91Ó°ÊÓ

The Uncertainty Principle, a cornerstone of quantum mechanics, states a fundamental limit on the precision with which certain pairs of physical properties, like energy and time, can be simultaneously known. The formula \[\Delta E \cdot \Delta t \approx \hbar\]expresses this relationship. Here, \(\Delta E\) is the uncertainty in energy and \(\Delta t\) is the time interval during which this uncertainty is present. This interval can be the lifetime of an excited state. For our problem, the lifetime (or \(\Delta t\)) of the atom's excited state is given as 10 nanoseconds (ns).To compute the energy uncertainty \(\Delta E\), we simply substitute the given values into the formula: - \(\hbar\), the reduced Planck's constant, is approximately \(1.0545718 \times 10^{-34}\, \text{Js}\).- \(\Delta t = 10 \times 10^{-9}\, \text{s}\).The calculation gives \(\Delta E = 1.0545718 \times 10^{-26}\, \text{J}\). This tells us that over the 10 ns period, there's an inherent uncertainty in the energy measurement of the emitted photon by this amount.

Photon energy is determined by the photon's wavelength. The relationship between energy \(E\) and wavelength \(\lambda\) is given by the equation: \[E = \frac{hc}{\lambda}\]where:- \(h\) is Planck's constant, approximately \(6.62607015 \times 10^{-34}\, \text{Js}\).- \(c\) is the speed of light, about \(3 \times 10^8\, \text{m/s}\).- \(\lambda = 500\, \text{nm} = 500 \times 10^{-9} \text{m}\).Substituting these values into the equation, we find the energy of the photon to be approximately \(3.9724 \times 10^{-19}\, \text{J}\). This means each photon released by the atom when it falls from the excited state carries this amount of energy.It's crucial to understand how these small quantities of energy integrate into larger systems. In quantum mechanics, energy quantization underscores how fundamental processes scale up to affect macroscopic phenomena.

The uncertainty in the wavelength of a photon is intrinsically linked to the energy uncertainty through their inverse relationship. The formula linking them is:\[\Delta \lambda = \frac{\lambda^2 \Delta E}{hc}\]This equation shows how changes in energy can influence the precision in the measurement of wavelength.Given that \(\lambda = 500 \times 10^{-9} \text{m}\) and \(\Delta E = 1.0545718 \times 10^{-26} \text{J}\), we find that \(\Delta \lambda\) is approximately \(6.63 \times 10^{-12} \text{m}\).The fractional wavelength uncertainty is calculated as:\[\frac{\Delta \lambda}{\lambda} = \frac{6.63 \times 10^{-12} \text{m}}{500 \times 10^{-9} \text{m}} \approx 1.326 \times 10^{-5}\]This result reflects the small but significant influence that energy-time uncertainty can have on our ability to measure photon characteristics, highlighting the interconnectedness of quantum properties.