Certain Uncertainty

Ever wondered how the Heisenburg Uncertainty Principle works? I mean, after so many textbooks and explanations, have you ever wondered what the following statement means:While the first term (ΔE) refers to an uncertainty in energy of the system, the second term (Δt) refers to the natural lifetime of the system in that state. And how would you use it?

Well here’s a classic example that is not often used in textbooks – consider the ground state of a molecule, and given that the ground state is the lowest energy state of the molecule, then the ground state must correspondingly be the most stable energy state of the molecule. That is the most basic postulate in statistical mechanics, where the lowest energy state is the most stable state.

Now, if we are given a molecule in the ground state, and provided that there is no perturbation (in the form of energy given to it via radiation, heat, etc.), then we can be very sure that the molecule will always stay in the ground state. We therefore say that the molecule stays in the ground state for an indefinitely long time unless energy is given to it to excite it.

Now, this would just mean that the natural lifetime of the ground state is:

Yes, in other words, we say that the lifetime is infinitely long! So what does the Uncertainty Principle tell us? It tells us that:

Indeed, we conclude that the uncertainty in the energy of the ground state must be zero! That is, the ground state is exactly known with a well defined energy! Our first conclusion must therefore be: the ground state of any molecule must therefore be a state of definite energy.

With this first concluding statement in mind, let us consider a typical quantum mechanical transition between two energy levels:

Now, we know that the ground state has a definite energy (G), as explained earlier, but what about the excited state (E)? Does it possess also a definite energy? Now, let us consider a fact, an experimental fact: excited molecules always decay back to the ground state, either via thermal collisions or re-radiation of electromagnetic radiation, giving rise to emission spectra lines.

This means that we must insist that the molecules don’t have an infinite life time, because if they did, then the molecules once excited, would never decay. Once again, we turn to the use of the Uncertainty Principle, where we see that:

This time round, the uncertainty in energy isn’t zero, but of a certain magnitude, depending on the lifetime of the excited state. Now, we should more properly depict the transition as:

Interesting! The transition is no longer as properly defined as it was earlier! This explains why spectroscopic graphs don’t possess infinitely sharp peaks, because this natural line broadening phenomenon takes place.

However, this equation has very often been misused, where the student very frequently mistakes the term ΔE for the energy gap between the excited state and the ground state, and simply substitutes this into the Uncertainty Principle to calculate the lifetime – notice that this isn’t correct, because the term ΔE more accurately refers to the blurring of the excited state, rather than the energy of transition itself.

Nevertheless, we can still say that as the energy of transition increases, the excited state is higher in energy, and thus less stable, which still translates to a smaller lifetime – it is for this reason that students fail to make the proper distinction between energy of transition, and energy of state.