Derivation of the QED Hamiltonian

We describe here a rigorous derivation of atom-photon interactions. Our overall goal is to arrive at a Hamiltonian description for the energy of a system of atoms, photons, and atoms interacting with photons through the radiation field.

The basic interaction we will obtain is the dipole interaction Hamiltonian,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int} = -\vec{d}\cdot\vec{E} \,, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{d}} is the atom's dipole moment, and ${\vec {E}}$ is the electric field at the position of the atom. In the end, the electric field will be quantized, and described by operators $a$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^\dagger} .

This result is simple and can be obtained with less rigorous derivations. Here, we want to at least mention all the steps of the rigorous derivation, starting from classical Maxwell's equations.

Quantum electrodynamics

The classical Hamiltonian which describes one particle

H={\frac {p^{2}}{2m}}+V(r)\,,

and this is transformed into the quantum picture by enforcing the commutation relation $[r,p]=i\hbar$ . We will do the same for an atom interacting with light.

Let us begin with Maxwell's equations.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \nabla \cdot E(r,t) &=& \frac{1}{\epsilon_0} \rho(r,t) \\ \nabla \cdot B(r,t) &=& 0 \\ \nabla \times E(r,t) &=& -\partial_t B(r,t) \\ \nabla \times B(r,t) &=& \frac{1}{c^2} \partial_t E(r,t) + \frac{1}{\epsilon_0 c^2} j(r,t) \,. \end{array}}

How many of the field components are true degrees of freedom? We can understand this by taking a spatial Fourier transform. The component of the (vectorial) Fourier component, which is parallel to the k vector, is called the longitudinal part, the component orthogonal to the k vector is the transverse part. By transforming back to the spatial domain, we can distinguish the longitudinal and transverse parts. It turns out that the longitudinal electric field is not a free degree of freedom, i.e. it follows instantaneously the positions of the particles through the Coulomb potential.

Introducing the vector potential reduces the number of field components to four.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} E(r,t) &=& \nabla U(r,t) - \partial_t A(r,t) \\ B(r,t) &=& \nabla \times A(r,t) \,. \end{array}}

In the Coulomb gauge, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \cdot A =0} , i.e. the longitudinal component of A vanished. Furthermore, the scalar potential is

U(r,t)={\frac {1}{4\pi \epsilon _{0}}}\int d^{3}r'{\frac {\rho (r',t)}{|r-r'|}}\,.

Now we are down to two components, the transverse components of the vector poential A.

Keep in mind that the equation of motion is now a second order differential equation for the vector potential. All that is needed to specify the field evolution is thus the initial values of $A_{\perp }$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{A}_\perp} , the transverse vector potential. The differential equation is identical to the equation for r and p for an harmonic oscillator potential.

We can go further by decomposing the Fourier representation of the field and potential in terms of its normal modes,

{\begin{array}{rcl}\alpha (k,t)&=&\lambda (k)\left[E_{\perp }(k,t)-c{\frac {\vec {k}}{k}}\times B(k,t)\right]\\&=&\lambda (k)[-{\dot {A}}_{\perp }(k,t)+i\omega A_{\perp }(k,t)]\,.\end{array}}

Using this, we can represent the equations of motion for the field as

{\dot {\alpha }}(k,t)+i\omega \alpha (k,t)={\frac {i}{\sqrt {2\epsilon _{0}\hbar \omega }}}j_{\perp }(k,t)\,.

Note

{\vec {A}}_{\perp }(r)=\int d^{3}k\sum _{\epsilon }A_{\omega }[{\vec {\epsilon }}\alpha _{\epsilon }(k)e^{i2r}+h.c.]\,.

in the normal mode decomposition, where we now use ${\vec {\epsilon }}$ as the photon polarization, which carries the vector direction of the potential.

Let us now quantize the field. We identify an equivalence between $A$ and ${\dot {A}}$ with $r$ and $p$ , and quantize accordingly. The normal mode coordinates are a linear combination of $A$ and ${\dot {A}}$ from which we obtain the commutator

[a_{\epsilon }(k),a_{\epsilon '}^{\dagger }(k')]=\delta _{\epsilon \epsilon '}\delta (k-k')\,.

in full analogy to the raising and lowering operators for the simple harmonic oscillator.

