# 物理代写|电动力学代写electromagnetism代考|Level Shifts and Self-Interaction

## 物理代写|电动力学代写electromagnetism代考|Level Shifts and Self-Interaction

The diagram expansion is equally applicable to the perturbation theory approach to energy level shifts; recall from Chapter 6 that a discrete energy level of the reference system can be related formally to an energy level of the full problem by solving for the roots of the equation,
\begin{aligned} E_n & =E_n^0+\Delta_n(E) \ & \approx E_n^0+\Delta_n\left(E_n^0\right) \end{aligned}
where
$$\Delta_n\left(E_n^0\right)=\left\langle\Phi_n^0\left|\left(\mathrm{~V}+\mathrm{VG}^0\left(E_n^0\right) \mathrm{V}+\ldots\right)^{\prime}\right| \Phi_n^0\right\rangle$$
This is essentially the same expansion as for the T-matrix with the supplementary condition that $\Phi_n^0$ must be excluded from sums over complete sets of states.

A diagram in Figure 10.2 can also be repurposed as the building block of the diagrammatic expansion for the energy shift $\Delta E$ due to Van der Waals interactions of pairs of neutral atoms/molecules/chromophores, supposed electronically distinct, via the exchange of virtual photons. One can imagine making a copy of diagram (10.2iia) and, taking the two copies together, joining the external lines 1 to $1^{\prime}, 2$ to $2^{\prime}$; the composite diagram is then relabelled so that the initial and final states are the same, $(1,2)$, and the virtual intermediate states are $\left(1^{\prime}, 2^{\prime}\right)$. There are six topologically distinct diagrams that can be formed in this way when all time orderings of the vertices are allowed for (excluding interchange of the two particles); the diagrams have four vertices and so, if $\Phi_n^0$ in (10.206) is taken to be the tensor product of the ground state of the atomic system and the photon vacuum, the diagrams describe the van der Waals interaction to order $e^4$ (i.e. $\alpha^2$ ); within the electric dipole approximation they lead to [11], [72],
$$\Delta E \approx \frac{C}{R^7}, \quad R \gg \lambda ; \quad \Delta E \approx \frac{C^{\prime}}{R^6}, \quad R \ll \lambda$$ where $C, C^{\prime}$ are related to the polarisabilities of the two atomic groups that are separated by $R . \lambda$ is a typical optical excitation wavelength. The weakening of the force law for the potential energy as $R$ increases was the property that points to the involvement of the electromagnetic field, and thus quantum electrodynamics, because the interaction cannot be instantaneous [72].

## 物理代写|电动力学代写electromagnetism代考|Quantum Electrodynamics beyond

Perturbation theory as used in non-relativistic quantum electrodynamics in the conventional fashion was described in Chapter 10. The S-matrix description is predicated on the assumption that the reference Hamiltonian, $\mathrm{H}0$, and the full Hamiltonian, $\mathrm{H}$, are related by a unitary transformation, that is, they are operators on the same Hilbert space $\mathcal{H}$. In the time-dependent view of scattering this requires that $\mathrm{H}_0$ and $\mathrm{H}$ coincide at $t= \pm \infty$. As shown in Chapter 6 , in this framework the unitary transformation operator can be constructed approximately as a perturbation expansion in powers of the coupling constant. We start here with some general remarks about how and why problems can arise when a quantum field is involved. Let us first work in a Schrödinger representation in which $\mathrm{H}$ is time-independent. We can gain an insight into the construction in the following way. Given any vector in $\mathcal{H}$ chosen as an initial state $\left|u_0\right\rangle$, one may develop an orthonormal sequence of states from a three-term recurrence relation generated by the full Hamiltonian, $$\mathrm{H}\left|u_n\right\rangle=a_n\left|u_n\right\rangle+b{n+1}\left|u_{n+1}\right\rangle+b_{n-1}\left|u_{n-1}\right\rangle$$
with starting coefficients [1]
\begin{aligned} a_0 & =\left\langle u_0|\mathrm{H}| u_0\right\rangle, \ b_1 & =\left\langle u_1|\mathrm{H}| u_0\right\rangle, \quad b_{-1}=0, b_0=1, \ \left|u_1\right\rangle & =b_1^{-1}\left(\mathrm{H}-a_0\right)\left|u_0\right\rangle . \end{aligned}
Requiring normalisation of $\left|u_1\right\rangle$ shows that the first off-diagonal coefficient is determined by the variance of the Hamiltonian in the starting state $\left|u_0\right\rangle$,
$$b_1^2=\left\langle u_0\left|\mathrm{H}^2\right| u_0\right\rangle-a_0^2$$
The quantities $\left{a_n, b_n\right}$ give a tri-diagonal (Jacobi) representation of the Hamiltonian which may be brought to diagonal form by a further unitary transformation. The normalisation requires that the states $\left{u_n\right}$ are square integrable. ${ }^1$

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Level Shifts and Self-Interaction

$$E_n=E_n^0+\Delta_n(E) \quad \approx E_n^0+\Delta_n\left(E_n^0\right)$$

$$\Delta_n\left(E_n^0\right)=\left\langle\Phi_n^0\left|\left(\mathrm{~V}+\mathrm{VG}^0\left(E_n^0\right) \mathrm{V}+\ldots\right)^{\prime}\right| \Phi_n^0\right\rangle$$

$$\Delta E \approx \frac{C}{R^7}, \quad R \gg \lambda ; \quad \Delta E \approx \frac{C^{\prime}}{R^6}, \quad R \ll \lambda$$

## 物理代写|电动力学代写electromagnetism代考|Quantum Electrodynamics beyond

$$a_0=\left\langle u_0|\mathrm{H}| u_0\right\rangle, b_1 \quad=\left\langle u_1|\mathrm{H}| u_0\right\rangle, \quad b_{-1}=0, b_0=1,\left|u_1\right\rangle=b_1^{-1}\left(\mathrm{H}-a_0\right)\left|u_0\right\rangle .$$

$$b_1^2=\left\langle u_0\left|\mathrm{H}^2\right| u_0\right\rangle-a_0^2$$

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