We begin from the following equations for expectation values of the coordinate x and momentum p
 $m{\frac {d}{dt}}\langle x\rangle =\langle p\rangle ,\qquad {\frac {d}{dt}}\langle p\rangle =\langle U'(x)\rangle ,$
aka, Newton's laws of motion averaged over ensemble. With the help of the operator axioms, they can be rewritten as
 ${\begin{aligned}m{\frac {d}{dt}}\langle \Psi (t){\hat {x}}\Psi (t)\rangle &=\langle \Psi (t){\hat {p}}\Psi (t)\rangle ,\\{\frac {d}{dt}}\langle \Psi (t){\hat {p}}\Psi (t)\rangle &=\langle \Psi (t)U'({\hat {x}})\Psi (t)\rangle .\end{aligned}}$
Notice a close resemblance with Ehrenfest theorems in quantum mechanics. Applications of the product rule leads to
 ${\begin{aligned}\langle d\Psi /dt{\hat {x}}\Psi \rangle +\langle \Psi {\hat {x}}d\Psi /dt\rangle &=\langle \Psi {\hat {p}}/m\Psi \rangle ,\\\langle d\Psi /dt{\hat {p}}\Psi \rangle +\langle \Psi {\hat {p}}d\Psi /dt\rangle &=\langle \Psi U'({\hat {x}})\Psi \rangle ,\end{aligned}}$
into which we substitute a consequence of Stone's theorem $id\Psi (t)/dt\rangle ={\hat {L}}\Psi (t)\rangle$ and obtain
 ${\begin{aligned}im\langle \Psi (t)[{\hat {L}},{\hat {x}}]\Psi (t)\rangle &=\langle \Psi (t){\hat {p}}\Psi (t)\rangle ,\\i\langle \Psi (t)[{\hat {L}},{\hat {p}}]\Psi (t)\rangle &=\langle \Psi (t)U'({\hat {x}})\Psi (t)\rangle .\end{aligned}}$
Since these identities must be valid for any initial state, the averaging can be dropped and the system of commutator equations for the unknown ${\hat {L}}$ is derived

$im[{\hat {L}},{\hat {x}}]={\hat {p}},\qquad i[{\hat {L}},{\hat {p}}]=U'({\hat {x}}).$


(commutator eqs for L)

Assume that the coordinate and momentum commute $[{\hat {x}},{\hat {p}}]=0$. This assumption physically means that the classical particle's coordinate and momentum can be measured simultaneously, implying absence of the uncertainty principle.
The solution ${\hat {L}}$ cannot be simply of the form ${\hat {L}}=L({\hat {x}},{\hat {p}})$ because it would imply the contractions $im[L({\hat {x}},{\hat {p}}),{\hat {x}}]=0={\hat {p}}$ and $i[L({\hat {x}},{\hat {p}}),{\hat {p}}]=0=U'({\hat {x}})$. Therefore, we must utilize additional operators ${\hat {\lambda }}_{x}$ and ${\hat {\lambda }}_{p}$ obeying

$[{\hat {x}},{\hat {\lambda }}_{x}]=[{\hat {p}},{\hat {\lambda }}_{p}]=i,\quad [{\hat {x}},{\hat {p}}]=[{\hat {x}},{\hat {\lambda }}_{p}]=[{\hat {p}},{\hat {\lambda }}_{x}]=[{\hat {\lambda }}_{x},{\hat {\lambda }}_{p}]=0.$


(KvN algebra)

The need to employ these auxiliary operators arises because all classical observables commute. Now we seek ${\hat {L}}$ in the form ${\hat {L}}=L({\hat {x}},{\hat {\lambda }}_{x},{\hat {p}},{\hat {\lambda }}_{p})$. Utilizing KvN algebra, the commutator eqs for L can be converted into the following differential equations
 ^{[7]}^{[9]}
 $mL'_{\lambda _{x}}(x,\lambda _{x},p,\lambda _{p})=p,\qquad L'_{\lambda _{p}}(x,\lambda _{x},p,\lambda _{p})=U'(x).$
Whence, we conclude that the classical KvN wave function $\Psi (t)\rangle$ evolves according to the Schrödingerlike equation of motion

