Categories: Electromagnetism, Optics, Perturbation, Physics, Quantum mechanics.

Electric dipole approximation

Suppose that an electromagnetic wave is travelling through an atom, and affecting the electrons. The general Hamiltonian of an electron in an electromagnetic field is:

H^=(P^−qA)22m+qΦ=P^22m−q2m(A⋅P^+P^⋅A)+q2A22m+qΦ\begin{aligned} \hat{H} &= \frac{(\vu{P} - q \vb{A})^2}{2 m} + q \Phi \\ &= \frac{\vu{P}{}^2}{2 m} - \frac{q}{2 m} (\vb{A} \cdot \vu{P} + \vu{P} \cdot \vb{A}) + \frac{q^2 \vb{A}^2}{2m} + q \Phi \end{aligned}

Where q<0q < 0 is the electron’s charge, P^=−iℏ∇\vu{P} = - i \hbar \nabla is the canonical momentum operator, A\vb{A} is the magnetic vector potential, and Φ\Phi is the electric scalar potential. We start by fixing the Coulomb gauge ∇⋅A=0\nabla \cdot \vb{A} = 0 such that P^\vu{P} and A\vb{A} commute; let ψ\psi be an arbitrary test function:

[A,P^]ψ=(A⋅P^−P^⋅A)ψ=−iℏA⋅(∇ψ)+iℏ∇⋅(Aψ)=iℏ(∇⋅A)ψ=0\begin{aligned} \comm{\vb{A}}{\vu{P}} \psi &= (\vb{A} \cdot \vu{P} - \vu{P} \cdot \vb{A}) \psi \\ &= -i \hbar \vb{A} \cdot (\nabla \psi) + i \hbar \nabla \cdot (\vb{A} \psi) \\ &= i \hbar (\nabla \cdot \vb{A}) \psi \\ &= 0 \end{aligned}

Meaning A⋅P^=P^⋅A\vb{A} \cdot \vu{P} = \vu{P} \cdot \vb{A}. Furthermore, we assume that A\vb{A} is so small that A2\vb{A}{}^2 is negligible, so the Hamiltonian is reduced to:

H^≈P^22m−qmP^⋅A+qΦ\begin{aligned} \hat{H} &\approx \frac{\vu{P}{}^2}{2 m} - \frac{q}{m} \vu{P} \cdot \vb{A} + q \Phi \end{aligned}

We now split H^\hat{H} like so, where H^1\hat{H}_1 can be regarded as a perturbation to the “base” H^0\hat{H}_0:

H^=H^0+H^1H^0≡P^22m+qΦH^1≡−qmP^⋅A\begin{aligned} \hat{H} = \hat{H}_0 + \hat{H}_1 \qquad\qquad \hat{H}_0 \equiv \frac{\vu{P}{}^2}{2 m} + q \Phi \qquad\qquad \hat{H}_1 \equiv - \frac{q}{m} \vu{P} \cdot \vb{A} \end{aligned}

In an electromagnetic wave, A\vb{A} is oscillating sinusoidally in time and space:

A(x,t)=A0sin⁡(k⋅x−ωt)\begin{aligned} \vb{A}(\vb{x}, t) = \vb{A}_0 \sin(\vb{k} \cdot \vb{x} - \omega t) \end{aligned}

Mathematically, it is more convenient to represent this with a complex exponential, whose real part should be taken at the end of the calculation:

A(x,t)=−iA0exp⁡(ik⋅x−iωt)\begin{aligned} \vb{A}(\vb{x}, t) = - i \vb{A}_0 \exp(i \vb{k} \cdot \vb{x} - i \omega t) \end{aligned}

The corresponding perturbative electric field E\vb{E} is then given by:

E(x,t)=−∂A∂t=E0exp⁡(ik⋅x−iωt)\begin{aligned} \vb{E}(\vb{x}, t) = - \pdv{\vb{A}}{t} = \vb{E}_0 \exp(i \vb{k} \cdot \vb{x} - i \omega t) \end{aligned}

Where E0=ωA0\vb{E}_0 = \omega \vb{A}_0. Light in and around the visible spectrum has a wavelength 2π/∣k∣∼10−7 m2 \pi / |\vb{k}| \sim 10^{-7} \:\mathrm{m}, while an atomic orbital is several Bohr radii ∼10−10 m\sim 10^{-10} \:\mathrm{m}, so k⋅x\vb{k} \cdot \vb{x} is very small. Therefore:

E(x,t)≈E0exp⁡(−iωt)\begin{aligned} \boxed{ \vb{E}(\vb{x}, t) \approx \vb{E}_0 \exp(- i \omega t) } \end{aligned}

This is the electric dipole approximation: we ignore all spatial variation of E\vb{E}, and only consider its temporal oscillation. Also, since we have not used the word “photon”, we are implicitly treating the radiation classically, and the electron quantum-mechanically.

