PHYS212 : Lectures

Subsections

• 7.7.1 Electric Dipole (E1) Transitions
• 7.7.2 Selection Rules
• 7.7.3 Spontaneous Emission and Lifetimes
• 7.7.4 E1 Transitions of an Electron Oscillator

7.7 Transitions and Radiation

7.7.1 Electric Dipole (E1) Transitions

• Electromagnetic waves can interact with systems via both electric and magnetic interactions and can carry one or more units of angular momentum. There are electric dipole (E1), quadrupole (E2), octupole (E3) transitions, magnetic dipole (M1), quadrupole (M2) and octupole (M3) transitions, and so on.

• The most probable kind of electromagnetic transition between two states is the E1 transition, followed by M1, E2, M2, E3, M3, ... We will restrict our discussion to E1 transitions.

• We will consider the electric field of an electromagnetic wave of frequency $\omega$ traveling along $\hat{k}$ (see Eq. 5.8), and polarized along $\hat{\imath}$,
  

  $$\vec{\mathcal{E}} = Re\{ \mathcal{E}_o e^{i(kz - \omega t)} \hat{\imath} \}$$ (7.131)

  
  
  which leads to a perturbation in the potential energy of our one dimensional system at $z=0$ of
  

  $$\hat{U}(x,t) = - q \mathcal{E}_o x \cos(\omega t)$$ (7.132)

  
  

• The matrix elements $U_{mn}$ of the electric dipole potential can be written in terms of the matrix elements of $\hat{x}$
  

  $$X_{mn} = \int_{-\infty}^\infty \psi^*_m(x) \psi_n(x) x dx$$ (7.133)

  
  
  as
  

  $$U_{mn} = -q \mathcal{E}_o X_{mn} \cos(\omega t)$$ (7.134)

  
  

• The transition probability of Eq. 7.127 becomes
  

  $$P_{i\rightarrow f} (t) = \frac{q^2 \mathcal{E}^2_o}{\hbar... ...i (\omega_i-\omega_f) t'} \cos(\omega t) dt' \right\vert^2$$ (7.135)

  
  

• Defining the transition energy $\omega_o = \omega_f-\omega_i$, and writing $\cos(\omega t)$ in terms of complex exponentials, we can evaluate the integral
  

  $$\frac{1}{2} \int_0^t \left(e^{i (\omega_o + \omega) t'} + e^{i (\omega_o - \omega) t'} \right) dt' = \frac{1}{2} \left( \frac{e^{i (\omega_o + \omega) t'}}{i (\omega_... ...} + \frac{e^{i (\omega_o - \omega) t'}}{i (\omega_o - \omega)} \right\vert _0^t$$

  $$= \frac{1}{2} \left( \frac{e^{i (\omega_o + \omega) t} - 1}{i (\ome... ...omega)} + \frac{e^{i (\omega_o - \omega) t} - 1}{i (\omega_o - \omega)} \right)$$ (7.136)

  
  
  the first term of which is negligible, because its denominator $\omega_o + \omega$ is always very large. The second term is sharply peaked at $\omega = \omega_o$.

• The remaining term can be written
  

  $$\frac{e^{i (\omega_o - \omega) t} - 1}{2 i (\omega_o - \omega)} = \left( \frac{e^{i \frac{(\omega_o - \omega)}{2} t} - e^{-i \frac{... ...}{2} t}}{2 i (\omega_o - \omega)} \right) e^{i \frac{(\omega_o - \omega)}{2} t}$$

  $$= \frac{\sin \left[ \frac{(\omega_o - \omega)}{2} t \right]} {\omega_o - \omega} e^{i \frac{(\omega_o - \omega)}{2} t}$$ (7.137)

  
  

• The transition probability is then
  

  $$P_{i \rightarrow f} (t) = \frac{q^2 \mathcal{E}^2_o}{\hba... ...rac{(\omega_o - \omega)}{2} t \right]} {(\omega_o - \omega)^2}$$ (7.138)

  
  

• In order to proceed, we must consider the frequency composition of the electric field. Truly monochromatic sources are difficult to arrange. It would be useful to be able to describe the response of our system to a frequency distribution $g(\omega)$. The ``normalization'' of $g(\omega)$ is the energy density of the electric field,
  

  $$\int_0^\infty g(\omega) d\omega =\frac{1}{2} \epsilon_o \mathcal{E}^2_o$$ (7.139)

  
  
  Hence, $g(\omega)$ has units of energy per frequency per volume.

