这是一份ucl伦敦大学学院 PHAS0042作业代写的成功案
问题 1.
The density matrix is a matrix representation of the statistical operator in an arbitrary basis of Hilbert space. Its properties are:
- 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.
- It obeys the invariant inequality
$$
0<\operatorname{tr} \varrho^{2} \leq \operatorname{tr} \varrho=1
$$
证明 .
- It serves to characterize the quantum state by the following criteria:
- If $\operatorname{tr} \varrho^{2}=\operatorname{tr} \varrho=1$ the state is a pure state,
- if $\operatorname{tr} \varrho^{2}<\operatorname{tr} \varrho=1$ the state is a mixed state.
- 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, $$ \langle\mathcal{O}\rangle=\operatorname{tr}(\varrho \mathcal{O})=\sum{m, n} \mathcal{O}{m n} \varrho{n m} .
$$
PHAS0042 COURSE NOTES ：
$$
q(t)=\beta+\frac{\alpha}{m} t,
$$
so that the flunction $S$ and its time derivative are
$$
\begin{aligned}
&S(\boldsymbol{q}, \boldsymbol{\alpha}, t)=\frac{\alpha^{2}}{2 m} t+\boldsymbol{\alpha} \cdot \boldsymbol{\beta}+c, \
&\frac{\mathrm{d} S(\boldsymbol{q}, \boldsymbol{\alpha}, t)}{\mathrm{d} t}=\boldsymbol{\alpha} \cdot \dot{\boldsymbol{q}}-\frac{1}{2 m} \alpha^{2}=\frac{\alpha^{2}}{2 m} .
\end{aligned}
$$