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?

Answer 1

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