Physics · Ch 7 — Dual Nature of Radiation and Matter
Concept of quantization of energy
Concept of quantization of energy
Having established the five experimental laws of the photoelectric effect, the natural next step is to try to explain them using the classical wave theory of light -- and this attempt fails badly, on three separate counts.
First, according to wave theory the energy carried by light is spread out continuously and uniformly across the wavefront, so a brighter (more intense) light should deliver more energy to each electron and should therefore free electrons with a greater kinetic energy. But experiment (law 2) shows the maximum kinetic energy of the photoelectrons does not depend on intensity at all -- intensity only changes how many electrons are emitted, never how energetic each one is.
Second, wave theory places no restriction on frequency at all: given a sufficiently intense beam, electrons should eventually be liberated however low the frequency, since there is always enough energy available if you wait or turn up the intensity. But experiment (law 4) shows photoelectric emission simply does not occur below a definite threshold frequency, no matter how intense the light is made.
Third, since wave theory spreads the incident energy thinly and continuously over a large number of electrons across the wavefront, each individual electron should need a considerable amount of time -- calculations put this at several hours to several days for realistic intensities -- to slowly accumulate enough energy to escape. But experiment (law 5) shows photoelectric emission is essentially instantaneous, with a time lag of well under a nanosecond.
The resolution came from Max Planck, who in 1900 proposed -- originally to explain the shape of black-body radiation curves, not the photoelectric effect -- that matter is made of a huge number of oscillating atomic particles, each vibrating at its own characteristic frequency and each emitting or absorbing electromagnetic radiation only at that same frequency. Planck's key new idea was that such an oscillator's energy cannot take just any value; it is quantized, restricted to a discrete ladder of allowed values
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Worked out. For photoelectric emission from caesium, with a work function of 2.14 eV and an incident power per unit area of (just enough to give a measurable photocurrent), the worked solution shows precisely how the classical wave picture of light fails on all three fronts the text raises. First, treating the incident energy as spread uniformly and continuously over the wavefront and absorbed by a single atom's cross-sectional area over a time gives : since can simply be increased without limit, wave theory predicts should keep growing with intensity, with no natural threshold frequency at all -- directly contradicting the experimentally observed threshold-frequency law. Second, replacing by a quantity proportional to the squared electric-field amplitude in this same expression shows would not depend on frequency at all under wave theory, again contradicting the observed straight-line -versus- relation. Third, using a typical atomic radius m to estimate how long it would take a single atom to accumulate enough wave-spread energy to just overcome the work function gives a time delay of roughly s, about 79 days -- wildly inconsistent with the experimentally observed, essentially instant …