$i{\frac {d}{dt}}\Psi (t)\rangle ={\hat {L}}\Psi (t)\rangle ,\qquad {\hat {L}}={\frac {\hat {p}}{m}}{\hat {\lambda }}_{x}U'({\hat {x}}){\hat {\lambda }}_{p}.$


(KvN dynamical eq)

Let us explicitly show that KvN dynamical eq is equivalent to the classical Liouville mechanics.
Since ${\hat {x}}$ and ${\hat {p}}$ commute, they share the common eigenvectors

${\hat {x}}x,p\rangle =xx,p\rangle ,\quad {\hat {p}}x,p\rangle =px,p\rangle ,\quad A({\hat {x}},{\hat {p}})x,p\rangle =A(x,p)x,p\rangle ,$


(xp eigenvec)

with the resolution of the identity
$1=\int dxdp\,x,p\rangle \langle x,p.$
Then, one obtains from equation (KvN algebra)
 $\langle x,p{\hat {\lambda }}_{x}\Psi \rangle =i{\frac {\partial }{\partial x}}\langle x,p\Psi \rangle ,\qquad \langle x,p{\hat {\lambda }}_{p}\Psi \rangle =i{\frac {\partial }{\partial p}}\langle x,p\Psi \rangle .$
Projecting equation (KvN dynamical eq) onto $\langle x,p$, we get the equation of motion for the KvN wave function in the xprepresentation

$\left[{\frac {\partial }{\partial t}}+{\frac {p}{m}}{\frac {\partial }{\partial x}}U'(x){\frac {\partial }{\partial p}}\right]\langle x,p\Psi (t)\rangle =0.$


(KvN dynamical eq in xp)

The quantity $\langle x,\,p\Psi (t)\rangle$ is the probability amplitude for a classical particle to be at point $x$ with momentum $p$ at time $t$. According to the axioms above, the probability density is given by
$\rho (x,p;t)=\left\langle x,p\Psi (t)\rangle \right^{2}$. Utilizing the identity
 ${\frac {\partial }{\partial t}}\rho (x,p;t)=\langle \Psi (t)x,p\rangle {\frac {\partial }{\partial t}}\langle x,p\Psi (t)\rangle +\langle x,p\Psi (t)\rangle \left({\frac {\partial }{\partial t}}\langle x,p\Psi (t)\rangle \right)^{*}$
as well as (KvN dynamical eq in xp), we recover the classical Liouville equation

$\left[{\frac {\partial }{\partial t}}+{\frac {p}{m}}{\frac {\partial }{\partial x}}U'(x){\frac {\partial }{\partial p}}\right]\rho (x,p;t)=0.$


(Liouville eq)

Moreover, according to the operator axioms and (xp eigenvec),
 ${\begin{aligned}\langle A\rangle &=\langle \Psi (t)A({\hat {x}},{\hat {p}})\Psi (t)\rangle =\int dxdp\,\langle \Psi (t)x,p\rangle A(x,p)\langle x,p\Psi (t)\rangle \\&=\int dxdp\,A(x,p)\langle \Psi (t)x,p\rangle \langle x,p\Psi (t)\rangle =\int dxdp\,A(x,p)\rho (x,p;t).\end{aligned}}$
Therefore, the rule for calculating averages of observable $A(x,p)$ in classical statistical mechanics has been recovered from the operator axioms with the additional assumption $[{\hat {x}},{\hat {p}}]=0$. 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 doubleslit experiment^{[6]}^{[10]}^{[11]} as well as Aharonov–Bohm effect^{[12]} is established in the KvN mechanics.
Projecting KvN dynamical eq onto the common eigenvector of the operators ${\hat {x}}$ and ${\hat {\lambda }}_{p}$ (i.e., $x\lambda _{p}$representation), one obtains classical mechanics in the doubled configuration space,^{[13]} whose generalization leads
^{[13]}
^{[14]}
^{[15]}
^{[16]}
^{[17]}
to the phase space formulation of quantum mechanics.