# 物理代写|理论力学作业代写Theoretical Mechanics代考|Physics2513

#### Doug I. Jones

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## 物理代写|理论力学作业代写Theoretical Mechanics代考|Classical ensembles with “Uncertainty Principle”

Feynman’s program, however, is still based on the conventional formalism of QM: state vectors in Hilbert space or wave functions in coordinates’ space. In fact, Wigner’s function $W(q, p)$ (pseudoprobability density for sharp values $q, p$ of incompatible variables $q$ and $p$ ) is defined by the expression
$$W(q, p)=\int \text { dy exp }(-i p y) \psi(q+(1 / 2) y) \psi^{*}(q-(1 / 2) y)$$
which contains explicitly the wave function of the state..In Feynman’sapproach waves are therefore still needed to start with, because pseudoprobabilities are first expressed in terms of wave functions, and then forgotten. We will show, however, that it is possible to express Quantum Mechanics from first principles in terms of pseudoprobabilities without ever introducing the concept of probability amplitudes. This program has been recently carried on [Cini 1999] by generalizing the formalism of classical statistical mechanics in phase space with the introduction of two postulates (uncertainty and discreteness), which impose mathematical constraints on the set of quantum variables in terms of which any physical quantity can be expressed. $\mathrm{QM}$ is therefore reformulated in terms of expectation values of quantum variables as a generalization of the correspondent classical varibles of classical statistical mechanics, with the introduction of a single quantum postulate.

This goal will be attained in two steps. The first step is the formulation of a classical Uncertainty Principle. We consider all the classical ensembles of particles in phase space with coordinate $\mathbf{q}$ and momentum $\mathbf{p}$ in which a given variable $\mathbf{A}(\mathbf{q}, \mathbf{p})$ has a well determined value $\alpha$ and its conjugate variable $\mathbf{B}(\mathbf{q}, \mathbf{p})$ is completely undetermined 1 . Only ensembles of this kind in fact are the classical limit of the quantum states.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|The quantum postulate

The second, essential, step is to introduce the quantum into this scheme. This is done by imposing the fulfilment of a second postulate, based on the assumption that the founding stone of quantum theory is the experimental fact that physical quantities exist (the action of periodic motions, the angular momentum, the energy of bound systems..) whose possible values form a discrete set, invariant under canonical transformations, characteristic of each variable in question. This means that we should request that a belongs to a discrete spectrum independent of the phase space variables.

This feature can only be ensured if eq. (8) for the classical characteristic function $\mathrm{Ca}{a}(\mathrm{k}, \mathrm{x})$, which yields a continuous spectrum a for the values of the classical variable $\mathbf{A}$, is modified to become a true Fredholm homogeneous integral equation for the quantum characteristic function $C{i}(k, y)$ with a nonseparable kernel $g(k y-l u x)$, alluwing for the existence of a discrete set of eigenvalues $\alpha_{i}$.
$$\iint \mathrm{dx} d \mathrm{~h} \mathrm{a}(\mathrm{h}-\mathrm{k}, \mathrm{y}-\mathrm{x}) \mathrm{g}(\mathrm{ky}-\mathrm{hx}) C_{\mathrm{i}}(\mathrm{h}, \mathrm{y})=\mathrm{a}{\mathrm{i}} \mathrm{Ci}(\mathrm{k}, \mathrm{y})$$ Similarly, eq.(12) expressing the uncertainty principle between the classical variables $\mathbf{A}$ and $B$ should be changed into $$\iint \mathrm{dy} d h a(h-k, y-x) f(k y-h x) C{i}(h, y)=0$$
for the quantum characteristic function $C_{\mathrm{i}}(\mathrm{k}, \mathrm{x})$ of the ensemble caracterized by one of the values $a_{i}$ of the quantum variable $A$ and by the complete indeterminacy of its quantum conjugate variable $\boldsymbol{B}$. The functions $g()$ and $f()$ should be determined by imposing new self consistent rules for the quantum variables involved.

The two eqs (13) (14), however, cannot be obtained from (7) and (11) as in the classical case by ordinary commuting numbers. In fact the only way to obtain (13) (14) is to replace the classical characteristic variables $C(k, x)$ obeying the standard rule of multiplication of exponentials with quantum variables $C(\mathrm{k}, \mathrm{x})$ having the property
$$\begin{gathered} (1 / 2)[\mathcal{C}(\mathrm{k}, \mathrm{x}) \mathcal{C}(\mathrm{h}, \mathrm{y})+\mathcal{C}(\mathrm{h}, \mathrm{y}) \mathcal{C}(\mathrm{k}, \mathrm{x})]= \ \mathrm{g}(\mathrm{ky}-\mathrm{hx}) \mathcal{C}(\mathrm{k}+\mathrm{h}, \mathrm{x}+\mathrm{y}) \end{gathered}$$
and to replace their classical Poisson bracket with the Quantum Poisson Bracket
$${C(\mathrm{k}, \mathrm{x}), C(\mathrm{~h}, \mathrm{y})}_{\mathrm{QPB}}=\mathrm{f}(\mathrm{ky}-\mathrm{hx}) \mathcal{C}[(\mathrm{k}+\mathrm{h}),(\mathrm{y}+\mathrm{x})]$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Classical ensembles with “Uncertainty Principle”

$$W(q, p)=\int \mathrm{dy} \exp (-i p y) \psi(q+(1 / 2) y) \psi^{*}(q-(1 / 2) y)$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|The quantum postulate

$$\iint \mathrm{dx} d \mathrm{ha}(\mathrm{h}-\mathrm{k}, \mathrm{y}-\mathrm{x}) \mathrm{g}(\mathrm{ky}-\mathrm{hx}) C_{\mathrm{i}}(\mathrm{h}, \mathrm{y})=\mathrm{aiCi}(\mathrm{k}, \mathrm{y})$$

$$\iint \mathrm{dy} d h a(h-k, y-x) f(k y-h x) C i(h, y)=0$$

$$(1 / 2)[\mathcal{C}(\mathrm{k}, \mathrm{x}) \mathcal{C}(\mathrm{h}, \mathrm{y})+\mathcal{C}(\mathrm{h}, \mathrm{y}) \mathcal{C}(\mathrm{k}, \mathrm{x})]=\mathrm{g}(\mathrm{ky}-\mathrm{hx}) \mathcal{C}(\mathrm{k}+\mathrm{h}, \mathrm{x}+\mathrm{y})$$

$$C(\mathrm{k}, \mathrm{x}), C(\mathrm{~h}, \mathrm{y})_{\mathrm{QPB}}=\mathrm{f}(\mathrm{ky}-\mathrm{hx}) \mathcal{C}[(\mathrm{k}+\mathrm{h}),(\mathrm{y}+\mathrm{x})]$$

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