12.9: Radiation from Harmonic Oscillator

Consider an electron in a one-dimensional harmonic oscillator potential aligned along the $x$-axis. According to Section [sosc], the unperturbed energy eigenvalues of the system are

$$E_n = (n+1/2)\,\hbar\,\omega_0,$$ where $\omega_0$ is the frequency of the corresponding classical oscillator. Here, the quantum number $n$ takes the values $0,1,2,\cdots$. Let the $\psi_n(x)$ be the (real) properly normalized unperturbed eigenstates of the system.

Suppose that the electron is initially in an excited state: that is, $n>0$. In principle, the electron can decay to a lower energy state via the spontaneous emission of a photon of the appropriate frequency. Let us investigate this effect. Now, according to Equation ([e3.115]), the system can only make a spontaneous transition from an energy state corresponding to the quantum number $n$ to one corresponding to the quantum number $n'$ if the associated electric dipole moment

$$(d_x)_{n,n'} = \langle n|e\,x|n'\rangle = e\int_{-\infty}^{\infty} \psi_n(x)\,x\,\psi_{n'}(x)\,dx$$ is non-zero [because $d_{if}\equiv (d_x)_{n,n'}^{\,2}$ for the case in hand]. However, according to Equation ([e5.xxx]), $$\int_{-\infty}^\infty \psi_n\,x\,\psi_{n'}\,dx =\sqrt{\frac{\hbar}{2\,m_e\,\omega_0}}\left(\sqrt{n}\,\delta_{n,n'+1} + \sqrt{n'}\,\delta_{n,n'-1}\right).$$ Because we are dealing with emission, we must have $n>n'$. Hence, we obtain $$(d_x)_{n,n'} = e\,\sqrt{\frac{\hbar\,n}{2\,m_e\,\omega_0}}\,\delta_{n,n'+1}.$$ It is clear that (in the electric dipole approximation) we can only have spontaneous emission between states whose quantum numbers differ by unity. Thus, the frequency of the photon emitted when the $n$th excited state decays is $$\omega_{n,n-1} = \frac{E_n - E_{n-1}}{\hbar} = \omega_0.$$ Hence, we conclude that, no matter which state decays, the emitted photon always has the same frequency as the classical oscillator.

According to Equation ([e3.115]), the decay rate of the $n$th excited state is given by

$$w_n = \frac{\omega_{n,n-1}^{\,3}\,(d_x)_{n,n-1}^{\,2}}{3\pi\,\epsilon_0\,\hbar\,c^{\,3}}.$$ It follows that $$w_n = \frac{n\,e^{\,2}\,\omega_0^{\,2}}{6\pi\,\epsilon_0\,m_e\,c^{\,3}}.$$ The mean radiated power is simply $$\label{e13.126} P_n = \hbar\,\omega_0\,w_n = \frac{e^{\,2}\,\omega_0^{\,2}}{6\pi\,\epsilon_0\,m_e\,c^{\,3}}\,[E_n -(1/2)\,\hbar\,\omega_0].$$ Classically, an electron in a one-dimensional oscillator potential radiates at the oscillation frequency $\omega_0$ with the mean power $$P= \frac{e^{\,2}\,\omega_0^{\,2}}{6\pi\,\epsilon_0\,m_e\,c^{\,3}}\,E,$$ where $E$ is the oscillator energy. It can be seen that a quantum oscillator radiates in an almost exactly analogous manner to the equivalent classical oscillator. The only difference is the factor $(1/2)\,\hbar\,\omega_0$ in Equation ([e13.126])—this is needed to ensure that the ground-state of the quantum oscillator does not radiate.

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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