# Does the energy-time uncertainty principle require energy levels to have finite width?

The uncertainty principle also has the form: $\Delta$$E$$\Delta$$t>h/2\pi Now this should mean that the thickness of the lines we draw in the energy level diagrams to show energy change undergone by atoms and electrons cannot be a single straight line. Then that should mean that E have a specific value so \Delta$$E=0$ and so $\Delta$$t tends to infinity. Which is rather ridiculous cause that would mean that the particle(atom or electron) forever stays in some specified state and such transitions which are mentioned in the Bohr’s postulates, would be impossible. So such lines must be of certain thickness to illustrate the fact that \Delta$$E$ should have a finite value, as in, E cannot be specified at a given time.

Is my conclusion following the reasoning correct?

• Of course most atoms are stable if not disturbed, i.e. they will not spontaneously decay and that would make very very thin lines. – anna v Apr 16 '14 at 18:05
• It would be great if you could explain what is it you mean by $E$ and $\Delta E$. If $E$ is Hamiltonian eigenvalue, this is a real number and has no "width". If $E$ is average expected value of Hamiltonian and $\Delta E$ is square root of its variance for some function $\psi$, it is still not clear what is $\Delta t$ and how your inequality was derived. If you mean something else, could you please explain what it is? – Ján Lalinský Apr 16 '14 at 19:49

Yes, you're right ! The uncertainty principle tells us that the thickness of the energy state $\Delta E$ is linked to the typical decay time of this energy level. If $\Delta E$ is large then $\Delta t \sim \tau$ (decay constant of the energy level) is small and then this energy state is very unstable.