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Quantum Hamiltonian Mechanics on Symplectic Manifolds

  • Maciej Błaszak
Chapter

Abstract

In the previous chapter we presented the general theory of quantum deformations of classical Poisson algebras. In the following chapter we develop a deformation procedure applied to classical statistical Hamiltonian mechanics (described in Sect.  3.3) in order to construct its quantum analogue on the phase space. First, we define quantum states as appropriate deformations of classical states and their time development through the respective deformation of the classical Liouville equation. Then we introduce quantum Hamiltonian equations of motion being a deformation of classical Hamiltonian equations and time development of quantum observables. With particular care we present the theory of quantum flow and quantum trajectories on a phase space together with a wide range of examples which illustrate the presented formalism. Such constructed quantum theories (each related with an appropriate quantum algebra) reduce to a common classical counterpart as deformation parameter \(\hslash \) tends to zero: \(\hslash \rightarrow 0.\)

In the previous chapter we presented the general theory of quantum deformations of classical Poisson algebras. In the following chapter we develop a deformation procedure applied to classical statistical Hamiltonian mechanics (described in Sect.  3.3) in order to construct its quantum analogue on the phase space. First, we define quantum states as appropriate deformations of classical states and their time development through the respective deformation of the classical Liouville equation. Then we introduce quantum Hamiltonian equations of motion being a deformation of classical Hamiltonian equations and time development of quantum observables. With particular care we present the theory of quantum flow and quantum trajectories on a phase space together with a wide range of examples which illustrate the presented formalism. Such constructed quantum theories (each related with an appropriate quantum algebra) reduce to a common classical counterpart as deformation parameter \(\hslash \) tends to zero: \(\hslash \rightarrow 0.\)

7.1 General Theory of Quantization

7.1.1 Quantum States

By definition, by an analogy with the classical case (cf. Sect.  3.3.1), quantum states related to quantum Poisson algebra Open image in new window are those functions ρ ∈ L2(M, d Ωħ) which satisfy the following conditions
  1. 1.

    ρ = ρ (self-conjugation),

     
  2. 2.

    Open image in new window (normalization),

     
  3. 3.

    Open image in new window for \(f\in C_{0}^{\infty }(M)\) (positive-definiteness).

     
Quantum states form a convex subset of the Hilbert space L2(M, d Ωħ). Pure states are defined as extreme points of the set of states, i.e. as those states which cannot be written as convex linear combinations of some other states. Thus ρpure is a pure state if and only if there do not exist two different states ρ1 and ρ2 such that ρpure = 1 + (1 − p)ρ2 for some p ∈ (0, 1). A state which is not pure is further called a mixed state.
For certain symplectic manifolds M pure states can be alternatively characterized as functions ρpure ∈ L2(M, d Ωħ) which are self-conjugated, normalized, and idempotent (cf. Sect.  3.3.1): Mixed states ρmix ∈ L2(M, d Ωħ) can be characterized as convex linear combinations, possibly infinite, of pure states \(\rho _{\text{pure}}^{(\lambda )}\) where pλ ≥ 0 and ∑λpλ = 1.
The interpretation of pure and mixed states is similar as in classical mechanics. When we have the full knowledge of the state of the system, then the system is described by a pure state. If we only know that the system is in some pure state with some probability, then the system must be described by a mixed state. The quantum states ρ are the analogue of the classical distribution functions representing states of the classical Hamiltonian system and considered in Sect.  3.3.1. The difference between classical and quantum distribution functions is that the latter do not have to be non-negative everywhere. Thus, ρ(x, p) cannot be interpreted as a probability density of finding a particle in a point (x, p) of the phase space. This is a consequence of the fact that x and p coordinates do not commute with respect to the ⋆ -multiplication, which yield, from the Heisenberg uncertainty principle, that it is impossible to measure simultaneously the position and momentum of a particle like in classical mechanics. Hence, the point position of a particle in the phase space does not make sense anymore. On the other hand it is possible to introduce the so called marginal distributions
$$\displaystyle \begin{aligned} P(x)=\int (S^{-1}\rho )(x,p)d\mu (p),\ \ \ \ P(p)=\int (S^{-1}\rho )(x,p)d\mu (x), {} \end{aligned} $$
(7.1.3)
which are probabilistic distribution functions and can be interpreted as probability densities that a particle in the phase space has position x or momentum p (see Sect.  8.1.2). In (7.1.3) S-operator links a given ⋆ -product with the Moyal one. The result is not surprising as each marginal distribution depends on commuting coordinates only. Note, however, that only in the case of the ⋆M-product the marginal distributions are received by simple integration of a distribution function with respect to x or p variable. In general, the distribution function first has to be transformed with the isomorphism S.
Finally, for a given observable Open image in new window and state ρ the expectation value of the observable A in the state ρ is defined by being the analogue of respective classical notion ( 3.3.5).

