[Q] Time uncertainty and particle energy similar to photon

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SUMMARY

The discussion centers on the relationship between time uncertainty and particle energy, drawing parallels to photon energy. The user applied the operator \(i\hbar \frac{\partial}{\partial x}\) to an energy eigenstate \(e^{(i(kx-wt)}\), yielding an energy eigenvalue of \(\hbar w\), which mirrors the energy of a photon. The conversation also touches on the time-energy uncertainty principle, specifically the expression \(\Delta E \Delta t \geq \frac{\hbar}{2}\), questioning the implications of time uncertainty in quantum mechanics and the dispersion relation for massive particles versus photons.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly energy eigenstates.
  • Familiarity with the time-energy uncertainty principle in quantum physics.
  • Knowledge of the mathematical representation of wave functions, specifically \(e^{(i(kx-wt)}\).
  • Basic grasp of the dispersion relation for both massless and massive particles.
NEXT STEPS
  • Study the implications of the time-energy uncertainty principle in quantum mechanics.
  • Explore the differences in dispersion relations for photons and massive particles.
  • Investigate the role of electromagnetic fields as energy eigenstates in quantum theory.
  • Review Henk's lecture on quantum mechanics for deeper insights into energy eigenstates.
USEFUL FOR

Students and professionals in quantum mechanics, physicists exploring particle energy and time uncertainty, and anyone interested in the foundational concepts of quantum theory and its applications.

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Hi i applied i\hbar \frac{\partial}{\partial x} to energy eigen state which is

evolving in time, e^{(i(kx-wt)} and i get energy eigen value \hbar w <br /> <br />

but i was surprised it is very analoguos to energy of photon!

How can is it happened? Energy of photon is depending on time frequency of

electormagnetic field and Every of free particle is depending on time frequency of its energy

eigen state in time evolution.

Even i thought electromagnetic field is energy eigen state of photon. Is it right?


And i have second question about result [i\hbar \frac{\partial}{\partial x},t] = \hbar^2 and corresponding uncertainty \Delta E \Delta t \geq \frac{\hbar}{2}

What does means time uncertainty? time for what? We are not able to determine exact time

current state is in? we can't conclude t of e^{(i(kx-wt)} ?
 
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I think the dispersion relation is different. For a photon w~k, but for a massive particle w~k2, because the Hamiltonian contains the second derivative of position. See Henk's lecture 2: http://www1.mpi-halle.mpg.de/~henk/
 
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