Time-Energy Relation

Question

The uncertainty of energy and time and their inter-relatedness is derived in the second form of Heisenberg's Uncertainty Principle where ∆T∆E ≥ h/2π. In quantum scales, energy can be exchanged for time. Thus implying something more fundamental that balances these two concepts.

If the "likelihood" (as defined in statistics) of an event increases with either greater energy for work (Boltzmann distribution, LHC, i.e higher probability (P)) or greater time (i.e. higher outcomes (x))... Do time and energy share something fundamental?

(Not directly related but remarkable, the equation ∆T∆E ≥ h/2π changes in value under different conditions. Which could also imply a third hidden variable that shares a hypothetical fundamental common denominator. Could it possibly be quantum information?)

Answer 1

Perhaps the first thing for you should be to get a classical intuition for this. Wavefunctions in quantum mechanics have many similarities to classical oscillating waves, including the typical time dependence factor $e^{i \omega t}$. This factor describes a wave oscillating at angular frequency $\omega$. Also remember, in quantum mechanics, the energy of a wave is directly related to its frequency: $E=\hbar \omega$.

Now if you are familiar with Fourier transforms, this will be easy to understand. The term $e^{i \omega t}$ describes a single frequency of oscillation (i.e. $\Delta E=0$), but if you plot this wave, you'll notice that it oscillates forever in time ($\Delta t= \infty$). Generally speaking, if you want to describe any wave which is a short pulse in time (meaning $\Delta t$ is small), you need to sum up a large number of waves with a broad range of frequencies (meaning $\Delta E$ is large), and vice versa. For waves, there is a fundamental trade-off between time and frequency. As mentioned before, this is a consequence of Fourier transforms. Since quantum mechanics describes particles as waves oscillating in time, they have the same mathematical property, resulting in the uncertainty principle.

This is the fundamental connection between time and energy: $E=\hbar \omega$ along with the time dependence factor $e^{i \omega t}$.