The energy levels of a quantum harmonic oscillator are given by En = n + 1 2 ¯h ω where 2π ¯h = h and h is Planck’s Constant. The energy o

Question

The energy levels of a quantum harmonic oscillator are given by En = n + 1 2 ¯h ω where 2π ¯h = h and h is Planck’s Constant. The energy of a quantum of electromagnetic radiation (a photon) is Eλ = h c λ where λ is the wavelength of the radiation and c is the speed of light. If a harmonic oscillator transitions down one energy level, what is the wavelength of electromagnetic radiation emitted?

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MichaelMet 4 years 2021-07-13T15:19:46+00:00 1 Answers 2 views 0

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  1. Tryphena
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    2021-07-13T15:21:14+00:00
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    Answer:

    2πc/w

    Explanation:

    To find the wavelength you take into account the difference in energy of two adjacent states n+1 and n:

    E_{n+1,n}=\hbar \omega((n+1)+\frac{1}{2})+\hbar \omega(n+\frac{1}{2})\\\\E_{n+1,n}=\hbar \omega(1)

    hbar = h/2π

    this energy is also the energy of an emitted photon in the transition, that is:

    E_{\lambda}=h\frac{c}{\lambda}   (2)

    you equal the equations (1) and (2) and compute the wavelength:

    E_{\lambda}=E_{n+1,n}\\\\h\frac{c}{\lambda}=\frac{h}{2\pi}\omega\\\\\lambda=\frac{2\pi c}{\omega}

    hence, the wavelength of the emitted photon is 2πc/w

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