Quantum characteristics are phase-space trajectories that arise in the phase space formulation of quantum mechanics through the Wigner transform of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton equations in quantum form and play the role of characteristics in terms of which time-dependent Weyl's symbols of quantum operators can be expressed. In the classical limit, quantum characteristics reduce to classical trajectories. The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics.
In Hamiltonian dynamics, classical systems with
n
2n
\xii=(x1,\ldots,xn,p1,\ldots,pn)\in\R2n,
\{\xik,\xil\}=-Ikl.
Ikl
\left\|I\right\|= \begin{Vmatrix} 0&-En\ En&0 \end{Vmatrix},
where
En
n x n
In quantum mechanics, the canonical variables
\xi
\hat{\xi}i=(\hat{x}1,\ldots,\hat{x}n,\hat{p}1,\ldots,\hat{p}n)\in\operatorname{Op}(L2(\Rn)).
These operators act in Hilbert space and obey commutation relations
[\hat{\xi}k,\hat{\xi}l]=-i\hbarIkl.
Weyl’s association rule[1] extends the correspondence
\xii → \hat{\xi}i
A one-sided association rule
f(\xi)\to\hat{f}
\hat{f}=f(\hat{\xi})\equiv
infty | |
\sum | |
s=0 |
1 | |
s! |
\partialsf(0) | ||||||||||
|
i1 | |
\hat{\xi} |
\ldots
is | |
\hat{\xi} |
.
The operators
\hat{\xi}
f(\xi)
\hat{f}
Under the reverse association
f(\xi)\leftarrow\hat{f}
A refined version of the Weyl–Wigner association rule was proposed by Groenewold[3] and Stratonovich.[4]
The set of operators acting in the Hilbert space is closed under multiplication of operators by
c
V
\left.\begin{array}{c} \begin{array}{c} \left.\begin{array}{ccc} f(\xi)&\longleftrightarrow&\hat{f}\ g(\xi)&\longleftrightarrow&\hat{g}\ c x f(\xi)&\longleftrightarrow&c x \hat{f}\ f(\xi)+g(\xi)&\longleftrightarrow&\hat{f}+\hat{g} \end{array} \right\} vectorspace V \end{array} \ \begin{array}{ccc} {f(\xi)\starg(\xi)}&{\longleftrightarrow}& {\hat{f}\hat{g}} \end{array} \end{array} \right\}{algebra
f(\xi)
g(\xi)
\hat{f}
\hat{g}
The elements of basis of
V
\xii\in(-infty,+infty)
\hat{B}(\xi)=\int
d2nη | |
(2\pi\hbar)n |
\exp(-
i | |
\hbar |
ηk(\xi-\hat{\xi})k)\inV.
The Weyl–Wigner two-sided association rule for function
f(\xi)
\hat{f}
f(\xi)=\operatorname{Tr}[\hat{B}(\xi)\hat{f}],
\hat{f}=\int
d2n\xi | |
(2\pi\hbar)n |
f(\xi)\hat{B}(\xi).
The function
f(\xi)
\hat{f}
\hat{B}(\xi)
\int
d2n\xi | |
(2\pi\hbar)n |
\hat{B}(\xi)\operatorname{Tr}[\hat{B}(\xi)\hat{f}]=\hat{f},
\operatorname{Tr}[\hat{B}(\xi)\hat{B}(\xi\prime)]=(2\pi\hbar)n\delta2n(\xi-\xi\prime).
Alternative operator bases are discussed also.[5] The freedom in choice of the operator basis is better known as the operator ordering problem. The coordinates of particle trajectories in phase space depend on the operator basis.
The set of operators Op(L2(Rn)) is closed under the multiplication of operators. The vector space
V
f(\xi)=Tr[\hat{B}(\xi)\hat{f}]~~and~~g(\xi)=Tr[\hat{B}(\xi)\hat{g}],
f(\xi)\starg(\xi)=Tr[\hat{B}(\xi)\hat{f}\hat{g}]
\star
f(\xi)\starg(\xi)=f(\xi)\exp(
i\hbar | |
2 |
l{P})g(\xi),
l{P}=-{I}kl\overleftarrow{
\partial | } \overrightarrow{ | |
\partial\xik |
\partial | |
\partial\xil |
\star
f\starg=f\circg+
i\hbar | |
2 |
f\wedgeg.
