Unit 18: Quantum Physics — Short Questions

Max Planck proposed that electromagnetic radiation is emitted or absorbed as discrete packets of energy called quanta (photons), rather than continuously. The energy of each quantum is directly proportional to its frequency f: E = hf, where h is Planck's constant (6.63 × 10⁻³⁴ J s). Thus a higher-frequency photon carries more energy, independent of the intensity (number) of photons.

The photoelectric current is directly proportional to the intensity of the incident light (for light above the threshold frequency), because a greater intensity means more photons strike the surface per second, ejecting more photoelectrons. However, the maximum kinetic energy of the photoelectrons, and hence the stopping potential, does not change with intensity — it stays the same for both low and high intensity beams of the same frequency.

A photon of UV radiation has more energy. Since photon energy is E = hf (or E = hc/λ), and UV radiation has a higher frequency (shorter wavelength) than yellow light, its photons carry greater energy.

Yes. The work function φ is the minimum energy needed by an electron to escape the metal surface, and this minimum energy is supplied exactly at the threshold frequency f₀ (the lowest frequency that can still cause photoemission, with zero kinetic energy left over). Thus φ = hf₀.

The de Broglie wavelength of a particle is λ = h/mv, inversely proportional to its mass (for comparable speeds). An electron has an extremely small mass, so its wavelength is large enough (comparable to atomic spacings, e.g. in a crystal) to produce observable diffraction and interference effects. A tennis ball has an enormous mass compared to an electron, so its de Broglie wavelength is immeasurably small, and no wave effects (diffraction/interference) can be detected for it.

Different metals have different work functions, and hence different threshold frequencies f₀ = φ/h. If the frequency of the radiation is above the threshold frequency of the first metal but below that of the second metal (which has a larger work function), photoelectrons will be emitted from the first metal but not from the second, however intense the light is made.

No. The number of photoelectrons emitted (the photoelectric current) depends on the intensity of the light, i.e. the number of photons striking the surface per second, not on its frequency. High-frequency light gives each emitted electron greater kinetic energy, but a low-frequency light (above threshold) of higher intensity can eject a greater number of electrons than a less intense high-frequency beam.

Light behaves as a wave when it propagates from source to detector and when phenomena such as interference and diffraction are observed (e.g. passing through a diffraction grating or double slit). Light behaves as a stream of particles (photons) when it interacts with matter, as in the photoelectric effect and the Compton effect. Both aspects are real but are never revealed simultaneously in the same experiment — this is the principle of wave-particle duality.

Two of the spectral series of atomic hydrogen are the Lyman series, which lies in the ultraviolet region (transitions terminating at n = 1), and the Balmer series, which lies in the visible region (transitions terminating at n = 2). (A third series, the Paschen series, lies in the infrared region, terminating at n = 3.)

In a solid, atoms are packed closely together and interact strongly, so their discrete energy levels broaden and overlap into continuous energy bands; transitions between these bands can release photons of almost any energy, producing a continuous spectrum. In a hot vapour (gas), the atoms are far apart and behave as isolated systems with sharp, discrete energy levels; electrons can only make transitions between these specific levels, emitting photons of only certain specific energies (frequencies), which appear as a set of discrete bright lines — a line spectrum.

This shows that light possesses a dual wave-particle nature. The photoelectric effect (electron emission) can only be explained by treating light as a stream of discrete energy packets, photons, with E = hf — demonstrating its particle nature. Diffraction, on the other hand, is a spreading/bending of light around obstacles that can only be explained by treating light as a wave. Both aspects are genuine properties of light; which one is revealed depends on the type of experiment performed, but both are never observed simultaneously in a single experiment (principle of complementarity).

Visible light photons are produced by transitions of outer (valence) electrons, where the energy gaps between levels are small, giving photons of relatively low energy (low frequency, long wavelength). X-rays, however, are produced by transitions of inner-shell electrons (after an inner electron is knocked out and an outer electron falls into the vacancy), where the energy gaps are far larger. Since E = hf, these much larger energy transitions produce photons of much higher frequency and energy and much shorter wavelength than visible light, giving X-rays their characteristic properties such as high penetrating power.

Yes, energy is conserved. When an electron in an atom falls from a higher energy level E_i to a lower energy level E_f, the atom loses energy equal to the difference between the two levels. This exact amount of energy is carried away by the emitted photon, so that hf = E_i − E_f. The total energy (internal energy of the atom plus the energy of the radiation field) before and after emission remains the same.

Although a single hydrogen atom has only one electron, a sample of hydrogen gas (as in a discharge tube) contains an enormous number of atoms. Collisions in the discharge excite the electrons of different atoms to different higher energy levels (n = 2, 3, 4, ...). Each excited electron can then fall back to a lower level either directly or through intermediate levels, and each possible transition (pair of energy levels) emits a photon of a specific, different frequency. With so many atoms undergoing all the different possible transitions simultaneously, the result is a spectrum with many distinct spectral lines, even though each individual electron can only occupy one level at a time.

The scattered X-ray has a longer wavelength than the incident X-ray (λ′ > λ, the Compton shift Δλ = λ′ − λ > 0). Since f = c/λ, a longer wavelength means a lower frequency (f′ < f), and since E = hf, the scattered photon has lower energy (E′ < E) — the lost energy is transferred to the recoiling electron. The speed of the scattered X-ray photon is unchanged and still equal to the speed of light, c, since all electromagnetic radiation, including scattered X-rays, travels at speed c in vacuum.

Compton treated the collision between an X-ray photon and a loosely bound (nearly free) electron as an elastic collision between two particles, like billiard balls. Applying the law of conservation of energy (equating the photon's energy loss to the kinetic energy gained by the electron) and the law of conservation of momentum (separately for the x- and y-components, since momentum is a vector) to this collision allowed the Compton shift formula Δλ = (h/m₀c)(1 − cos θ) to be derived. This derived formula matched experimental results exactly, which was strong confirmation that photons behave as particles with definite energy and momentum.

When an excited gas emits light, only specific, discrete wavelengths (frequencies) are observed, appearing as separate bright lines rather than a continuous band of colour. Since each emitted photon's energy is given by E = hf, and this energy equals the difference between two electron energy levels in the atom (hf = E_i − E_f), the fact that only a limited, discrete set of frequencies is emitted implies that the electron can only occupy a limited, discrete set of energy levels within the atom — if the levels could take any value, a continuous range of frequencies (a continuous spectrum) would be observed instead of a line spectrum.

A photon has no rest mass, but it is never at rest — it always travels at the speed of light, c. From Einstein's relativistic energy relation applied to the photon, its momentum is given by p = E/c, and since E = hf = hc/λ, this gives p = hf/c = h/λ. Thus the photon's momentum arises from its energy and its motion at speed c (a purely relativistic relation), not from the ordinary formula p = mv used for particles with rest mass.