Tag Archives: Quantum Mechanics代写

量子力学 Quantum Mechanics MATH325

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这是一份liverpool利物浦大学MATH326的成功案例

量子力学 Quantum Mechanics MATH325

$$
\langle x \mid p\rangle=h^{-1 / 2} e^{i x p / \hbar}
$$
are improper eigenstates ${ }^{2}$ of $\widehat{p}$ and $\widehat{H}{0}$, normalized according to Dirac’s delta function, $$ \left\langle p \mid p^{\prime}\right\rangle=\delta\left(p-p^{\prime}\right) . $$ Closure relations (or resolutions of the unit operator $\hat{1}$ ) may therefore be written in momentum or coordinate representation as $$ \widehat{1}=\int{-\infty}^{\infty} d x|x\rangle\left\langle x\left|=\int_{-\infty}^{\infty} d p\right| p\right\rangle\langle p|
$$

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MATH325 COURSE NOTES ：

The properties of $T(p)$ as a function of the complex momentum $p$ are of importance for many application. Let the potential function $V(x)$ be such that
$$
\int_{-\infty}^{\infty} d x|V(x)|\left(1+x^{2}\right)<\infty . $$ Then $T(p)$ is meromorphic in $\operatorname{Im} p>0$ with a finite number $n_{b}$ of simple poles $i \beta_{1}, i \beta_{2}, \ldots,, i \beta_{n}, \beta_{j}>0$ on the imaginary axis. The numbers $-\beta_{j}^{2} /(2 m)$ are the eigenvalues of $H$. Moreover,
$$
T(p)=1+O(1 / p) \text { as }|p| \rightarrow \infty, \quad \operatorname{Im} p \geq 0,
$$
and there can only be a zero at the real axis, at $p=0$,
$$
|T(p)|>0 \quad \operatorname{Im} p \geq 0, p \neq 0 .
$$
In the generic case $T(0)=0$, and
$$
T(p)=\gamma p+o(p), \gamma \neq 0, \text { as } p \rightarrow 0, \quad \operatorname{Im} p \geq 0 .
$$








量子力学|Quantum Mechanics代写 PHAS0042

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这是一份ucl伦敦大学学院 PHAS0042作业代写的成功案

量子力学|Quantum Mechanics代写 PHAS0042
问题 1.

The density matrix is a matrix representation of the statistical operator in an arbitrary basis of Hilbert space. Its properties are:

  1. It is hermitean $\varrho^{\dagger}=\varrho$, its eigencalues are real, positive-semidefinite numbers between 0 and $1,0 \leq w_{j} \leq 1$, i. e. $\varrho$ is a positive matrix.
  2. It obeys the invariant inequality
    $$
    0<\operatorname{tr} \varrho^{2} \leq \operatorname{tr} \varrho=1
    $$


证明 .

  1. It serves to characterize the quantum state by the following criteria:
  2. If $\operatorname{tr} \varrho^{2}=\operatorname{tr} \varrho=1$ the state is a pure state,
  3. if $\operatorname{tr} \varrho^{2}<\operatorname{tr} \varrho=1$ the state is a mixed state.
  4. Expectation values of an observable $\mathcal{O}$, in the $B$-representation, are given by the trace of the product of $\varrho$ and the matrix representation $\mathcal{O}{p q}$ of the observable, $$ \langle\mathcal{O}\rangle=\operatorname{tr}(\varrho \mathcal{O})=\sum{m, n} \mathcal{O}{m n} \varrho{n m} .
    $$

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PHAS0042 COURSE NOTES ：

$$
q(t)=\beta+\frac{\alpha}{m} t,
$$
so that the flunction $S$ and its time derivative are
$$
\begin{aligned}
&S(\boldsymbol{q}, \boldsymbol{\alpha}, t)=\frac{\alpha^{2}}{2 m} t+\boldsymbol{\alpha} \cdot \boldsymbol{\beta}+c, \
&\frac{\mathrm{d} S(\boldsymbol{q}, \boldsymbol{\alpha}, t)}{\mathrm{d} t}=\boldsymbol{\alpha} \cdot \dot{\boldsymbol{q}}-\frac{1}{2 m} \alpha^{2}=\frac{\alpha^{2}}{2 m} .
\end{aligned}
$$