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# 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Einstein coefficients revisited

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## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Einstein coefficients revisited

In quantum mechanics we usually study a single electron in a background potential $V(x)$. In quantum field theory, the background (e.g. the electromagnetic system) is dynamical, so all kinds of new phenomena can be explained. We already saw one example in Chapter 1. We can now be a little more explicit about what the relevant Hamiltonian should be for Dirac’s calculation of the Einstein coefficients.
We can always write a Hamiltonian as
$$H=H_0+H_{\mathrm{int}},$$
where $H_0$ describes some system that we can solve exactly. In the case of the two-state system discussed in Chapter 1, we can take $H_0$ to be the sum of the Hamiltonians for the atom and the photons:
$$H_0=H_{\text {atom }}+H_{\text {photon }}$$
The eigenstates of $H_{\text {atom }}$ are the energy eigenstates $\left|\psi_n\right\rangle$ of the hydrogen atom, with energies $E_n . H_{\text {photon }}$ is the Hamiltonian in Eq. (2.65) above:
$$H_{\text {photon }}=\int \frac{d^3 k}{(2 \pi)^3} \omega_k\left(a_k^{\dagger} a_k+\frac{1}{2}\right) .$$
The remaining $H_{\mathrm{int}}$ is hopefully small enough to let us use perturbation theory.
Fermi’s golden rule from quantum mechanics says the rate for transitions between two states is proportional to the square of the matrix element of the interaction between the two states:
$$\Gamma \propto\left|\left\langle f\left|H_{\text {int }}\right| i\right\rangle\right|^2 \delta\left(E_f-E_i\right)$$
and we can treat the interaction semi-classically:
$$H_{\text {int }}=\phi H_I$$

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Hamiltonians and Lagrangians

A classical field theory is just a mechanical system with a continuous set of degrees of freedom. Think about the density of a fluid $\rho(x)$ as a function of position, or the electric field $\vec{E}(x)$. Field theories can be defined in terms of either a Hamiltonian or a Lagrangian, which we often write as integrals over all space of Hamiltonian or Lagrangian densities:
$$H=\int d^3 x \mathcal{H}, \quad L=\int d^3 x \mathcal{L}$$
We will use a calligraphic script for densities and an italic script for integrated quantities. The word “density” is almost always omitted.

Formally, the Hamiltonian (density) is a functional of fields and their conjugate momenta $\mathcal{H}[\phi, \pi]$. The Lagrangian (density) is the Legendre transform of the Hamiltonian (density). Formally, it is defined as
$$\mathcal{L}[\phi, \dot{\phi}]=\pi[\phi, \dot{\phi}] \dot{\phi}-\mathcal{H}[\phi, \pi[\phi, \dot{\phi}]],$$
where $\dot{\phi}=\partial_t \phi$ and $\pi[\phi, \dot{\phi}]$ is implicitly defined by $\frac{\partial \mathcal{H}[\phi, \pi]}{\partial \pi}=\dot{\phi}$. The inverse transform is
$$\mathcal{H}[\phi, \pi]=\pi \dot{\phi}[\phi, \pi]-\mathcal{L}[\phi, \dot{\phi}[\phi, \pi]],$$
where $\dot{\phi}[\phi, \pi]$ is implicitly defined by $\frac{\partial \mathcal{L}[\phi, \dot{\phi}]}{\partial \dot{\phi}}=\pi$.

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Einstein coefficients revisited

$$H=H_0+H_{\mathrm{int}},$$

$$H_0=H_{\text {atom }}+H_{\text {photon }}$$
$H_{\text {atom }}$的本征态为氢原子的能量本征态$\left|\psi_n\right\rangle$，其中能量$E_n . H_{\text {photon }}$为上式(2.65)中的哈密顿量:
$$H_{\text {photon }}=\int \frac{d^3 k}{(2 \pi)^3} \omega_k\left(a_k^{\dagger} a_k+\frac{1}{2}\right) .$$

$$\Gamma \propto\left|\left\langle f\left|H_{\text {int }}\right| i\right\rangle\right|^2 \delta\left(E_f-E_i\right)$$

$$H_{\text {int }}=\phi H_I$$

## 物理代考|量子场论代考QUANTUM FIELD THEORY代考|Hamiltonians and Lagrangians

$$H=\int d^3 x \mathcal{H}, \quad L=\int d^3 x \mathcal{L}$$

$$\mathcal{L}[\phi, \dot{\phi}]=\pi[\phi, \dot{\phi}] \dot{\phi}-\mathcal{H}[\phi, \pi[\phi, \dot{\phi}]],$$

$$\mathcal{H}[\phi, \pi]=\pi \dot{\phi}[\phi, \pi]-\mathcal{L}[\phi, \dot{\phi}[\phi, \pi]],$$

## MATLAB代写

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