Explain the connection between Planck's hypothesis of energy quanta and the energies of the quantum harmonic oscillator.

The connection between Planck's hypothesis of energy quanta and the energies of the quantum harmonic oscillator lies in the quantization of energy. Planck's hypothesis states that energy can only exist in discrete packets called quanta, formulated as \(E = h\nu\), where \(h\) is Planck's constant and \(\nu\) is the frequency of radiation. The quantum harmonic oscillator, which models oscillatory systems, has quantized energy levels described by \(E_n = \hbar\omega (n + \frac{1}{2})\), where \(\hbar\) is the reduced Planck's constant, \(\omega\) is the angular frequency, and \(n\) is the energy level. The quantized energy levels in the quantum harmonic oscillator are consistent with Planck's hypothesis, as both concepts involve specific discrete energy levels, with Planck's constant playing a crucial role. When a quantum harmonic oscillator transitions between energy levels, it absorbs or emits energy in discrete quanta, aligning with Planck's hypothesis of energy quantization.