Why is Schrodinger's Equation complex?

  • #26
Delta2
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Nice dickforce, a more accurate derivation of what i originally posted. What we didnt notice is that if we look at the equation
[tex]
\begin{array}{rcl}

- i \, \left(\nabla \times \mathbf{X}\right) & = & \frac{1}{c} \, \frac{\partial \mathbf{X}}{\partial t} + \, \mathbf{J}_{X}
\end{array}
[/tex]

and we rewrite it by multiplying both sides by i and take the case of free space where the current density is zero we have :

[tex]

\nabla \times \mathbf{X}\ & = & i \, \frac{1}{c} \, \frac{\partial \mathbf{X}}{\partial t}

[/tex]

If we now replace the curl operator with the hamiltonian operator so that [tex] \mathbf{H} & = & c \hbar \nabla \times [/tex]

the equation becomes the schrodinger equation [tex] \mathbf{H} \mathbf{X}\ & = & i \, \hbar \frac{\partial \mathbf{X}}{\partial t} [/tex]

So schrodinger equation seems to be a straightforward generalization of the equation of electromagnetic field, with the hamiltonian operator replacing as a more generic operator the curl operator.
 
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  • #27
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So, there, I reduced the number of Maxwell's equations to half it's original number.
I'm quite surprised to see anyone else here on PF has taken a look at complex Maxell equations.

The number of Maxwell's equations can be reduced to zero where all that is left is the assignment of variables to measurable things, which is already implied in the original equations anyway. However, once you introduce complex fields, how do you expect to make the magnetic monopoles go away to correspond with know physical reality, except by demanding the phase in the complex plane is fixed?
 
  • #28
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Actually all six Maxwell equations can be rewritten as just one equation - by using the real Clifford algebra, for instance Cl(3,0).
 
  • #29
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Six?
 
  • #30
Delta2
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He probably refers to
[tex] \mathbf{F} = \left ( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right ) q [/tex]

[tex] \mathbf{J} = \sigma \mathbf{E} [/tex]

as the additional 2 equations.
 
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  • #32
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Six or twelve, six complex, twelve real, twenty two "original" - what's the difference, the same order :smile:
No, F_{uv} are simply the electric and magnetic field strengths.

Define G=*F. Then dG=J, and dF=0 are Maxwell's equations. G and F are skew symmetric two-forms. d is the exterior derivative, * is the Hodge duality operator. If, however, F is defined as dA, then dF=0 is a mathematical identity (All exact forms are closed.), and the number of independent tensor equations reduces to one.


But it hardly can be less than one.
You'll have to try harder to find the reduction.

An otherwise arbitrary trice differentiable 4-vector field, A is imposed on a pseudo-Riemann manifold. A is the electromagnetic dual 4-vector potential, dA are the electric and magnetic fields, d*dA are the electric current 4-densities, ddA are the magnetic 4-current densities. There are no equations here.

d*dA expresses Ampere's Law and Gauss' Law, and ddA expresses Gauss' law for magnetism and the Maxwell-Faraday equation under the correct identification of the tensor element derivatives of A.

For all this to hang together a subtle ontological distinction is made. It is tacitly assumed that no equivalence relationship is implied between distinct physical elements such as J==d*dA, but that current and charge density are simply aspects of the vector potential.

Where A is complex, additional terms are introduced for currents, d*dA --> d(*+i)dA, where i=sqrt(-1). The wave equation and charge continuity also change.
 
  • #33
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Did you forget the Lorentz condition d*A=0?

And another comment: I understand that for you General Relativity has no equations at all (thus less than one), because Einstein's equations are nothing else but the definition of the RHS? Right?
 
  • #34
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Did you forget the Lorentz condition d*A=0?
Again, with the wild cards. Why don't you tell why you think the Lorentz gauge is relevant. Recall that the expressions I gave above are differential not integral.
 
