Question

How to relate the wavenumber to the momentum and the energy of a electromagnetic wave?

Answer 1

When we are solving the electromagnetic wave equation,

∂2E∂x2=1c2∂2E∂t2,∂2E∂x2=1c2∂2E∂t2,

by separation of variables, that is, by assuming that the solution has the form E(x,t)=χ(x)T(t)E(x,t)=χ(x)T(t), we have to introduce a constant kk:

∂2χ(x)∂x2−k2χ(x)=0∂2χ(x)∂x2−k2χ(x)=0

∂2T(t)∂t2−k2c2T(t)=0∂2T(t)∂t2−k2c2T(t)=0

and we end up with solutions of the form exp[ik(x±ct)]exp⁡[ik(x±ct)]. This constant kk is given by k=2π/λk=2π/λ is related to the energy ϵϵ of the wave through

k=ϵℏc.k=ϵℏc.

How is this relation established? What additional information do you have to bring in to relate this constant you just brought up as a helping hand in solving the wave equation to the momentum and the energy of the wave

Thank you very much.

Answer 2

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✔️✔️∂2χ(x)∂x2−k2χ(x)=0∂2χ(x)∂x2−k2χ(x)=0 ∂2T(t)∂t2−k2c2T(t)=0∂2T(t)∂t2−k2c2T(t)=0

or


we end up with solutions of the form exp[ik(x±ct)]exp⁡[ik(x±ct)].

This constant kk is given by k=2π/λk=2π/λ is related to the energy ϵϵ of the wave through k=ϵℏc.k=ϵℏc.


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