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According to quantum mechanics, energy of a state is uncertain within a small range in hydrogen atom. But we also know that energy of a state is quantized which is contradictory to the first. Which one is correct?

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You can see the energy uncertainty in the following practical way: the energy of a level has a real and imaginary part. They both are "sharp" values. The time evolution of the given state describes a transition rate into another state. – Vladimir Kalitvianski Dec 7 '11 at 17:05
You're using the word "state" in two different ways here. – Harry Johnston Dec 7 '11 at 19:32
I meant a decaying factor $exp(-\gamma_{12})$ at the initial state $|\psi_1 |^2$ and a growing factor $[1-exp(-\gamma_{12})]$ at the final state $|\psi_2 |^2$ or so. – Vladimir Kalitvianski Dec 7 '11 at 22:10

The uncertainty refers only to states which stick around for a limited time. If you have a hydrogen atom form by capturing an electron, you have to wait a while before the H atom is in a sharply defined energy state. At early times, it is in a quantum superposition of all possible states. In order for a state to have definite energy, it must be undisturbed for a long time. But then the energy uncertainty goes to zero. The quantization is of this asymptotic long-time energy.

This topic is called the "time-energy uncertainty principle", and it has a long and inglorious history. The principle was the obvious time analog to the position-momentum uncertainty principle, and was appreciated as such by Bohr and Einstein. But the principle has some false formulations, like this one:

  • If a state has a definite energy, the time at which it has this energy is uncertain.

Landau said: "I can violate the time energy uncertainty principle by measuring the energy of a system very precisely, and then looking at my watch!" I think this is a very funny thing to say. It is a deep joke, because the time that a system is at is not a property of the system, but of Landau's watch. People sometimes tell this joke in a less droll way by saying "time is a parameter, not an observable", but this of course is in a particular formalism in which the parameter refers to watches, and the observables refer to system, and particle-time is matched to external proper time. But there are other formalisms in the world, some of which associate an internal ticking clock to the system.

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