We begin from the following equations for expectation values of the coordinate x and momentum p
- [math]\displaystyle{
m\frac{d}{dt} \langle x \rangle = \langle p \rangle, \qquad \frac{d}{dt} \langle p \rangle =\langle -U'(x) \rangle,
}[/math]
aka, Newton's laws of motion averaged over ensemble. With the help of the operator axioms, they can be rewritten as
- [math]\displaystyle{
\begin{align}
m\frac{d}{dt} \langle \Psi(t) | \hat{x} | \Psi(t) \rangle &= \langle \Psi(t) | \hat{p} | \Psi(t) \rangle, \\
\frac{d}{dt} \langle \Psi(t) | \hat{p} | \Psi(t) \rangle &= \langle \Psi(t) | -U'(\hat{x}) | \Psi(t) \rangle.
\end{align}
}[/math]
Notice a close resemblance with Ehrenfest theorems in quantum mechanics. Applications of the product rule leads to
- [math]\displaystyle{
\begin{align}
\langle d\Psi/dt | \hat{x} | \Psi \rangle + \langle \Psi | \hat{x} | d\Psi/dt \rangle &= \langle \Psi | \hat{p}/m | \Psi \rangle, \\
\langle d\Psi/dt | \hat{p} | \Psi \rangle + \langle \Psi | \hat{p} | d\Psi/dt \rangle & = \langle \Psi | -U'(\hat{x}) | \Psi \rangle,
\end{align}
}[/math]
into which we substitute a consequence of Stone's theorem [math]\displaystyle{ i | d\Psi(t)/dt \rangle = \hat{L} | \Psi(t) \rangle }[/math] and obtain
- [math]\displaystyle{
\begin{align}
im \langle \Psi(t) | [\hat{L}, \hat{x} ] | \Psi(t) \rangle &= \langle \Psi(t)| \hat{p} |\Psi(t)\rangle, \\
i \langle \Psi(t) | [\hat{L}, \hat{p}] | \Psi(t)\rangle &= - \langle \Psi(t)| U'(\hat{x}) |\Psi(t)\rangle.
\end{align}
}[/math]
Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown [math]\displaystyle{ \hat{L} }[/math] is derived
-
[math]\displaystyle{ im [\hat{L}, \hat{x}] = \hat{p} , \qquad i [\hat{L}, \hat{p}] = -U'(\hat{x}). }[/math]
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(commutator eqs for L)
|
Assume that the coordinate and momentum commute [math]\displaystyle{ [ \hat{x}, \hat{p} ] = 0 }[/math]. This assumption physically means that the classical particle's coordinate and momentum can be measured simultaneously, implying absence of the uncertainty principle.
The solution [math]\displaystyle{ \hat{L} }[/math] cannot be simply of the form [math]\displaystyle{ \hat{L} = L(\hat{x}, \hat{p}) }[/math] because it would imply the contractions [math]\displaystyle{ im [L(\hat{x}, \hat{p}), \hat{x}] = 0 = \hat{p} }[/math] and [math]\displaystyle{ i [L(\hat{x}, \hat{p}), \hat{p}] = 0 = -U'(\hat{x}) }[/math]. Therefore, we must utilize additional operators [math]\displaystyle{ \hat{\lambda}_x }[/math] and [math]\displaystyle{ \hat{\lambda}_p }[/math] obeying
-
[math]\displaystyle{
[ \hat{x}, \hat{\lambda}_x ] = [ \hat{p}, \hat{\lambda}_p ] = i, \quad [\hat{x}, \hat{p}] = [ \hat{x}, \hat{\lambda}_p ] = [ \hat{p}, \hat{\lambda}_x ] = [ \hat{\lambda}_x, \hat{\lambda}_p ] = 0.
}[/math]
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(KvN algebra)
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The need to employ these auxiliary operators arises because all classical observables commute. Now we seek [math]\displaystyle{ \hat{L} }[/math] in the form [math]\displaystyle{ \hat{L} = L(\hat{x}, \hat{\lambda}_x, \hat{p}, \hat{\lambda}_p) }[/math]. Utilizing KvN algebra, the commutator eqs for L can be converted into the following differential equations:[14][16]
- [math]\displaystyle{
m L'_{\lambda_x} (x, \lambda_x, p, \lambda_p) = p, \qquad L'_{\lambda_p} (x, \lambda_x, p, \lambda_p) = -U'(x).
}[/math]
Whence, we conclude that the classical KvN wave function [math]\displaystyle{ |\Psi(t)\rangle }[/math] evolves according to the Schrödinger-like equation of motion
-
[math]\displaystyle{
i\frac{d}{dt} |\Psi(t)\rangle = \hat{L} |\Psi(t)\rangle,
\qquad \hat{L} = \frac{\hat{p}}{m} \hat{\lambda}_x - U'(\hat{x}) \hat{\lambda}_p.
