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The time-independent Shrödinger equation (TISE) is

$$\dfrac{-\hbar^2}{2m} \nabla^2 \psi + V \psi = E \psi,$$

where $m$ is the mass of the particle.

I just had a thought: If $m$ is the particle's mass, then the TISE is invalid for photons, since they're massless! But I know that the Shrödinger equation is used to model photons. So how can this be?

The Pointer
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  • This is the non-relativistic (time-independent) Schrodinger equation, photons are a relativistic phenomena, one uses relativistic quantum mechanics/quantum field theory to deal with photons. – bolbteppa Jun 03 '20 at 18:21
  • So non-relativistic quantum mechanics cannot model light? And are you saying that relativistic quantum mechanics and quantum field theory are actually synonymous? – The Pointer Jun 03 '20 at 18:24
  • quantum field theory obeys the postulates of quantum mechanics and uses the relativistic version of the eguations (Dirac, Klein Gordon) to model the fields. It extends to many particle interactions instead of just potentials. – anna v Jun 03 '20 at 18:38
  • But I know that the Shrödinger equation is used to model photons. Where did you learn that? – G. Smith Jun 03 '20 at 19:03
  • @G.Smith I think the language from my textbook confused me: "In our case we are interested in both electrons and photons as particles. For photons this description is roughly equivalent to standard electromagnetic theory where the wavefunction is analogous to a normalized electric field. Maxwell’s equations give the description of photon fields. In this appendix we shall focus more specifically on the properties of electrons." This was written in the section on quantum mechanics, so it seemed to me that it implied that the equations were valid for photons. – The Pointer Jun 03 '20 at 19:13
  • It explicitly says “Maxwell’s equations give the description of photon fields” so I don’t see why you would think instead that the Schrodinger equation does. – G. Smith Jun 03 '20 at 19:20
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    @ThePointer A comment about language: Often the name "Schrödinger equation" refers specifically to a strictly non-relativistic model of the type shown in the question. Such a model cannot account for photons. But where I come from, the name "Schrödinger equation" is also used more generally for any equation of the form $id/dt|\psi(t)\rangle=H|\psi(t)\rangle$, where $|\psi(t)\rangle$ is the state-vector and $H$ is a Hamiltonian. This general equation holds even in relativistic QFTs (in the Schrödinger picture, with an appropriate Hamiltonian), and it can model photons. – Chiral Anomaly Jun 03 '20 at 22:54
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    @ChiralAnomaly, I believe Sredniki calls $H|\psi(t)\rangle=i\frac{d}{dt}|\psi(t)\rangle$ the abstract SE which does, I think, help to keep things straight. – Alfred Centauri Jun 04 '20 at 15:59

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The Schrödinger equation can not be used to model photons. In order to understand why, I think it is useful to use the effective field theory machinery here, since Shcrödinger-like equations can be obtained as the "equation of motion" of Non-Relativistic Quantum Electrodynamics theories (NRQED). In these theories, in the reference frame where the massive particle is at rest, the leading order operator is of the form (in the abscence of EM field):

$$i\dfrac{\partial}{\partial t}+\dfrac{\hbar^2}{2m}\nabla^2$$

Which is the Kinetic part of the Schrödinger equation. You can also add interactions. I recommend you look at any recent review of NRQED at the arxiv, such as this one.

vin92
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