• Calculating a weighted average of Eq. 7.138 over the frequency distribution, we find
  

  $$P_{i\rightarrow f} (t) = \frac{2 q^2}{\epsilon_o \hbar^2} ... ..._o - \omega)}{2} t \right]} {(\omega_o - \omega)^2} d\omega$$ (7.140)

  
  

• Eq. 7.138 is very sharply peaked at $\omega_o$, and requiring that $g(\omega)$ varies more gradually near $\omega_o$, can be approximated as
  
  $$P_{i\rightarrow f} (t) = \frac{2 q^2}{\epsilon_o \hbar^2} ... ..._o - \omega)}{2} t \right]} {(\omega_o - \omega)^2} d\omega$$
  
  
  yielding
  

  $$P_{i\rightarrow f} (t) = \frac{\pi q^2}{\epsilon_o \hbar^2} \left\vert X_{fi} \right\vert^2 g(\omega_o) t$$ (7.141)

  
  
  where we have used the elementary form
  
  $$\int_0^\infty \frac{\sin^2 x}{x^2} dx = \frac{\pi}{2}$$
  
  

• For a three dimensional system, we would also average over possible polarization states of the electromagnetic wave. For our 1D system, we'll assume the electric field is aligned with our system.

• The transition probability varies linearly with time. Hence, the transition rate in units of particles per time is a constant given by
  

  $$r_{i\rightarrow f} = \frac{\pi q^2}{\epsilon_o \hbar^2} \left\vert X_{fi} \right\vert^2 g(\omega_o)$$ (7.142)

  
  

• A note about units - for electrons and protons, $q^2 = e^2$, and we can express Eq. 7.142 in terms of the fine structure constant
  
  $$\alpha = \frac{e^2}{4 \pi \epsilon_o \hbar c} = 0.0072974 \approx \frac{1}{137}$$
  
  
  as
  

  $$r_{i\rightarrow f} = 4\pi^2 \frac{\alpha c}{\hbar} \left\vert X_{fi} \right\vert^2 g(\omega_o)$$ (7.143)

  
  

7.7.2 Selection Rules

• The expression for transition probability (Eq. 7.138) is proportional to the matrix element $U_{mn}$.

• Matrix elements can be zero. Usually, more are zero than non-zero. If we can find the patterns that the non-zero $U_{mn}$ follow, we can find selection rules describing allowed transitions.

• For electric dipole (E1) transitions, selection rules are determined by the $X_{mn}$ in Eq. 7.141.

• Let's consider, for example, E1 transitions between simple harmonic oscillator states. In this case, we have a particularly nice way to resolve the $X_{mn}$, because the operator $\hat{x}$ can be written in terms of the raising and lowering operators $\hat{a}_+$ and $\hat{a}_-$ as
  

  $$\hat{x} = \frac{1}{i \sqrt{2m} \omega} (\hat{a}_+ - \hat{a}_-)$$ (7.144)

  
  
  which gives
  

  $$X_{mn} = \frac{1}{i \sqrt{2m} \omega} \int_{-\infty}^\infty \psi^*_m (\hat{a}_+ - \hat{a}_-) \psi_n dx$$ (7.145)

  
  

• Without actually carrying out any integration, we find that the nature of the raising and lowering operators (Eq. 7.114) along with the orthogonality of the stationary states (Eq. 7.116) allow us to conclude that
  

  $$X_{mn} \propto \delta_{{m,n\pm1}}$$ (7.146)

  
  
  which leads to the selection rule $\Delta n = \pm 1$ for E1 transitions between harmonic oscillator states.

7.7.3 Spontaneous Emission and Lifetimes

• There are three competing transition modes we must consider - absorption, stimulated emission, and spontaneous emission.

• The E1 transition probability (Eq. 7.141) and transition rate (Eq. 7.143) derived above describe absorption and stimulated emission - transitions initiated by an external field.

• Einstein showed^7.11that for a system in thermal equilibrium with its environment, the transition rates for absorption and stimulated emission are identical, and that for spontaneous emission,^7.12
  

  $$r_{i\rightarrow f} = \frac{q^2}{\pi \epsilon_o \hbar} \frac{\omega^3}{c^3} \left\vert X_{fi} \right\vert^2$$ (7.147)

  
  

• For one dimensional electron systems, the rate of spontaneous emission can be written as
  

  $$r^{(s)}_{i\rightarrow f} = 4 \alpha \frac{\omega_o^3}{c^2} ... ...rac{E_o}{\hbar c}\right)^3 \left\vert X_{fi} \right\vert^2$$ (7.148)

  
  
  where $E_o$ is the transition energy.

