We use time independent schrodinger equation to find Stationary state solution for some potentials. My question is that, these Stationary state solutions are physically reliable or not? I am asking this because these solutions provide us state of definite energy i.e, we can measure energy with zero uncertainty. But physically there must be some non-zero uncertainty associated with them.
There are many kinds of physical uncertainties, not all of them (directly) related to quantum mechanics. To name just a few:
- the fundamental quantum uncertainty
- the uncertainty due to the fact that our model only approximates the essential elements of the system that we study
- the uncertainty due to some objective limitations, like impossibility to write down and solve the equations for all the particles in the system (the subject of statistical physics), the fact that energy measurements done withing finite time, etc.
- Finally, there are uncertainties due to the limitations of the existing equipment (like the biggest available particle accelerator)
The fundamental difference between quantum mechanics and classical mechanics is in treating the first type of uncertainty. Where in classical mechanics measurements are theoretically possible with infinite precision (assuming that we have overcome other uncertainties), in quantum mechanics this is possible only in some situations. Stationary state is such a situation. This does not prevent other uncertainties from disturbing our measurements:
- the interactions mean that the actual energy is not the one that we calculated and that the stationary state decays
- the finite measurement time means that we are limited by the Heisenberg uncertainty principle ($\Delta E\Delta t \geq \hbar/2$).
- the imperfect equipment adds additional error bars