  • #35
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The number of Maxwell's equations can be reduced to zero where all that is left is the assignment of variables to measurable things, which is already implied in the original equations anyway.
Please justify this statement.

However, once you introduce complex fields, how do you expect to make the magnetic monopoles go away to correspond with know physical reality, except by demanding the phase in the complex plane is fixed?
The phase of what?
 
  • #36
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Please justify this statement.
I think I did that in post #32. If there is anything there that needs clarification, I might supplement. But I think what Delta2 brought up in post #26 is more interesting and to the topic at hand.

The phase of what?
The phase of the complex 4-potential. Going to a complex 4-vector potential we can gauge fix the phase of the vector to be real valued over all spacetime. This is a global gauge fixing. Then the complex equations reduce to Maxwell's equations and magnetic charge is fixed to zero, everywhere (and everywhen).

I believe a global phase factor is an unmeasurable quantity. A = A' <-- A ephi
 
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  • #37
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Again, with the wild cards. Why don't you tell why you think the Lorentz gauge is relevant. Recall that the expressions I gave above are differential not integral.
Lorentz gauge is not necessarily relevant in general. But here we are discussing Maxwell equations in the context of wave equations. Wave equations, similar to Schrodinger's equation, usually describe quantum mechanics of elementary particles. Photon supposedly is a spin 1 particle. Without Lorentz gauge we also have longitudinal spin zero states.
 
  • #38
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I think I did that in post #32. If there is anything there that needs clarification, I might supplement. But I think what Delta2 brought up in post #26 is more interesting and to the topic at hand.
For all this to hang together a subtle ontological distinction is made. It is tacitly assumed that no equivalence relationship is implied between distinct physical elements such as J==d*dA, but that current and charge density are simply aspects of the vector potential.
I'm afraid you had mistaken the cause and effect. J IS the source for d*dA, not the other way around. All currents are produced by moving charged particles in a unique manner. Consequently, external fields act on these particles with a Lorentz force. Your model fails when you try to write an equation of motion for a charged particle in an external field.

The phase of the complex 4-potential. Going to a complex 4-vector potential we can gauge fix the phase of the vector to be real valued over all spacetime. This is a global gauge fixing. Then the complex equations reduce to Maxwell's equations and magnetic charge is fixed to zero, everywhere (and everywhen).

I believe a global phase factor is an unmeasurable quantity. A = A' <-- A ephi
If you had looked at the equations more carefully, you would have noticed that the sourceless equations (Gauss' Law and Faraday's Law) now have a non-zero right hand side (due to monopoles). This is why the concept of electrodynamic potentials in the usual sense of the word does not have a straightforward generalization.
 
  • #39
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I'm afraid you had mistaken the cause and effect. J IS the source for d*dA, not the other way around. All currents are produced by moving charged particles in a unique manner. Consequently, external fields act on these particles with a Lorentz force. Your model fails when you try to write an equation of motion for a charged particle in an external field.
It is convenient to begin with a distribution of charge and work out the fields. This convenience does not imply a physically measurable cause and effect such as "charge causes electromagnetic fields," or that "the electromagnetic fields cause the vector potential." I'm open to proof or motivation to this conjecture.

If you had looked at the equations more carefully, you would have noticed that the sourceless equations (Gauss' Law and Faraday's Law) now have a non-zero right hand side (due to monopoles). This is why the concept of electrodynamic potentials in the usual sense of the word does not have a straightforward generalization.
No, a complex vector potential admits magnetic monopoles.
 
  • #40
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It is convenient to begin with a distribution of charge and work out the fields. This convenience does not imply a physically measurable cause and effect such as "charge causes electromagnetic fields," or that "the electromagnetic fields cause the vector potential." I'm open to proof or motivation to this conjecture.
But, it's not the question whether electromagnetic fields cause a vector potential, but whether electromagnetic fields cause charges and currents. If you don't think this stance is absurd, I don't have what to say anymore.

No, a complex vector potential admits magnetic monopoles.
Please show your work.
 

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