这是一份bath巴斯大学PH10048作业代写的成功案

where $c_{\in}$ is a complex constant to be chosen such that $U_{E}$ is unitary. Begin with the relativistic particle action, where
$$
\psi\left(x^{\prime}, \tau+\epsilon\right)=\int d^{D} x \mu_{\epsilon} \exp \left[-c_{\epsilon} m \sqrt{-\eta_{\mu \nu} \Delta x^{\mu} \Delta x^{\nu}}\right] \psi(x, \tau)
$$
with measure
$$
\mu_{\epsilon}^{-1}=\int d^{D} x \exp \left[-c_{\epsilon} m \sqrt{-\eta_{\mu \nu} \Delta x^{\mu} \Delta x^{\nu}}\right]
$$
so that $U_{E} \rightarrow 1$ as $\in \rightarrow 0$ Comparing to the corresponding expression for a free nonrelativistic particle
$$
\psi\left(x^{\prime}, \tau+\epsilon\right)=\int d^{D} x \mu_{\epsilon} \exp \left[m \frac{\delta_{i j} \Delta x^{i} \Delta x^{j}}{(-i \epsilon \hbar)}\right] \psi(x, t)
$$

PH10048 COURSE NOTES ：

which is simply the Klein-Gordon equation
$$
\left[\eta^{a b} p_{a} p_{b}+m_{e f f}^{2}\right] \psi=0
$$
with a mass term
$$
m_{e f f}^{2}=\frac{2 \mathcal{E}}{D+1} m^{2}
$$
Non-stationary states are superpositions of Klein-Gordon states with different values of $m_{e f f}^{2}$. It can be shown that as long as we only superimpose states with $\varepsilon>0$, i.e. $m_{e f f}^{2}>0$ non-tachyonic, the wavepacket follows a timelike trajectory.
Next, consider minisuperspace models of the form

这是一份bath巴斯大学PH10001作业代写的成功案

$$
E_{k}^{1} \equiv \bar{V}{k k}, $$ and $$ \psi{k}^{1}=-\sum_{j \neq{k}} \frac{\psi_{j}^{0}}{E_{k}^{0}-E_{j}^{0}} V_{j k}
$$
Here, ${k}$ denotes the modified degenerate states.
The second order correction is obtained by solving Eq. (59),
$$
\psi_{k}^{2}=-\frac{1}{E_{k}^{0}-H_{0}}\left(E_{k}^{0} \psi_{k}^{0}-V \psi_{k}^{1}\right)
$$

PH10001 COURSE NOTES ：

$$
\Psi=\sum_{l} C_{l} \Phi_{l}
$$
The individual “configuration” Slater determinants, $\Phi_{l}$, are constructed from spin-orbitals, $\phi_{j}$, given by
$$
\phi_{j}=|\alpha\rangle \sum_{\lambda} C_{\lambda j} \chi_{\lambda} \text { or }|\beta\rangle \sum_{\lambda} C_{\lambda j} \chi_{\lambda} \text {, }
$$
where $\chi_{\lambda}$ is a spatial basis function. The $\chi_{\lambda}$ generally fall into two classes:

• Slater-type orbitals (STOs)
  $$
  \chi_{\lambda}=N_{n l m \zeta} Y_{l m}(\theta, \phi) r^{n-1} \exp (-\zeta r),
  $$