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Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.
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Where does a LED use energy other than emitting light?
What your formula is actually meant to give is the emission wavelength as a function of the material bandgap:
$$\lambda \approx \frac{\rm1240\,eV\,nm}{E_{gap}}.$$
In your example, when carriers reco …