}[/math]
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(KvN dynamical eq)
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Let us explicitly show that KvN dynamical eq is equivalent to the classical Liouville mechanics.
Since [math]\displaystyle{ \hat{x} }[/math] and [math]\displaystyle{ \hat{p} }[/math] commute, they share the common eigenvectors
-
[math]\displaystyle{
\hat{x} |x,p\rangle = x |x,p\rangle, \quad \hat{p} |x,p\rangle = p |x,p\rangle , \quad
A(\hat{x}, \hat{p}) |x,p\rangle = A(x,p) |x,p\rangle,
}[/math]
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(xp eigenvec)
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with the resolution of the identity
[math]\displaystyle{ 1 = \int dx \, dp \, |x,p\rangle \langle x,p|. }[/math]
Then, one obtains from equation (KvN algebra)
[math]\displaystyle{
\langle x,p| \hat{\lambda}_x | \Psi \rangle = -i \frac{\partial}{\partial x} \langle x,p | \Psi \rangle, \qquad
\langle x,p| \hat{\lambda}_p | \Psi \rangle = -i \frac{\partial}{\partial p} \langle x,p | \Psi \rangle.
}[/math]
Projecting equation (KvN dynamical eq) onto [math]\displaystyle{ \langle x,p| }[/math], we get the equation of motion for the KvN wave function in the xp-representation
-
[math]\displaystyle{
\left[ \frac{\partial }{\partial t} + \frac{p}{m} \frac{\partial}{\partial x} - U'(x) \frac{\partial}{\partial p} \right] \langle x,p | \Psi(t) \rangle = 0.
}[/math]
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(KvN dynamical eq in xp)
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The quantity [math]\displaystyle{ \langle x,\, p |\Psi(t) \rangle }[/math] is the probability amplitude for a classical particle to be at point [math]\displaystyle{ x }[/math] with momentum [math]\displaystyle{ p }[/math] at time [math]\displaystyle{ t }[/math]. According to the axioms above, the probability density is given by
[math]\displaystyle{ \rho(x,p;t) = \left| \langle x, p |\Psi(t) \rangle \right|^2 }[/math]. Utilizing the identity
[math]\displaystyle{
\frac{\partial }{\partial t} \rho(x,p;t) = \langle \Psi(t) | x, p \rangle \frac{\partial }{\partial t} \langle x, p |\Psi(t) \rangle
+ \langle x, p |\Psi(t) \rangle \left( \frac{\partial }{\partial t} \langle x, p |\Psi(t) \rangle \right)^*
}[/math]
as well as (KvN dynamical eq in xp), we recover the classical Liouville equation
-
[math]\displaystyle{
\left[ \frac{\partial }{\partial t} + \frac{p}{m} \frac{\partial}{\partial x} - U'(x) \frac{\partial}{\partial p} \right] \rho(x,p;t) = 0.
}[/math]
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(Liouville eq)
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Moreover, according to the operator axioms and (xp eigenvec),
[math]\displaystyle{
\begin{align}
\langle A \rangle &= \langle \Psi (t)| A(\hat{x}, \hat{p}) | \Psi(t) \rangle
= \int dx \, dp \, \langle \Psi (t)| x,p\rangle A(x, p) \langle x,p | \Psi(t) \rangle \\
& = \int dx \, dp \, A(x, p) \langle \Psi (t)| x,p\rangle \langle x,p | \Psi(t) \rangle
= \int dx \, dp \, A(x, p) \rho(x,p;t).
\end{align}
}[/math]
Therefore, the rule for calculating averages of observable [math]\displaystyle{ A(x,p) }[/math] in classical statistical mechanics has been recovered from the operator axioms with the additional assumption [math]\displaystyle{ [ \hat{x}, \hat{p} ] = 0 }[/math]. As a result, the phase of a classical wave function does not contribute to observable averages. Contrary to quantum mechanics, the phase of a KvN wave function is physically irrelevant. Hence, nonexistence of the double-slit experiment[13][17][18] as well as Aharonov–Bohm effect[19] is established in the KvN mechanics.
Projecting KvN dynamical eq onto the common eigenvector of the operators [math]\displaystyle{ \hat{x} }[/math] and [math]\displaystyle{ \hat{\lambda}_p }[/math] (i.e., [math]\displaystyle{ x\lambda_p }[/math]-representation), one obtains classical mechanics in the doubled configuration space,[20] whose generalization leads
[20]
[21]
[22]
[23]
[24]
to the phase space formulation of quantum mechanics.