# 物理代写|量子力学代写quantum mechanics代考|Radiative Decay

## 物理代写|量子力学代写quantum mechanics代考|Radiative Decay

We now have enough information to compute the rate for the radiative decay $\psi_i \rightarrow \psi_f+\gamma$. With the interaction in Eqs. (6.77) and (6.78), it is the appropriate creation operator in the vector potential that makes the transition to the final photon state of $\left|1_{\vec{k} s}\right\rangle$. The decay rate then follows as in Eq. (6.30)
\begin{aligned} R_{f i} d n_f= & \left(\frac{\hbar e^2}{2 \omega_k \varepsilon_0 \Omega}\right)\left(\frac{2 \pi}{\hbar}\right)\left|\vec{e}{\vec{k} s} \cdot \int d^3 x e^{-i \vec{k} \cdot \vec{x}} \psi_f^(\vec{x}) \frac{\vec{p}}{m} \psi_i(\vec{x})\right|^2 \times \ & \delta\left(E_f-E_i+\hbar \omega_k\right)\left[\frac{L^3}{(2 \pi)^3} d^3 k\right] \end{aligned} The volume element $\Omega=L^3$ cancels. Now use $d^3 k=k^2 d k d \Omega_k$, and $$\frac{d k}{d\left(\hbar \omega_k\right)}=\frac{1}{\hbar c}$$ This gives the photon emission rate \begin{aligned} \omega{f i} & =R_{f i} d n_f \ & =\frac{\alpha}{2 \pi c^2} \omega_k\left|\vec{e}_{\vec{k} s} \cdot \int d^3 x e^{-i \vec{k} \cdot \vec{x}} \psi_f^(\vec{x}) \frac{\vec{p}}{m} \psi_i(\vec{x})\right|^2 d \Omega_k \end{aligned}
This is a powerful result. We have calculated the rate for photon emission by a charged particle making a transition in any quantum system!

## 物理代写|量子力学代写quantum mechanics代考|Schr¨odinger Picture

In the Schrödinger picture, the operators are time-independent, and all the time dependence is put into the wave function. Thus the Schrödinger equation for a non-relativistic particle in a potential $V(r)$, in the presence of additional electromagnetic fields with vector and scalar potentials $(\vec{A}, \Phi)$, in the Schrödinger picture is given by
\begin{aligned} i \hbar \frac{\partial \Psi(\vec{x}, t)}{\partial t} & =H \Psi(\vec{x}, t) \quad ; \text { Schrödinger picture } \ H & =\frac{1}{2 m}[\vec{p}-e \vec{A}(\vec{x})]^2+e \Phi(\vec{x})+V(r) \end{aligned}
Upon quantization, in order to satisfy the basic commutation relation
$$\left[p_i, x_j\right]=\frac{\hbar}{i} \delta_{i j}$$ we continue to employ
$$\vec{p}=\frac{\hbar}{i} \vec{\nabla} \quad ; \text { canonical momentum }$$
The vector potential $\vec{A}(\vec{x})$ is quantized in Eq. (6.77), where it is $\vec{A}(\vec{x}, 0)$. Note that one then has a full interacting quantum field theory.

Other pictures, in particular the interaction picture where the free-field time dependence is put into the quantum field operators [see Eqs. (6.14)] are discussed in Sec. 9.7. ${ }^{14}$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Radiative Decay

$$\left.R_{f i} d n_f=\left(\frac{\hbar e^2}{2 \omega_k \varepsilon_0 \Omega}\right)\left(\frac{2 \pi}{\hbar}\right) \mid \vec{e} \vec{k} s \cdot \int d^3 x e^{-i \vec{k} \cdot \vec{x}} \psi_f^{(} \vec{x}\right)\left.\frac{\vec{p}}{m} \psi_i(\vec{x})\right|^2 \times \quad \delta\left(E_f-E_i\right.$$

$$\frac{d k}{d\left(\hbar \omega_k\right)}=\frac{1}{\hbar c}$$

$$\left.\omega f i=R_{f i} d n_f \quad=\frac{\alpha}{2 \pi c^2} \omega_k \mid \vec{e}_{\vec{k} s} \cdot \int d^3 x e^{-i \vec{k} \cdot \vec{x}} \psi_f^{(} \vec{x}\right)\left.\frac{\vec{p}}{m} \psi_i(\vec{x})\right|^2 d \Omega_k$$

## 物理代写|量子力学代写quantum mechanics代考|Schr¨odinger Picture

$$i \hbar \frac{\partial \Psi(\vec{x}, t)}{\partial t}=H \Psi(\vec{x}, t) \quad ; \text { Schrödinger picture } H \quad=\frac{1}{2 m}[\vec{p}-e \vec{A}(\vec{x})]^2+e \Phi(\vec{x})$$

$$\left[p_i, x_j\right]=\frac{\hbar}{i} \delta_{i j}$$

$$\vec{p}=\frac{\hbar}{i} \vec{\nabla} \quad ; \text { canonical momentum }$$

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