Statistical interpretation of Quantum mechanics

Last updated Jan 9, 2023

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Basic statistical formulation:

1. An ensemble $\mathcal{E}$ of identically prepared quantum mechnical systems $S^{(i)}$ is fully characterized by a state $\ket{\psi}\in\mathcal{H}$
2. Observables are hermitian operators, whose measurement on the ensemble $\mathcal{E}$ gives rise to a random variable with the cum. distribution $F(r) = \braket{\psi|E_r|\psi}$, with the spectral family $E_r$

In general one can obtain the probability $P$ for an event $B$ with $$P(B) = \braket{\psi|\int_B dE_r|\psi}$$

States described above are pure states.

Mixed states describe a total ensemble $\mathcal{E} = {\mathcal{E}_1, …, \mathcal{E}_M}$ where every ensemble $\mathcal{E}_i$ describes an ensemble of identical quantum systems, given by a state vector $\ket{\psi_i}\in\mathcal{H}$.

The statistics of the total ensemble $\mathcal{E}$ are given by the number of systems in a single ensemble $N_\alpha$. Weighting the influence of a single ensemble by $w_\alpha = \frac{N_\alpha}{\sum_\alpha N_\alpha}$, which leads to $$F(r) = \sum_\alpha w_\alpha \braket{\psi_\alpha|E_r|\psi_\alpha}$$ and expectation values $$E[\hat{R}] = \sum_\alpha w_\alpha \braket{\psi_\alpha|\hat{R}|\psi_\alpha}$$

Definition: Density operator The density matrix or density operator $\rho$ is defined as $$\rho = \sum_\alpha w_\alpha \ket{\psi_\alpha}\bra{\psi_\alpha}$$ It satisfies:

1. hermetian: $\rho^\dagger = \rho$
2. positive: $\rho \geq 0$ (eigenvalues $\geq 0$)
3. probability perserved: $\text{tr}(\rho)=1$ We can always diagonalize $\rho$, that is, write with respect to a basis of $\mathcal{H}$

Definition: Trace The trace of an operator $A$ is defined as $$\text{tr}(A) = \sum_i \braket{\phi_i|A|\phi_i}$$ where ${\ket{\phi_i}}$ is any basis of the hilbert space

With the trace and density operators we get the expressions $$F(r) = \text{tr}(E_r\rho)$$ $$E(\hat{R}) = \braket{\hat{R}} = \text{tr}(\hat{R}\rho)$$ $$\text{Var}(\hat{R}) = \text{tr}(\hat{R}^2\rho)- \left[\text{tr}(\hat{R}\rho)\right]^2$$

Definition: Pure state The denstiy matrix of a pure state can be written as $\rho = \ket{\psi}\bra{\psi}$ A state is pure if $$\text{tr}(\rho^2) = \text{tr}(\rho) = 1$$ A pure state is defined up to a phase $\exp(i\chi)$ and corresponds to an ensemble $\mathcal{E}_1$ with only same states. Therefore splitting $\mathcal{E}$ into two sub ensembles $\mathcal{E}_1$ and $\mathcal{E}_2$ results in the same statistics for the sub ensembles

Definition: Statistical mixture A Statistical mixture or mixed state has $$\text{tr}(\rho^2) < \text{tr}(\rho) = 1$$

• There exists no dispersion free ensemble $\text{Var}(\rho)=0$

Theorem: Nelson’s theorem For two operators $\hat{R}_1$ and $\hat{R}_2$ with associated random variables and distributions $$F_i(r_i)=\braket{\psi|E^i_{r_i}|\psi}$$ there exists a joint probability distribution $$F(r_1,r_2)=\braket{\psi|E^1_{r_1}E^2_{r_2}|\psi}$$ in all states $\ket{\psi}\in\mathcal{H}$ if and only if the operators commute $[\hat{R}_1,\hat{R}_2 = 0$.