A certain atom remains in an excited state for about 51.7 ns before emitting a 2.15-eV photon and transitioning to the ground state. What is the uncertainty in the frequency of the photon in Hz?

The uncertainty in the frequency of a photon can be determined using the Heisenberg Uncertainty Principle. In this case, the uncertainty in energy is related to the uncertainty in time. Calculating the uncertainty in frequency using the given data, the uncertainty in frequency is approximately 2.42 x 10^14 Hz.

Explanation:

The uncertainty in the frequency of a photon can be determined using the Heisenberg Uncertainty Principle. The principle states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as energy and time, can be measured simultaneously. In this case, the uncertainty in energy is related to the uncertainty in time. The minimum uncertainty in energy is given by the equation: ΔE ≥ h / Δt, where ΔE is the uncertainty in energy, h is Planck's constant, and Δt is the uncertainty in time.

In the given question, the atom remains in an excited state for about 51.7 ns before emitting a photon. The uncertainty in time (Δt) is therefore 51.7 ns. To find the uncertainty in frequency, we can use the fact that the energy of a photon is given by E = hf, where E is the energy, h is Planck's constant, and f is the frequency. Rearranging this equation, we have f = E / h. Substituting the given energy of the photon (2.15 eV) into the equation, we can find the frequency. Finally, the uncertainty in frequency (Δf) can be determined using the equation Δf = ΔE / h.

Calculating the uncertainty in frequency using the given data, the uncertainty in time is 51.7 ns (or 51.7 x 10-9 s). The energy of the photon is 2.15 eV. Substituting these values into the equations, we get:

E = 2.15 x 1.602 x 10-19 J
ΔE = E = 2.15 x 1.602 x 10-19 J
f = E / h = (2.15 x 1.602 x 10-19 J) / (6.626 x 10-34 J s)
Δf = ΔE / h = (2.15 x 1.602 x 10-19 J) / (6.626 x 10-34 J s)

Calculating these values, we find that the uncertainty in frequency is approximately 2.42 x 1014 Hz.

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