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# 物理代写|量子力学代写Quantum mechanics代考|PHYS3001 Time Evolution of the Density Operator

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## 物理代写|量子力学代写Quantum mechanics代考|Time Evolution of the Density Operator

Equation (14), the Schrödinger equation, gives the time evolution of the state vector in quantum mechanics. As we have seen in Notes 3, however, the state of a quantum system is described in general by a density operator, not a state vector. Therefore we require an equation of evolution for the density operator. We work here in the Schrödinger picture, and consider the density operator for a discrete ensemble of pure states $\left|\psi_i\right\rangle$ with statistical weights $f_i$, as in Eq. (3.16). The density operator $\hat{\rho}$ is a function of time because the states $\left|\psi_i\right\rangle$ are functions of time. We put a hat on $\hat{\rho}$ to distinguish it from the classical density $\rho$ discussed in Sec. B.24.
At the initial time $t_0$ the density operator is given by
$$\hat{\rho}\left(t_0\right)=\sum_i f_i\left|\psi_i\left(t_0\right)\right\rangle\left\langle\psi_i\left(t_0\right)\right|,$$
while at the final time it is
\begin{aligned} \hat{\rho}(t) & =\sum_i f_i\left|\psi_i(t)\right\rangle\left\langle\psi_i(t)\right|=U\left(t, t_0\right)\left(\sum_i f_i\left|\psi_i\left(t_0\right)\right\rangle\left\langle\psi_i\left(t_0\right)\right|\right) U\left(t, t_0\right)^{\dagger} \ & =U\left(t, t_0\right) \hat{\rho}\left(t_0\right) U\left(t, t_0\right)^{\dagger} \end{aligned}

## 物理代写|量子力学代写Quantum mechanics代考|Commutators and Poisson Brackets

Comparing the quantum equations (22) and (33) with their classical counterparts (B.101) and (B.112), respectively, we see that one goes into the other if we map commutators into Poisson brackets according to the rule,
$$[A, B] \rightarrow i \hbar{A, B}$$
This association was first noticed by Dirac, and it helped him to establish the formal structure of quantum mechanics on the classical model. In important cases it is an exact correspondence, that is, quantum commutators are the same as classical Poisson brackets, apart from the factor $i \hbar$ and a reinterpretation of the symbols as operators (quantum observables) instead of classical observables. Compare, for example, the classical canonical Poisson bracket relations (B.108) with the HeisenbergBorn commutation relations (4.69). Another example is the classical Poisson bracket relations for the components of orbital angular momentum, $\mathbf{L}=\mathbf{x} \times \mathbf{p}$,
$$\left{L_i, L_j\right}=\epsilon_{i j k} L_k$$
which may be compared to the quantum commutation relations for the components of the angular momentum operator,
$$\left[L_i, L_j\right]=i \hbar \epsilon_{i j k} L_k .$$

## 物理代写|量子力学代写Quantum mechanics代考|Time Evolution of the Density Operator

$$\hat{\rho}\left(t_0\right)=\sum_i f_i\left|\psi_i\left(t_0\right)\right\rangle\left\langle\psi_i\left(t_0\right)\right|$$

$$\hat{\rho}(t)=\sum_i f_i\left|\psi_i(t)\right\rangle\left\langle\psi_i(t)\right|=U\left(t, t_0\right)\left(\sum_i f_i\left|\psi_i\left(t_0\right)\right\rangle\left\langle\psi_i\left(t_0\right)\right|\right) U\left(t, t_0\right)^{\dagger} \quad=U\left(t, t_0\right) \hat{\rho}\left(t_0\right) U\left(t, t_0\right)^{\dagger}$$

## 物理代写|量子力学代写Quantum mechanics代考|Commutators and Poisson Brackets

$$[A, B] \rightarrow i \hbar A, B$$

〈left 缺少或无法识别的分隔符

$$\left[L_i, L_j\right]=i \hbar \epsilon_{i j k} L_k \text {. }$$

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