Dual Nature of Radiation and Matter — Questions & Answers

Sample questions from Dual Nature of Radiation and Matter

1. Q1. The photoelectric work function of a metal is 2.0 eV. Light containing TWO wavelengths, 400 nm and 700 nm, of comparable intensity falls on it. The observed stopping potential is (hc = 1240 eV nm)

   • A.the sum of the values for the two wavelengths
   • B.set by the 700 nm component alone
   • C.1.1 V, set by the 400 nm component alone✓
   • D.the average of the values for the two wavelengths
   Solution

   The 700 nm photons carry 1.77 eV, below the work function, so they eject nothing at all. The 400 nm photons carry 3.1 eV and give K_max = 1.1 eV, so V0 = 1.1 V. Stopping potential is fixed by the most energetic electron present, never by a sum or an average.

2. Q2. A sodium surface (work function 2.3 eV) is illuminated by a laser of wavelength 800 nm at an intensity high enough to melt the metal in a few seconds. The photocurrent measured before melting is (hc = 1240 eV nm)

   • A.zero, because each photon carries only about 1.55 eV✓
   • B.small but non-zero, because a few photons still exceed 2.3 eV
   • C.large, because high intensity always liberates electrons
   • D.zero only until the surface heats up, then large
   Solution

   Photon energy = 1240/800 = 1.55 eV, below the 2.3 eV work function, and emission is a one-photon-one-electron process, so no intensity produces photoelectrons. Monochromatic light has no higher-energy photons in it. Any current after melting would be thermionic emission, a different process.

3. Q3. Considering both the photoelectric effect (light behaving as particles/photons) and Young's double-slit experiment (light behaving as waves), a SINGLE photon passing through a double-slit apparatus, detected one at a time (with a very low-intensity source, so photons arrive individually, well-separated in time), when many such individual photon-detection events are accumulated over time on a screen, will show:

   • A.An interference pattern (alternating bright/dark bands) gradually built up from the accumulated positions of many INDIVIDUAL, discretely-detected photon events, even though each single photon is detected as one discrete, localized point/event on the screen✓
   • B.No pattern at all, individual photons detected one at a time would show completely random, uniform scattering with no discernible pattern whatsoever
   • C.A pattern only if TWO OR MORE photons pass through the slits at EXACTLY the same time, interfering with each other directly
   • D.The double-slit experiment cannot be performed with individual, well-separated single photons at all, it requires a continuous, classical wave of light
   Solution

   This is one of the most profound and genuinely puzzling results in quantum mechanics: even when photons are sent through a double-slit apparatus ONE AT A TIME (each individual photon detected as a single, discrete, localized point/event on the screen, exactly as a 'particle' would be expected to behave) -- when MANY such individual detection events are accumulated over a sufficiently long time, the resulting overall PATTERN of accumulated detection points gradually builds up to form the SAME interference pattern (alternating bright and dark bands) predicted by the WAVE theory for double-slit interference -- this remarkable result shows that EACH INDIVIDUAL photon somehow 'knows about' and is influenced by BOTH slits (behaving, in some sense, as if it were a wave passing through both slits simultaneously and interfering with itself), even though it is ultimately DETECTED as a single, localized particle-like event -- crucially, this pattern does NOT require multiple photons to be present simultaneously/interfering with EACH OTHER (ruling out option C) -- rather, it reflects each single photon's own inherent quantum-mechanical wave-like behaviour/probability distribution, a deeply counter-intuitive but experimentally well-confirmed feature of quantum mechanics that goes beyond simple classical notions of either 'wave' or 'particle' behaviour alone. A student who assumes individual, well-separated photons would show no pattern at all (a reasonable-seeming but experimentally INCORRECT classical-particle-based intuition) picks B, missing this genuinely profound quantum result.

4. Q4. The stopping potential in a photoelectric experiment is doubled when the incident frequency is changed from nu to 2nu. The threshold frequency of the cathode, in terms of nu, is

   • A.nu/2
   • B.nu/4
   • C.3nu/4
   • D.zero✓
   Solution

   Writing e V0 = h nu - h nu0 and 2 e V0 = 2h nu - h nu0 and subtracting gives e V0 = h nu, so h nu - h nu0 = h nu and nu0 = 0. Such a cathode is unphysical, which is exactly why doubling the frequency never simply doubles the stopping potential for a real metal.

5. Q5. A photon and an electron are found to carry exactly the SAME momentum, of magnitude 3.3x10^-27 kg m/s. The ratio (photon energy) : (electron kinetic energy) is of the order of

   • A.1
   • B.10^2
   • C.10^-5
   • D.10^5✓
   Solution

   For the photon E = pc = 3.3x10^-27 x 3x10^8 = 9.9x10^-19 J = 6.2 eV. For the electron K = p^2/2m = (3.3x10^-27)^2/(2 x 9.1x10^-31) = 6.0x10^-24 J = 3.8x10^-5 eV. The ratio is about 1.6x10^5, i.e. of order 10^5, because photon energy is linear in p while non-relativistic kinetic energy is quadratic.

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