A certain atom has an energy level 3.50 eV above the ground...

A certain atom has an energy level 3.50 eV above the ground state. When excited to this state, it remains $4.0 \mu s,$ on the average, before emitting a photon and returning to the ground average, before emitting a photon and returning to the ground state. (a) What is the energy of the photon? What is its wavelength? (b) What is the smallest possible uncertainty in energy of the photon?

Key Concepts

Quantum Energy Levels

In quantum mechanics, atoms have electrons that occupy discrete energy levels rather than a continuous spectrum of energies. Transitions between these quantized levels occur when an electron absorbs or emits a specific amount of energy, often resulting in the emission or absorption of a photon. This concept forms the basis for understanding atomic emission spectra and the energy changes involved during electron transitions.

Photon Energy and Wavelength Relationship

The energy of a photon is directly related to its frequency and inversely related to its wavelength. This relationship is articulated by the equation E = hc/?, where h is Planck's constant and c is the speed of light. It allows one to convert between the energy of an emitted or absorbed photon and its corresponding wavelength, a fundamental principle in the study of electromagnetic radiation.

Time-Energy Uncertainty Principle

The Heisenberg uncertainty principle, when applied to time and energy, indicates that there is a fundamental limit to the precision with which the energy of a quantum state can be known if the lifetime of that state is finite. This principle is expressed as ?E?t ? ?/2, meaning that a shorter-lived excited state will inherently have a greater uncertainty in its energy. This concept is critical in understanding phenomena such as spectral line broadening in atomic transitions.

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