Heisenberg uncertainty principle (28-7)

Question 1 of 3

Question

The energy of a phenomenon is necessarily uncertain by an amount $\Delta{E}$ .

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Question

$\hbar = \frac{h}{2 \pi}$ = Planck’s constant divided by 2 $\pi$

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Question

The shorter the duration $\Delta{t}$ of the phenomenon, the greater the energy uncertainty $\Delta{E}$ .

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Review

Another way to interpret Equation 28-7 is to say that it places limits on how precisely the law of conservation of energy must be obeyed. Suppose a system undergoes some kind of process that lasts for a time $\Delta{t}$ . Equation 28-7 says that the energy of the system is necessarily uncertain during that process, and the minimum energy uncertainty is given by $\Delta{E} \Delta{t} = \hbar / 2$ or $\Delta{E} = \hbar / (2 \Delta{t})$ . So it’s fundamentally impossible to measure the energy of the system with an uncertainty less than $\hbar / (2 \Delta{t})$ . This means that during the process, the energy of the system could actually vary by as much as $\hbar / (2 \Delta{t})$ , and there would be no way that we could tell that the energy had changed value—that is, that the energy was not conserved. The shorter the duration $\Delta{t}$ of the process, the greater the amount $\hbar / (2 \Delta{t})$ by which energy conservation can be (temporarily) violated during that time $\Delta{t}$ . Stated another way, it’s acceptable to violate the law of conservation of energy by an amount $\Delta{E}$ , provided the duration of time during which the law is violated is no more than $\Delta{t} = \hbar / (2 \Delta{E})$ .