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- Interferometry and Synthesis in Radio Astronomy;
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Even for perfect measuring devices, these uncertainties would remain because they originate in the wave-like nature of matter. Interestingly, the Gaussian function or bell-curve distribution gives the minimum value of the uncertainty product:.
Principles of Modern Physics
Unlike the position uncertainty for the electron, the position uncertainty for the bowling ball is immeasurably small. Hint : According to early experiments, the size of a hydrogen atom is approximately 0. The ground-state wavefunction of this system is a half wave.
Note that this function is very similar in shape to a Gaussian bell curve function. This average energy of a particle is related to its average of the momentum squared, which is related to its momentum uncertainty. For the Gaussian function, the uncertainty product is.
The estimated ground state energy is therefore. Based on early estimates of the size of a hydrogen atom and the uncertainty principle, the ground-state energy of a hydrogen atom is in the eV range.
The ionization energy of an electron in the ground-state energy is approximately 10 eV, so this prediction is roughly confirmed. Another kind of uncertainty principle concerns uncertainties in simultaneous measurements of the energy of a quantum state and its lifetime,.
The energy-time uncertainty principle does not result from a relation of the type expressed by Equation for technical reasons beyond this discussion. Nevertheless, the general meaning of the energy-time principle is that a quantum state that exists for only a short time cannot have a definite energy. The reason is that the frequency of a state is inversely proportional to time and the frequency connects with the energy of the state, so to measure the energy with good precision, the state must be observed for many cycles.
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To illustrate, consider the excited states of an atom. The finite lifetimes of these states can be deduced from the shapes of spectral lines observed in atomic emission spectra.
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Each time an excited state decays, the emitted energy is slightly different and, therefore, the emission line is characterized by a distribution of spectral frequencies or wavelengths of the emitted photons. As a result, all spectral lines are characterized by spectral widths. The average energy of the emitted photon corresponds to the theoretical energy of the excited state and gives the spectral location of the peak of the emission line.