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When only a few photons are observed, classical physics fails. When many are observed, classical physics is valid. Which of these two facts is consistent with the correspondence principle?

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2. A free particle (a particle that has zero potential energy) has mass 8 eV/c² and total energy 10 eV and is traveling to the right. At x = 0, the potential jumps from zero to Vo = 5 eV and remains at this value for all positive x. (a) In classical mechanics, what happens to the particle when it reaches x = 0? (b) What is the wavenumber of the quantum particle in the region x > 0? (c) Find the reflection coefficient R and the transmission coefficient T for the quantum particle. (d) If one million particles with this same momentum and energy are incident on this poten- tial step, how many particles are expected to continue along in the positive x direction? How does this compare with the classical prediction?
In quantum interpretation of the electromagnetic waves in vacuum the photon has the energy E = hw/2n and the momentump= hk/2n. So the ratio E of the energy to the momentum is = c, the speed of light in k vacuum. Similar relation can be obtained in classical electrodynamics as follows. Consider the time-averaged energy density of the electromagnetic field ɛ = B² /2µ0 + €0 E² /2 for a plane wave propagating in vacuum along the z-direction. Calculate the time-averaged energy flux (Poynting flux) for such a wave S = E × H and confirm that its ratio to the time-averaged energy density ratio is equal to the speed of light in vacuum, S/e = c.
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