Next, we want to rewrite H^1\hat{H}_1 to use the electric field E\vb{E} instead of the potential A\vb{A}. To do so, we use that momentum P^≡m dx^/dt\vu{P} \equiv m \: \idv{\vu{x}}{t} and evaluate this in the interaction picture:

P^=mdx^dt=miℏ[H^0,x^]\begin{aligned} \vu{P} &= m \dv{\vu{x}}{t} = m \frac{i}{\hbar} \comm{\hat{H}_0}{\vu{x}} \end{aligned}

Taking the off-diagonal inner product with the two-level system’s states ∣1⟩\Ket{1} and ∣2⟩\Ket{2} gives:

⟨2∣P^∣1⟩=miℏ⟨2∣H^0x^−x^H^0∣1⟩=miω0⟨2∣x^∣1⟩\begin{aligned} \matrixel{2}{\vu{P}}{1} &= m \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vu{x} - \vu{x} \hat{H}_0}{1} \\ &= m i \omega_0 \matrixel{2}{\vu{x}}{1} \end{aligned}

Where ω0≡(E2 ⁣− ⁣E1)/ℏ\omega_0 \equiv (E_2 \!-\! E_1) / \hbar is the resonance of the energy gap, close to which we assume that A\vb{A} and E\vb{E} are oscillating, i.e. ω≈ω0\omega \approx \omega_0. Therefore, P^/m=iω0x^\vu{P} / m = i \omega_0 \vu{x}, so we get:

H^1(t)=−qmP^⋅A=−(−ii)qω0x^⋅A0exp⁡(−iωt)≈−p^⋅E0exp⁡(−iωt)\begin{aligned} \hat{H}_1(t) &= - \frac{q}{m} \vu{P} \cdot \vb{A} \\ &= - (- i i) q \omega_0 \vu{x} \cdot \vb{A}_0 \exp(- i \omega t) \\ &\approx - \vu{p} \cdot \vb{E}_0 \exp(- i \omega t) \end{aligned}

Where p^≡qx^\vu{p} \equiv q \vu{x} is the transition dipole moment operator of the electron, hence the name electric dipole approximation. Finally, we take the real part, yielding:

H^1(t)=−p^⋅E(t)=−qx^⋅E0cos⁡(ωt)\begin{aligned} \boxed{ \begin{aligned} \hat{H}_1(t) &= - \vu{p} \cdot \vb{E}(t) \\ &= - q \vu{x} \cdot \vb{E}_0 \cos(\omega t) \end{aligned} } \end{aligned}

If this approximation is too rough, E\vb{E} can always be Taylor-expanded in (ik⋅x)(i \vb{k} \cdot \vb{x}):

E(x,t)=E0(1+(ik⋅x)+12(ik⋅x)2+ ...)exp⁡(−iωt)\begin{aligned} \vb{E}(\vb{x}, t) = \vb{E}_0 \Big( 1 + (i \vb{k} \cdot \vb{x}) + \frac{1}{2} (i \vb{k} \cdot \vb{x})^2 + \: ... \Big) \exp(- i \omega t) \end{aligned}

Taking the real part then yields the following series of higher-order correction terms:

E(x,t)=E0(cos⁡(ωt)+(k⋅x)sin⁡(ωt)−12(k⋅x)2cos⁡(ωt)+ ...)\begin{aligned} \vb{E}(\vb{x}, t) = \vb{E}_0 \Big( \cos(\omega t) + (\vb{k} \cdot \vb{x}) \sin(\omega t) - \frac{1}{2} (\vb{k} \cdot \vb{x})^2 \cos(\omega t) + \: ... \Big) \end{aligned}

References

  1. M. Fox, Optical properties of solids, 2nd edition, Oxford.
  2. D.J. Griffiths, D.F. Schroeter, Introduction to quantum mechanics, 3rd edition, Cambridge.