Energy in the radiation field

It is helpful to go back to consider for a moment what the energy in the field is. Recall that

{\begin{array}{rcl}H&=&{\frac {\epsilon _{0}}{2}}\int d^{3}r({\vec {E}}^{2}+c^{2}{\vec {B}}^{2})\\&=&\int d^{3}rE_{\parallel }^{2}+{\frac {\epsilon _{0}}{2}}\int d^{3}r(E_{\perp }^{2}+c^{2}B^{2})\end{array}}

where we can identify

\int d^{3}rE_{\parallel }^{2}={\frac {1}{8\pi \epsilon _{0}}}\int \int d^{3}rd^{3}d'{\frac {\rho (r)\rho (r')}{|r-r'|}}

as the energy of the Coulomb field due to the charge configuration. This part of the energy is static, with respect to time evolution of the field. The second term is the energy is the transverse component of the field, $H_{\rm {trans}}$ , the radiation energy, which we can understand by introducing again our expression for the vector potential.

{\begin{array}{rcl}H_{\rm {trans}}=\epsilon _{0}\int d^{3}k\left[{\frac {\Pi _{\epsilon }^{2}\Pi _{e}p}{\epsilon _{0}^{2}}}+k^{2}c^{2}A_{\epsilon }^{*}A_{\epsilon }\right]\,,\end{array}}

where $\Pi _{\epsilon }(k)=\epsilon _{0}{\dot {A}}_{\epsilon }(k)$ is identified as the conjugate momentum. This looks much like a simple harmonic oscillator Hamiltonian. Now introduce normal modes, using

\alpha \sim [-{\dot {A}}_{\perp }+i\omega E_{\perp }]\,.

This gives

{\begin{array}{rcl}H_{\rm {trans}}=\int d^{3}k\sum _{\epsilon }{\frac {\hbar }{2}}[\alpha _{\epsilon }^{*}\alpha _{e}p+\alpha _{\epsilon }\alpha ^{*}\epsilon ]\,,\end{array}}

a purely classical expression for energy in the radiation field. Where does $\hbar$ come from? It enters in the constant relating the normal modes with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} . It is just a unit used in the definitions at this moment, which is convenient to use because later on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar} appears in the quantum expression of the energy.

This expression for energy is really identical to that for the classical simple harmonic oscillator,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = () p^2 + () q^2 \,, }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\sim \dot{q}} . For this classical oscillator, it is helpful to introduce a variable describing superpositions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = (q+ip)/\sqrt{2}} , giving

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \frac{\hbar\omega}{2} (p^2 + q^2) = \frac{\hbar}{2}(a^* a + a a^*) \,. }

When quantized, this becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \hbar\omega ( a^\dagger a + 1/2)} . We conclude that the radiation field is just a bunch of oscillators, with one per Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} vector, and each one is described in its quantized form by the Hamiltonian Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \hbar\omega ( a^\dagger _k a_k + 1/2)} . Beware, however, that not all expressions for the energy are valid models for the radiation field; it is essential that they also correspond to a valid Lagrangian for the system. We have also not provide a relativistically covariant formulation of the radiation field.

Quantum description of the radiation field

Let us now go back to the quantum description of the radiation field. Keep in mind that the naive approach of using Cartesian coordinates, without eliminating the longitudinal field, would fail miserably, because

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [ A_i(r) , \Pi_j(r')] \neq i\hbar \delta_{ij}(r-r') \,. }

The treatment above, using the vector potential, is thus essential. We now have as the field definitions, in terms of the quantum operators,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} A_\perp(r) &=& \int d^3k \sum_ \epsilon A_\omega [ \epsilon a_ \epsilon (k) e^{ik\cdot r} + \epsilon a_ \epsilon ^\dagger(k) e^{-ik\cdot r} ] \\ E_\perp(r) &=& \int d^3k \sum_ \epsilon i E_\omega [ \epsilon a_ \epsilon (k) e^{ik\cdot r} - \epsilon a_ \epsilon ^\dagger(k) e^{-ik\cdot r} ] \\ B(r) &=& \int d^3k \sum_ \epsilon i B_\omega [(k\times \epsilon ) a_ \epsilon (k) e^{ik\cdot r} - (k\times \epsilon ) a_ \epsilon ^\dagger(k) e^{-ik\cdot r} ] \,. \end{array}}

The particle operators are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \rho &=& \sum_\alpha q_\alpha \delta(r-r_\alpha) \end{array}}

