Complex Trajectories: Quantum Mechanics and Lagrangians

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In summary, the Schrodinger equation can be expressed as F=expi/hbar(S+hbar/iLn[R]), where R is a real number formed by setting R=F*.F. This implies the relationship Pq[F>=Pc+hbar/igra[R][R>, where Pq is the quantum operator associated with momentum, Pc is the classic momentum, and hbar is Planck's constant h/2pi. From this, it can be concluded that particles have trajectories in the complex plane and these trajectories are given by extremizing the Lagrangian L=L0+V+hbar/igra[R]/[R], where L0 is the free Lagrangian and V is the potential. The concept of complex
  • #1
eljose79
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in fact you can e that any solution of the schroedinguer equation can be put in the form F=expi/hbar(S+hbar/iLn[R] where R is the real number formed by setting R=F*.F so this implies the relationship
Pq[F>=Pc+hbar/igra[R][R> for any R so we would have the equality
Pq= quantum operator asociated to momentum
Pc=classic momentum
hbar= Planck,s h/2pi

Pq=Pc+hbar/igra[R] so with this i conclude that:
a)particles have trajectories in the complex plane
b)the trajectories are given by extremizing the lagrangian
L=L0+V+hbar/igra[R]/[R], with L0 the free lagrangian V the potential

What do you think?.
 
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I didn't check your math, so I don't know if what you did is valid. The question of complex velocities has arisen before. It brings up an interesting possibility. There really is no ban on superluminal velocities. The "impossibility" is movement at the speed of light. It has always been assumed that superluminal velocities are also banned because an object would need to accelerate through c to get to them. But if we allow complex velocities, one could accelerate "around" c to go superluminal.

Njorl
 
  • #3
If there is no ban on super-liminal velocities, would a particle be constrained to either super-liminal, or sub liminal velocity. Or could it tunnel across the light barrier ? It cannot cross the light barrier because it needs infinite mass or infinite accelleration.

If subatomic particles were traveling at above light speed, then at sqrt(2)c their mass would be j, compared to 1 at v=0. In fact we should consider their mass/energy to be a vector with angle 0, and their mass/energy above light speed to be a vector with angle pi/2.

Would we oberve the mass/energy of a super-liminal particle as the length of its mass vector, or would be unable to observe it ?

Actually such a particle would have complex mass, travel a complex distance, take a complex amount of time (actually from Einstein's equation I don't get negative time which is what I grew up beleiving) but it would travel at a real velocity which happens to be above c, between two points in real time space.
 

1. What is the significance of complex trajectories in quantum mechanics?

In quantum mechanics, particles are described by wave functions that can have complex values. These complex wave functions can be used to describe the probability of a particle being in a certain location at a certain time. This allows for the prediction of complex trajectories, which are the paths that particles may take through space and time.

2. How are complex trajectories related to Lagrangians?

Lagrangians are mathematical functions that describe the dynamics of a system. In the context of quantum mechanics, they can be used to describe the complex trajectories of particles. The Lagrangian is a key component in the Schrodinger equation, which is the fundamental equation of quantum mechanics.

3. Can complex trajectories be observed in experiments?

No, complex trajectories cannot be directly observed in experiments. In quantum mechanics, the concept of measurement collapses the wave function, meaning that the particle's exact position and trajectory cannot be observed. However, the probability of a particle being in a certain location can be measured, which is related to the complex trajectory.

4. Are there any real-world applications of complex trajectories?

Yes, complex trajectories have several real-world applications, particularly in the fields of quantum computing and quantum cryptography. Complex trajectories are also utilized in the design of quantum algorithms and in understanding the behavior of quantum systems.

5. How do complex trajectories differ from classical trajectories?

In classical mechanics, the trajectories of particles are described by Newton's laws of motion, which are deterministic and can be predicted with certainty. However, in quantum mechanics, the complex trajectories of particles are described by the probabilistic nature of wave functions, meaning that the exact path of a particle cannot be determined. Additionally, classical trajectories are described by real numbers, while complex trajectories involve complex numbers.

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