Finding an energy of a particle in an infinite potential well

Question

This question arises from a discussion I recently had with my friend. We were talking about a particle in an infinite potential well. The particle is in an arbitrary wavefunction $\Psi$. When one measures the energy of the particle, then he gets a specific value $E_1$, which is an eigen value corresponding to an eigen-state of the particle $\psi_1$, to which the system has been forced to be in by making the measurement. The part that tricked us is: How can anyone know with definite certainty what the value of energy is for a quantum particle? Shouldn't there exist some uncertainty in the value of $E$?

Answer 1

Do you refer to the energy-time principle involving the product $\Delta E \Delta t?$ Here, $\Delta E$ is the resulting standard deviation you obtain on measuring $E$ in an infinite number of identically prepared quantum systems. In the case of the system in an eigenstate of energy $E$ (through e.g a collapse as per the Copenhagen interpretation), you will always measure $E$ and so here the standard deviation $\Delta E = 0$.

Now, to satisfy the principle constraint it means $\Delta t$ is infinite. As emphasised within this thread,

What is $\Delta t$ in the time-energy uncertainty principle?

this is a reflection that there is no finite time in which one will measure a deviation in the energy of the state.

Answer 2

My advice is to remember at all times that quantum mechanics is just a set of rules for building mathematical models of reality, so some of its predictions may be features of the model rather than features of reality.

The QM model of a particle in an infinite potential well predicts a set of exact allowed energy values, corresponding to a set of eigenfunctions. The Copenhagen interpretation says that a measurement of the energy of the particle will cause its wave function to be one of the allowed eigenfunctions, and its energy to be the corresponding one of the exact energy values- no uncertainty about that whatsoever. Of course your experiment might not be exact so the reading you see on a dial somewhere might be slightly off the exact allowed value. And, of course, when you set-up your potential well, you might have made the width of the well slightly more or less than the value you plugged into the formula to calculate the allowed energy values. So the theory predicts exact answers, but reality is something else.