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## 物理代写|量子力学代写Quantum mechanics代考|Analytic continuation to imaginary time

It is useful to consider the related problem obtained by an analytic continuation to imaginary time, $t \rightarrow-i \tau$. We saw in section $2.8$ that this problem is related to statistical physics. We will now work out one example that will be very instructive.

Formally, upon analytic continuation $t \rightarrow-i \tau$, the matrix element of the time evolution operator becomes
$$\left\langle q_{f}\left|e^{-\frac{i}{\hbar} H\left(t_{f}-t_{i}\right)}\right| q_{i}\right\rangle \rightarrow\left\langle q_{f}\left|e^{-\frac{1}{\hbar} H\left(\tau_{f}-\tau_{i}\right)}\right| q_{i}\right\rangle$$
Let us choose
$$\tau_{i}=0 \quad \tau_{f}=\beta \hbar$$
where $\beta=1 / T$, and $T$ is the temperature (in units of $k_{B}=1$ ). Hence, we find that
$$\left\langle q_{f},-i \beta / \hbar \mid q_{i}, 0\right\rangle=\left\langle q_{f}\left|e^{-\beta H}\right| q_{i}\right\rangle$$
The operator $\hat{\rho}$
$$\hat{\rho}=e^{-\beta H}$$
is the density matrix in the canonical ensemble of statistical mechanics for a system with Hamiltonian $H$ in thermal equilibrium at temperature $T$.
It is customary to define the partition function $\mathcal{Z}$,
$$\mathcal{Z}=\operatorname{tr} e^{-\beta H} \equiv \int d q\left\langle q\left|e^{-\beta H}\right| q\right\rangle$$
where I inserted a complete set of eigenstates of $\hat{q}$. Using the results that were derived above, we see that the partition function $\mathcal{Z}$ can be written as a (Euclidean) Feynman path integral in imaginary time, of the form
\begin{aligned} \mathcal{Z} &=\int \mathcal{D} q[\tau] \exp \left{-\frac{1}{\hbar} \int_{0}^{\beta \hbar} d \tau\left[\frac{1}{2} m\left(\frac{\partial q}{\partial \tau}\right)^{2}+V(q)\right]\right} \ & \equiv \int \mathcal{D} q[\tau] \exp \left{-\int_{0}^{\beta} d \tau\left[\frac{m}{2 \hbar^{2}}\left(\frac{\partial q}{\partial \tau}\right)^{2}+V(q)\right]\right} \end{aligned}
where, in the last equality, we have rescaled $\tau \rightarrow \tau / \hbar$.

## 物理代写|量子力学代写Quantum mechanics代考|The functional determinant

Let us now compute the determinant in $\mathcal{Z}^{(2)}$ given in eq. (5.52). We will do the calculation in imaginary time and then carry out the analytic continuation to real time. We follow closely the method as explained in detail in Sidney Coleman’s book (Coleman, 1985).
We want to compute
$$D=\operatorname{Det}\left[-\frac{m}{\hbar^{2}} \frac{d^{2}}{d \tau^{2}}+V^{\prime \prime}\left(q_{c}(\tau)\right)\right]$$
subject to the requirement that the space of functions that the operator acts on obeys specific boundary conditions in (imaginary) time. We are interested in two cases: (a) vanishing boundary conditions, which are useful for studying quantum mechanics at $T=0$, and (b) periodic boundary conditions with period $\beta=1 / T$. The approach is somewhat different in the two situations.

A: Vanishing boundary conditions We define the (real) variable $x=\frac{\hbar}{m} \tau$. The range of $x$ is the interval $[0, L]$, with $L=\hbar \beta / \sqrt{m}$. Let us consider the following eigenvalue problem for the Schrödinger operator $-\partial^{2}+W(x)$,
$$\left(-\partial^{2}+W(x)\right) \psi(x)=\lambda \psi(x)$$
subject to the boundary conditions $\psi(0)=\psi(L)=0$. Formally, the determinant is given by
$$D=\prod_{n} \lambda_{n}$$

where $\left{\lambda_{n}\right}$ is the spectrum of eigenvalues of the operator $-\partial^{2}+W(x)$ for a space of functions satisfying a given boundary condition.

Let us define an auxiliary function $\psi_{\lambda}(x)$, with $\lambda$ a real number not necessarily in the spectrum of the operator, such that the following requirements are met:
1) $\psi_{\lambda}(x)$ is a solution of eq. (5.68), and
2) $\psi_{\lambda}$ obeys the initial conditions, $\psi_{\lambda}(0)=0$ and $\partial_{x} \psi_{\lambda}(0)=1$.

## 物理代写|量子力学代写Quantum mechanics代考|Analytic continuation to imaginary time

$$\left\langle q_{f}\left|e^{-\frac{i}{\hbar} H\left(t_{f} t_{i}\right)}\right| q_{i}\right\rangle \rightarrow\left\langle q_{f}\left|e^{-\frac{1}{\hbar} H\left(\tau_{f}-\tau_{i}\right)}\right| q_{i}\right\rangle$$

$$\tau_{i}=0 \quad \tau_{f}=\beta \hbar$$

$$\left\langle q_{f},-i \beta / \hbar \mid q_{i}, 0\right\rangle=\left\langle q_{f}\left|e^{-\beta H}\right| q_{i}\right\rangle$$

$$\hat{\rho}=e^{-\beta H}$$

$$\mathcal{Z}=\operatorname{tr} e^{-\beta H} \equiv \int d q\left\langle q\left|e^{-\beta H}\right| q\right\rangle$$

〈left 的分隔符缺失或无法识别

## 物理代写|量子力学代写Quantum mechanics代考|The functional determinant

$$D=\operatorname{Det}\left[-\frac{m}{\hbar^{2}} \frac{d^{2}}{d \tau^{2}}+V^{\prime \prime}\left(q_{c}(\tau)\right)\right]$$

A: 消失的边界条件我们定义 (实际) 变量 $x=\frac{h}{m} \tau$. 的范围 $x$ 是区间 $[0, L]$ ，和 $L=\hbar \beta / \sqrt{m}$. 让我们考虑薛定谔算子的以下特 征值问题 $-\partial^{2}+W(x)$,
$$\left(-\partial^{2}+W(x)\right) \psi(x)=\lambda \psi(x)$$

$$D=\prod_{n} \lambda_{n}$$

1) $\psi_{\lambda}(x)$ 是 eq 的解。(5.68) 和
2) $\psi_{\lambda}$ 服从初始条件， $\psi_{\lambda}(0)=0$ 和 $\partial_{x} \psi_{\lambda}(0)=1$.

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