How to Derive the Wave Equations for Photons?

[jb646]

Homework Statement

Use the relativistic expression for energy E^2=p^2c^2+(m_0)^2(c)^4 to find a wave equations for photons. Find a solution for ψ and compare to the electric field (hint: photons are massless, E_op=ih(d/dt) and p_op=h/i(d/dx)

Homework Equations

the only equations i know are the ones given in the problem

The Attempt at a Solution

if somebody could please point me in the right direction, i do not have the mental power to understand what i should even try to do first. Thanks for any help you can provide.

 

[dextercioby]

Photons are quanta of fields, it doesn't really work the way your problem suggests.

The differential equation for the quantized electromagnetic field is obtained by applying a canonical quantization* to the classical electrodynamic equation for the one-forms describing the field at a classical level.

* very tricky issue.

 

[Redbelly98]

Homework Statement

Use the relativistic expression for energy E^2=p^2c^2+(m_0)^2(c)^4 to find a wave equations for photons. Find a solution for ψ and compare to the electric field (hint: photons are massless, E_op=ih(d/dt) and p_op=h/i(d/dx)

Homework Equations

the only equations i know are the ones given in the problem

The Attempt at a Solution

if somebody could please point me in the right direction, i do not have the mental power to understand what i should even try to do first. Thanks for any help you can provide.

Okay, if E_op=ih(d/dt), then what would E_op² be?

Photons are quanta of fields, it doesn't really work the way your problem suggests.

The differential equation for the quantized electromagnetic field is obtained by applying a canonical quantization* to the classical electrodynamic equation for the one-forms describing the field at a classical level.

* very tricky issue.

I fail to see how that is helpful.

 

[dextercioby]

I fail to see how that is helpful.

The problem is wrong. The photon has no wave equation. The electric field has a field equation, but in the classical sense, as it can be deduced from Maxwell's equations (for simplicity, in vacuum).

 

[paranormal]

I would approach this problem by first understanding the equations given in the problem. The relativistic expression for energy, E^2=p^2c^2+(m_0)^2(c)^4, is known as the energy-momentum relation and it describes the total energy of a particle (including its rest mass energy) in terms of its momentum and the speed of light. The other equations, E_op=ih(d/dt) and p_op=h/i(d/dx), are related to the properties of operators in quantum mechanics.

To find a wave equation for photons, we need to use the energy-momentum relation and the properties of operators. Since photons are massless particles, we can set m_0=0 in the energy-momentum relation. This gives us E=pc, where p is the momentum of the photon. We can also use the properties of operators to write the energy and momentum operators in terms of the electric field operator (E_op=ih(d/dt)) and the momentum operator (p_op=h/i(d/dx)).

Substituting these expressions into the energy-momentum relation, we get (E_op)^2=(p_op)^2c^2. Now, we can use the properties of operators to rewrite this equation as (ih(d/dt))^2=(h/i(d/dx))^2c^2. Simplifying this equation, we get -(d^2/dt^2)=(d^2/dx^2)c^2. This is the wave equation for photons, which is a second-order differential equation in both time and position.

To find a solution for ψ, we need to solve this wave equation. This can be done using standard techniques from differential equations. Once we have a solution for ψ, we can compare it to the electric field operator. This comparison will give us insights into the wave-like nature of photons and how they are related to the electromagnetic field.