In the classical limit, the
\circ
f\wedgeg
\star
\circ
The correspondence
\xi\leftrightarrow\hat{\xi}
\hat{U
\hat{U}=\expl(-
i | |
\hbar |
\hat{H}\taur),
\hat{H}
\begin{align} &{}\xi\stackrel{q}\longrightarrow\acute{\xi}\\ &{}\updownarrow \updownarrow\\ &{}\hat{\xi}\stackrel{\hat{U}}\longrightarrow\acute{\hat{\xi}} \end{align}
Quantum evolution transforms vectors in the Hilbert space and, under the Wigner association map, coordinates in the phase space. In the Heisenberg representation, the operators of the canonical variables transform as
\hat{\xi}i → \acute{\hat{\xi}i
\acute{\xi}i
\acute{\hat{\xi}i
\hat{B}(\xi)
\xii → \acute{\xi}i=qi(\xi,\tau)=Tr[\hat{B}(\xi)\hat{U}\dagger\hat{\xi}i\hat{U}],
qi(\xi,0)=\xii.
qi(\xi,\tau)
The set of operators of canonical variables is complete in the sense that any operator can be represented as a function of operators
\hat{\xi}
\hat{f} → \acute{\hat{f}}=\hat{U}\dagger\hat{f}\hat{U}
\begin{align} &{}f(\xi)\stackrel{q}\longrightarrow\acute{f}(\xi)=Tr[\hat{B}(\xi)\hat{U}\dagger\hat{f}\hat{U}]\\ &{}\updownarrow \updownarrow\\ &{}\hat{f} \stackrel{\hat{U}}\longrightarrow\acute{\hat{f}} =\hat{U}\dagger\hat{f}\hat{U} \end{align}
Using the Taylor expansion, the transformation of function
f(\xi)
f(\xi) → \acute{f}(\xi)\equivTr[\hat{B}(\xi)\hat{U\dagger
\star
The composition law differs from the classical one. However, the semiclassical expansion of
f(\starq(\xi,\tau))
f(q(\xi,\tau))
\hbar
qi(\xi,\tau)
The Wigner transform of the evolution equation for the density matrix in the Schrödinger representation leads to a quantum Liouville equation for the Wigner function. The Wigner transform of the evolution equation for operators in the Heisenberg representation,
\partial | |
\partial\tau |
\hat{f}=-
i | |
\hbar |
[\hat{f},\hat{H}],
\partial | |
\partial\tau |
f(\xi,\tau)=f(\xi,\tau)\wedgeH(\xi).
\star
f(\xi,\tau)=f(\starq(\xi,\tau),0).
W(\xi,\tau)=W(\starq(\xi,-\tau),0).
\omega2s
\omega2=Ikld\xik\curlywedged\xil
The Wigner function represents a quantum system in a more general form than the wave function. Wave functions describe pure states, while the Wigner function characterizes ensembles of quantum states. Any Hermitian operator can be diagonalized:
\hat{f}=\sumsλs|s\rangle\langles|
Those operators whose eigenvalues
λs
W(\xi)=\sumsλsWs(\xi),
with
λs\ge0
Ws(\xi)\starWr(\xi)=\deltasrWs(\xi)
The Quantum Hamilton's equations can be obtained applying the Wigner transform to the evolution equations for Heisenberg operators of canonical coordinates and momenta,
\partial | |
\partial\tau |
qi(\xi,\tau)=\{\zetai,H(\zeta)\}|\zeta.
The right-hand side is calculated like in the classical mechanics. The composite function is, however,
\star
\star
\tau
The antisymmetrized products of even number of operators of canonical variables are c-numbers as a consequence of the commutation relations. These products are left invariant by unitary transformations, which leads, in particular, to the relation
qi(\xi,\tau)\wedgeqj(\xi,\tau)=\xii\wedge\xij=-Iij.
In general, the antisymmetrized product
[i1 | |
q |
(\xi,\tau)\star
i2 | |
q |
(\xi,\tau)\star\ldots\star
i2s] | |
q |
(\xi,\tau)
is also invariant, that is, it does not depend on time, and moreover does not depend on the coordinate.
