Demonstrate that the wavefunction of a free particle without mass, e.g. photon, solves the time dependent wave equation (which is in Europe called the Helmholtz equation but fails to do so for the time dependent Schrödinger equation
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The wavefunction of a free particle without mass, such as a photon, is typically represented by a plane wave solution of the form: Ψ(x,t) = A * exp[i(kx - ωt)] where A is the amplitude, k is the wave number, ω is the angular frequency, x is the position, Show more…
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The time-independent Schrödinger equation for a nonrelativistic free particle of mass $m$ is obtained from the energy relationship $E=p^{2} /(2 m)$ by replacing $E$ and $p$ with appropriate derivative operators, as suggested by the de Broglie relations. Using this procedure, derive a quantum wave equation for a relativistic particle of mass $m,$ for which the energy relation is $E^{2}-p^{2} c^{2}=m^{2} c^{4},$ without taking any square root of this relation.
Prove that a free electron cannot absorb a photon photoelectrically by showing that the conservation of energy and the conservation of momentum cannot be satisfied simultaneously in such a process.
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