Lifetimes of States and the Uncertainty Principle

As discussed in Sect. 2.1.2, the position and momentum operators do not commute the combined action of the two operators gives different results, depending on which operator is used first  

With some algebra, it follows from Eq. (2.65) that the product of the uncertainties (root-mean-square deviations) in the expectation values for position and momentum must be Hjl [4]. This is a statement of Heisenberg s uncertainty principle. 

The potential energy of a particle can be specified precisely as a function of position. However, the Hamiltonian operator H also includes a term for kinetic energy. Because kinetic energy depends on momentum, H does not commute withf  

There is no uncertainty principle comparable to the one for momentum and position that links the energy of a state with the state s lifetime. Indeed, there is no quantum mechanical operator for the lifetime of a state. There is, nevertheless, a relationship between the lifetime and our ability to assign the state a definite energy. One way to view this relationship is to recall that the full wavefunction for a system with energy is an oscillating function of time, and that the oscillation frequency is proportional to the energy (Eq. 2.16)  

According to this expression, if is constant the probability of finding the system in the state is independent of time (P = T T = Conversely, if a system remains in one state indefinitely we can specify its oscillation frequency (EJh), and thus its energy, with arbitrarily high precision. But if the particle can make a transition to another state the probability density for the initial state clearly must decrease with time. 

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