Phase-space transformations induced by the evolution operator preserve the Moyal bracket and do not preserve the Poisson bracket, so the evolution map
\xi → \acute{\xi}=q(\xi,\tau),
is not canonical beyond O(τ).[8] The first order in τ defines the algebra of the transformation group. As previously noted, the algebra of canonical transformations of classical mechanics coincides with the algebra of unitary transformations of quantum mechanics. These two groups, however, are different because the multiplication operations in classical and quantum mechanics are different.
Transformation properties of canonical variables and phase-space functions under unitary transformations in the Hilbert space have important distinctions from the case of canonical transformations in the phase space.
Quantum characteristics can hardly be treated visually as trajectories along which physical particles move. The reason lies in the star-composition law
q(\xi,\tau1+\tau2)=q(\starq(\xi,\tau1),\tau2),
The energy conservation implies
H(\xi)=H(\starq(\xi,\tau)),
H(\xi)=Tr[\hat{B}(\xi)\hat{H}]
H(\xi)
The origin of the method of characteristics can be traced back to Heisenberg’s matrix mechanics. Suppose that we have solved in the matrix mechanics the evolution equations for the operators of the canonical coordinates and momenta in theHeisenberg representation. These operators evolve according to
\hat{\xi}i → \hat{\xi}i(\tau)=\hat{U}\dagger\hat{\xi}i\hat{U}.
\hat{f}
\hat{f}
f(\hat{\xi})
\hat{f}
\hat{f}(\tau)=\hat{U}\dagger\hat{f}\hat{U}=\hat{U}\daggerf(\hat{\xi})\hat{U}=f(\hat{U}\dagger\hat{\xi}\hat{U})=f(\hat{\xi}(\tau)).
\hat{\xi}(\tau)
Liouville equation | ||||||||||
First-order PDE | Infinite-order PDE | |||||||||
\rho(\xi,\tau)=-\{\rho(\xi,\tau),l{H}(\xi)\} |
W(\xi,\tau)=-W(\xi,\tau)\wedgeH(\xi) | |||||||||
Hamilton's equations | ||||||||||
Finite-order ODE | Infinite-order PDE | |||||||||
ci(\xi,\tau)=\{\zetai,l{H}(\zeta)\} | _ |
qi(\xi,\tau)=\{\zetai,H(\zeta)\} | _ | |||||||
Initial conditions | Initial conditions | |||||||||
ci(\xi,0)=\xii | qi(\xi,0)=\xii | |||||||||
Composition law | ||||||||||
Dot-composition | \star | |||||||||
c(\xi,\tau1+\tau2)=c(c(\xi,\tau1),\tau2) | q(\xi,\tau1+\tau2)=q(\starq(\xi,\tau1),\tau2) | |||||||||
Invariance | ||||||||||
Poisson bracket | Moyal bracket | |||||||||
\{ci(\xi,\tau),cj(\xi,\tau)\}=\{\xii,\xij\} | qi(\xi,\tau)\wedgeqj(\xi,\tau)=\xii\wedge\xij | |||||||||
Energy conservation | ||||||||||
Dot-composition | \star | |||||||||
H(\xi)=H(c(\xi,\tau)) | H(\xi)=H(\starq(\xi,\tau)) | |||||||||
Solution to Liouville equation | ||||||||||
Dot-composition | \star | |||||||||
\rho(\xi,\tau)=\rho(c(\xi,-\tau),0) | W(\xi,\tau)=W(\starq(\xi,-\tau),0) |
Table compares properties of characteristics in classical and quantum mechanics. PDE and ODE indicate partial differential equations and ordinary differential equations, respectively. The quantum Liouville equation is the Weyl–Wigner transform of the von Neumann evolution equation for the density matrix in the Schrödinger representation. The quantum Hamilton equations are the Weyl–Wigner transforms of the evolution equations for operators of the canonical coordinates and momenta in the Heisenberg representation.
In classical systems, characteristics
ci(\xi,\tau)
qi(\xi,\tau)
qi(\xi,\tau)
f(\xi,\tau)
The quantum phase flow contains all of the information about the quantum evolution. Semiclassical expansion of quantum characteristics and
\star
de:Amand Fäßler
. Faessler, A. . 2007 . Semiclassical expansion of quantum characteristics for many-body potential scattering problem . Annalen der Physik . 519 . 9. 587–614 . 10.1002/andp.200610251 . 2007AnP...519..587K . nucl-th/0605015 .