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# 物理代写|量子力学代写Quantum mechanics代考|PHYS4141 Quantum Statistics and Hidden Variables

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## 物理代写|量子力学代写Quantum mechanics代考|Quantum Statistics and Hidden Variables

In quantum mechanics it is meaningful to talk about the value of a certain observable, if the system has been prepared in an eigenstate of that observable with some eigenvalue (which is done by measuring that observable and keeping only those systems with a particular outcome). Then subsequent measurements of that observable will return the given eigenvalue with $100 \%$ probability, as explained in Sec. 2.8. More generally, we can prepare a system in a simultaneous eigenstate of commuting observables, and we can talk about the values of those observables. If the observables in question are not constant in time (if they do not commute with the Hamiltonian), then the system will not remain in the given eigenstate, and in order to obtain the $100 \%$ probability quoted it will be necessary to make the subsequent measurements immediately after the preparation. In particular, the Hamiltonian for an isolated system commutes with itself, so if a system is measured to have a certain energy, then it is meaningful afterwards (assuming the system remains isolated) to talk about its energy.

There is, however, no role played in the orthodox interpretation of quantum mechanics for the simultaneous values of noncommuting observables. These are in principle not measurable. We are of course tempted by the analogy with classical mechanics to think in such terms, because in classical mechanics such simultaneous values of noncommuting observables are meaningful. But to do so means that we are using concepts for understanding physical reality that have no physical consequences. One is reminded of Newton’s ideas of absolute space and time, which likewise had no physical consequences, and which were eliminated from physics with the advent of relativity theory. The idea of basing quantum mechanics on strictly measurable quantities seems to be due to Heisenberg, who was apparently influenced by Einstein’s similar reasoning in his development of relativity theory.

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

We return now to the density operator and describe its characteristic properties, of which there are three. First, $\rho$ is Hermitian, as follows immediately from the definitions (15) and (16). Second, $\rho$ is nonnegative definite (see Eq. (1.65)), as follows by computing the expectation value of $\rho$ with respect to an arbitrary ket $|\phi\rangle$,
$$\langle\phi|\rho| \phi\rangle=\sum_i f_i\left|\left\langle\psi_i \mid \phi\right\rangle\right|^2 \geq 0,$$
where for simplicity we work with the discrete case. The third characteristic property of a density operator is that it has unit trace,
$$\operatorname{tr} \rho=1 .$$
This property is equivalent to the normalization condition on the probabilities, Eq. (11) or (13), as one can easily show. Conversely, as we shall show below, every nonnegative definite, Hermitian operator with unit trace can be interpreted as a density operator, that is, there exist kets and corresponding statistical weights such that the operator can be written in the form (15) or (16).

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

$$\langle\phi|\rho| \phi\rangle=\sum_i f_i\left|\left\langle\psi_i \mid \phi\right\rangle\right|^2 \geq 0,$$

$$\operatorname{tr} \rho=1 .$$

## MATLAB代写

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