Coupling of atoms and the radiation field

The total Hamiltonian for the radiation field and charges is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \sum_\alpha \frac{1}{2m_\alpha} \left[ p_\alpha - q_\alpha A_\perp(r_\alpha)\right]^2 + \sum_\alpha \left(-g \frac{q}{2m_\alpha}\right ) S_\alpha \cdot B(r_\alpha) + V_{\rm Coul} + H_{R} \,, }

where the second term, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_\alpha \cdot B(r_\alpha)} has been added by hand, and describes spin interacting with the magnetic field, which will be discussed later. It can be derived from first principles by starting with the Dirac equation, expressed in the non-relativistic limit. The important new term, compared with standard nonrelativistic quantum mechanics, is the replacement of momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p-qA} . The Coulomb interaction energy is standard. The radiation field energy is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_R \sum_i \hbar\omega_i ( a^\dagger _i a_i + 1/2)} . Simplifying this, we may write the total Hamiltonian as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = H_P + H_R + H_I \,, }

where $H_{P}$ is the particle Hamiltonian,

H_{P}=\sum _{\alpha }{\frac {p_{\alpha }^{2}}{2m_{\alpha }}}+V_{\rm {Coul}}\,,

$H_{R}$ is the Hamiltonian for the radiation field, and $H_{I}$ can be written as the sum of three parts, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_I = H_{I1} + H_{I2} + H_{I1}^S} , and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} H_{I1} &=& -\sum_\alpha \frac{q_\alpha}{m_\alpha} p_\alpha A_\perp(r_\alpha) \\ H_{I1}^S &=& -\sum_\alpha g_\alpha \frac{q_\alpha}{2m_\alpha} S_\alpha \cdot B(r_\alpha) A_\perp(r_\alpha) \\ H_{I2} &=& \sum_\alpha \frac{q_\alpha}{2m_\alpha} A^2_\perp(r_\alpha) \,. \end{array}}

For atoms, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{I1}\gg H_{I1}^S, H_{I2}} , typically.

The dipole approximation

Typically, for atomic physics, the wavelength of radiation is much much larger than the size of the atom, so that we may write the main interaction between atoms and the radiation field as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int} = H_{I1} = \frac{q}{m} \vec{A}_0 \langle 2| \vec{p} e^{ikr} |1{\rangle} \,, }

where $E=i\omega A$ . This simplifies to

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int} = \frac{iq\vec{E}_0}{m\omega} \langle 2| p |1{\rangle} \,, }

in the limit that the field wavelength is much larger than the atom, so we can take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} . Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [r,H] = (i\hbar/m) p} , and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle 2|[r,H]|1 \rangle = (E_1-E_2) \langle 2| r |1{\rangle} \,, }

then using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\omega_{12} = E_1-E_2} gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle 2|p|1 \rangle = -im\omega_{12} \langle 2|r|1{\rangle} \,. }

The interaction energy is thus

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int} = q\vec{E}_0 \frac{\omega_{12}}{\omega} \langle 2|\vec{r}|1{\rangle} \,, }

which in the limit of a resonant interaction, becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{int} = q\vec{E}_0 \langle 2|\vec{r}|1 \rangle \sim -\vec{d}\cdot\vec{E} \,. }

Questions that arise in this loose derivation include: what happens for off-resonant interactions? And what happens with the other interaction term we derived above, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{I2}} ?

A rigorous way to obtain the full solution, which is essentially the same as that sketched above, is given in API. It involves using a canonical transformation with the operator

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T = \exp \left[ -\frac{i}{\hbar} \vec{d}\cdot A_\perp \right] \,, }

which is just a displacement operator acting on momentum, that transforms Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p-qA(r)} into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , in a new frame of reference. We approximate $p-qA(r)$ with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p-qA(0)} , which is valid in the dipole approximation, in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \ll \lambda} . The Hamiltonian in this frame of reference is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} H' &=& THT^\dagger \\ &\sim& \frac{p^2}{2m} + V_{Coul} + \sum_j \frac{1}{2 \epsilon _0 L^3}( \epsilon _j\cdot d)^2 + \sum_j \hbar\omega_j ( a^\dagger _j a+1/2) - d\cdot \sum_j E_{\omega_j} [ia_j \epsilon _j - i a^\dagger _j \epsilon _j] \,. \end{array}}

In this frame, the dipole interaction energy appears explicitly. The transformed electric field is

{\begin{array}{rcl}E_{\perp }'(r)&=&TE_{\perp }T^{\dagger }\\&=&E_{\perp }(r)-{\frac {1}{\epsilon _{0}}}P_{\perp }(r)\,.\end{array}}

The interaction Hamiltonian is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_I' = -d \cdot D'(0)/ \epsilon _0 = -d \cdot E_\perp(0) \,. }

Note that this formulation already takes into account the polarizability of matter, and the relation between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D } .

References

Derivation of the QED Hamiltonian

Contents

Quantum electrodynamics

Energy in the radiation field

Quantum description of the radiation field

Coupling of atoms and the radiation field

The dipole approximation

References

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