the nuclear overlap threshold from the rough estimate is $T_N\simeq 211/A$ K, illustrating why the nuclear mass cannot be omitted from a blanket low-temperature claim.

At the same temperature, momenta increase as the square root of mass, so wavelengths decrease as its inverse square root. For a nucleus of mass $M_N\simeq A{m}_p$,At the electron overlap threshold, ${\lambda }_e=d$, the nuclear wavelength is about $7.0\mathrm{pm}/\sqrt{A}$, far below the $0.3$ nm spacing. Nuclear exchange effects are consequently much less important than electron exchange effects. This statement about nuclear overlap does not mean lattice vibrations are always classical.

Set $\lambda \simeq d$ with the convention of part b. ThenThus ordinary solid-state temperatures are far below this rough overlap scale. The electrons need quantum statistics and quantum states throughout that regime. The number is an order-of-magnitude wavelength criterion, not an exact Fermi temperature: electron density, electronic band structure, effective mass and the numerical convention all matter.

Using the given one-component equipartition estimate, $p^2=M{k}_{B}T$. Substituting into the de Broglie relation givesComparison with a confinement length or interparticle spacing $d$ gives the criterion $T\lesssim h^2/(M{k}_B{d}^2)$. Numerical factors depend on the convention: the statistical thermal de Broglie wavelength is usually $h/\sqrt{2\pi M{k}_{B}T}$, while a three-component root-mean-square momentum introduces a factor $\sqrt{3}$. The question's specified one-component convention is used consistently here. Once degeneracy becomes important, classical equipartition itself ceases to describe the electrons accurately.

The de Broglie hypothesis associates a wave phase with a material particle. Its momentum and energy setThis is not a literal extended classical lump of matter oscillating at every point. A wavefunction gives amplitudes for measurements, and a localized particle is represented by a superposition of momentum waves. Electron diffraction, for example, exhibits interference at angles set by the same wavelength relation.

A useful empirical profile is the Sérsic law,where $b_n$ is chosen so that $R_e$ encloses half the light. Classical bright elliptical galaxies are often approximated by the de Vaucouleurs's law case $n=4$, with $b_4\simeq 7.67$. Real ellipticals span a range of Sérsic indices, so $R^{1/4}$ is not universal.

Their classical luminosity-dispersion scaling is the Faber-Jackson relation,as an approximate empirical trend with intrinsic scatter and sample-dependent slope. Combining size, surface brightness and dispersion gives the more informative fundamental plane of elliptical galaxies.

The integrated spectrum combines stars moving toward and away from us within the galaxy, broadening their absorption by Doppler shifts. Assuming instrumental and individual-stellar widths have been removed or are negligible, the line-of-sight velocity dispersion isThe angular half-light radius at the low-redshift distance is $R_e\simeq 2.35$ kpc. For a pressure-supported, approximately isotropic equilibrium spheroid with a standard structural virial coefficient of order five, useThe coefficient represents assumed density profile, aperture and orbital structure; a different reasonable mass estimator changes the result by an order-unity factor. Treating $D$ as a luminosity distance and using $D_A=D_L/(1+z{)}^2$ would reduce the radius and mass by about four percent here.

The distance modulus is about $34.93$. The magnitude within $R_e$ therefore corresponds to $M_{V,\mathrm{half}}\simeq -19.23$ andDoubling the half-light luminosity givesThis ratio is compatible with an old stellar population plus possible dark matter. It is larger than a solar-like mass-to-light ratio, but does not by itself isolate a unique dark-matter fraction: stellar age, initial mass function, anisotropy and the virial coefficient are not specified. Counting the quoted half-light magnitude as the total would incorrectly double the inferred ratio.

A K-star-like integrated spectrum indicates that old or intermediate-age cool stars, often luminous red giants, dominate the observed optical/near-infrared light. It suggests little ongoing formation of very luminous hot O and B stars. It does not imply that every star is K type: faint low-mass dwarfs and remnants can contribute substantial mass while contributing little light.

Using the reference wavelength printed in the problem,The low-redshift Hubble estimate isPeculiar velocities and the distinction between luminosity and angular-diameter distance introduce percent-level or larger corrections to this simple estimate. The numerical result intentionally uses the supplied calcium reference rather than replacing it with a different tabulated line wavelength.

The sequence primarily orders effective temperature rather than chemical composition. Hot O spectra show strong ionized-helium features and highly ionized metals, with relatively weak neutral-hydrogen lines. B stars show neutral helium and increasingly prominent hydrogen. A stars have the strongest Balmer absorption. Through F and G, lower-ionization metal lines and calcium features become prominent. K stars have strong low-ionization metal absorption and cool-star features; M stars develop strong molecular bands such as TiO, while Balmer absorption is weak.

Line opacity depends on how many atoms occupy both the relevant ionization stage and the particular lower level. Boltzmann excitation controls the latter and Saha ionization the former, so equal elemental abundances need not give equal line strengths. Electron pressure, surface gravity, broadening and atmospheric temperature gradients also affect the spectrum.

For Balmer absorption the lower state is $n=2$, $10.2$ eV above the hydrogen ground state. In cool atmospheres its population is exponentially small. At higher temperatures it rises, but at still higher temperature ionization removes neutral hydrogen. The competition gives the maximum in A-type stars, with effective temperatures around $9000$-$10,000$ K. This is an excitation-and-ionization effect, not evidence that A stars have the largest hydrogen abundance.

Core-helium-burning stars populate the horizontal branch and red clump; hot subdwarf B stars provide examples with a helium-burning core of roughly half a solar mass and a very small remaining hydrogen envelope. Their high luminosity relative to mass, hot spectra and locations away from an ordinary hydrogen-burning main sequence are observational evidence for such evolved configurations.

A plausible history starts with a hydrogen-burning star. After central hydrogen exhaustion, hydrogen-shell burning builds a compact helium core and the envelope expands on the red giant branch. Sufficient core temperature ignites helium, often by a helium flash for a low-mass degenerate core. Strong mass loss or binary interaction can strip most of the hydrogen envelope, exposing a hot helium-burning remnant; retaining a substantial envelope instead gives an ordinary horizontal-branch or red-clump star.

The simple radiative Kramers scaling estimates a luminosity contrast but does not uniquely identify one evolutionary state or predict a real subdwarf luminosity precisely. A similarly low-mass helium white dwarf would be a cooling, degenerate object rather than the sustained high-temperature helium-burning configuration assumed here.