• If an initial excited state $i$ can de-excite to more than one final state, the total spontaneous transition rate is
  

  $$r^{(s)}_i = \sum_f r^{(s)}_{i\rightarrow f}$$ (7.149)

  
  

• If we have $N$ systems in state $i$, the rate in transitions per unit time at which they make spontaneous transitions to a final state $f$ is simply $r^{(s)} N$.
  

  $$\frac{dN}{dt} = -r^{(s)}_i N$$ (7.150)

  
  
  which we can integrate to find
  

  $$N(t) = N_o e^{-r^{(s)}_i t}$$ (7.151)

  
  

• The mean lifetime of state $i$ is
  

  $$\tau = \frac{1}{r^{(s)}_i}$$ (7.152)

  
  

• The mean lifetime is the analog of the time constant of the exponential decay in RC and RL circuits. It is the time it takes for the number of systems in state $i$ to fall by a fraction $\frac{1}{e} \approx 0.368$. It is related to the half-life via
  

  $$t_{\frac{1}{2}} = \ln(2) \tau \approx 0.693 \tau$$ (7.153)

  
  

7.7.4 E1 Transitions of an Electron Oscillator

• We found in Section 7.7.2 that E1 transitions between stationary states of the harmonic oscillator potential must change the quantum number $n$ by one.

• Any excited state $n$ can make a transition to state $n+1$ via absorption of energy $\hbar \omega$ from an external electromagnetic wave.

• Any excited state $n$ can make a transition to state $n-1$ via either spontaneous or stimulated emission.

• Since each excited state can de-excite to only one final state, calculating spontaneous transition rates and lifetimes involves only one calculation.

• Without proof^7.13, the constants $A_{n\pm}$ of Eq. 7.114 associated with raising and lowering are
  $$\begin{eqnarray*} A_{n+} &=& \sqrt{\hbar \omega (n+1)}\\ A_{n-} &=& \sqrt{\hbar \omega n} \end{eqnarray*}$$
  
  
  so that
  

  $$\hat{a}_+ \psi_n = \sqrt{\hbar \omega (n+1)} \psi_{n+1}$$

  $$\hat{a}_- \psi_n = \sqrt{\hbar \omega n} \psi_{n-1}$$ (7.154)

  
  

• With Eq. 7.154, Eq. 7.145 becomes
  
  $$X_{mn} = \frac{1}{i \sqrt{2m} \omega} \left[ \sqrt{n+1} \in... ...{n} \int_{-\infty}^\infty \psi^*_m \psi_{n-1} dx \right]$$
  
  
  which by orthogonality (Eq. 7.116) simplifies to
  

  $$X_{mn} = \frac{1}{i \sqrt{2m} \omega} \left[ \sqrt{n+1} \delta_{m,n+1} + \sqrt{n} \delta_{m,n-1} \right]$$ (7.155)

  
  

• Spontaneous transitions only happen to lower energy states, so only the second term in Eq. 7.155 contributes to the spontaneous transition rate (Eq. 7.148) giving
  

  $$r^{(s)}_{n\rightarrow n-1} = 4 \alpha \frac{\omega^3}{c^... ...ac{\hbar n}{2m\omega} = \frac{2 \alpha E^2 }{\hbar mc^2} n$$ (7.156)

  
  
  where $E = \hbar \omega$ is the transition energy.

• The electron oscillator example we considered in Section 7.5.3 had $E = 1$ eV, which corresponds to transition rates
  

  $$r^{(s)}_{n\rightarrow n-1} = (43.4 \mathrm{MHz}) n$$ (7.157)

  
  
  and lifetimes
  
  $$\tau_n = \frac{23.0 \mathrm{ns}}{n}$$
  
  

• Apparently, excited states with lower energy last longer than higher-lying states in the level scheme.

• It is interesting to note that, while transitions happen rapidly on our time scale, lifetimes are relatively long on the scale of the oscillations of the wavefunctions. For example, the ground state wavefunction oscillates with frequency
  

  $$\frac{\omega}{2} = 0.76 \mathrm{rad/fs}$$ (7.158)

  
  
  which has a period of 8.3 fs, 2.8 million times shorter than the mean lifetime of the first excited state.

• You will consider transitions between infinite square well states for homework.

Copyright © 2003-2009, Lewis A. Riley

Updated Wed Jan 18 09:51:28 2006