7.1.2 Time Evolution of Quantum Systems

The quantum time evolution of a system is governed by a quantum Hamilton function Open image in new window which is, like in classical mechanics, a distinguished observable, being a deformation of a classical Hamilton function HC and self-conjugated with respect to involution ∗ of respective ⋆ -algebra. Like in the classical theory, there are two equivalent points of view on the time evolution: quantum Schrödinger picture and quantum Heisenberg picture. In the Schrödinger picture states undergo time development while observables do not. An equation of motion for states which is the analogue of classical Liouville equation ( 3.3.11) takes the form The formal solution of (7.1.5) is of the form
$$\displaystyle \begin{aligned} \rho (t)=U(t)\star \rho (0)\star U(t)^{\ast }, {} \end{aligned} $$
(7.1.6)
where
$$\displaystyle \begin{aligned} U(t)=\exp _{\star }\left( -\frac{i}{\hbar }tH\right) =\sum_{k=0}^{\infty } \frac{1}{k!}\left( -\frac{i}{\hbar }t\right) ^{k}\underbrace{H\star \dotsm \star H}_{k} {} \end{aligned} $$
(7.1.7)
is a unitary function as H is self-conjugated
$$\displaystyle \begin{aligned} H=H^{\ast }\rightarrow U(t)^{\ast }= \overline{U(t)} \end{aligned}$$
and hence
$$\displaystyle \begin{aligned} U(t)\star \overline{U(t)}=\overline{U(t)}\star U(t)=1. {} \end{aligned} $$
(7.1.8)
Here \(\phi (t)\equiv \exp _{\star }tB\) is the noncommutative exponential solution of
$$\displaystyle \begin{aligned} \frac{\partial\phi}{\partial t}=B\star \phi=\phi\star B, \quad \phi(0)=1. \end{aligned}$$

In consequence, the time evolution of states can be expressed in terms of the one-parameter group of unitary functions U(t). Notice that quantum Liouville equation (von Neumann equation) on the symplectic manifold is represented like its classical analogue by a linear PDE.

States ρ which do not depend explicitly on time: \( \frac {\partial \rho }{\partial t}=0\), are called stationary states and hence fulfill the relation
$$\displaystyle \begin{aligned} \left[ H,\rho \right] =0. {} \end{aligned} $$
(7.1.9)
As will be shown in Sect.  8.1.2, if ρ is a pure state, then ( 7.1.9) is equivalent to a pair of ⋆ -genvalue problems Notice that E in (7.1.10) is equal to the expectation value of Hamiltonian H in a pure stationary state ρ, i.e. is equal to energy of a system in that state
$$\displaystyle \begin{aligned} \left\langle H\right\rangle _{\rho }=\int_{M}\left( H\star \rho \right) d\Omega _{\hbar }=E\int_{M}\rho d\Omega _{\hbar }=E. \end{aligned}$$
From (7.1.5) follows that time evolution of expectation value of observable \(A\in \mathcal {A}_{Q}\) in a state ρ(t), i.a. \(\left \langle A\right \rangle _{\rho (t)}\), fulfills the following equation of motion Indeed
In the Heisenberg picture states remain still whereas observables undergo the time evolution. The time evolution of an observable Open image in new window is given by the action of the unitary function U(t) from (7.1.7) on A: where and U(t) is given by (7.1.7).
Differentiating (7.1.12) with respect to t results in the evolution equation for A: Equation (7.1.13) is the quantum analogue of the classical equation (  3.3.13).
In particular, quantum Hamiltonian equations of motion are of the form and, as in the classical case, are nonlinear PDE’s and represent a quantum Hamiltonian transport. What is important, the system of PDE’s (7.1.14) is equivalent to the system of ordinary differential equations but in the space of ⋆  -functions. Indeed, any function \(A\in \mathcal {A}_{Q}\) can be expanded in a ⋆ -power series ( 6.1.47). In particular, any monomial xnpm can be expressed as an ⋆ -polynomial ( 6.2.16)
$$\displaystyle \begin{aligned} x^{n}p^{m}=\left( x^{n}p^{m}\right) \star 1=\left( x^{n}p^{m}\right) _{S}(x\star ,p\star )1. {} \end{aligned} $$
(7.1.15)
So, equations (7.1.14) can be written as the system of ODE’s in the space of ⋆ -functions in the form Notice, that according to Observation  12, equations (7.1.16) can be always transformed to the Moyal case in (x, p) coordinates and Weyl ordering, with transformed Hamiltonian H(ħ) = S(ħ)H.

Example 7.1

Consider the Hamiltonian system on \(M=\mathbb {R}^{2}\) with the classical Hamiltonian function in canonical coordinates (x, p)
$$\displaystyle \begin{aligned} H(x,p)=\kappa x^{2}p^{2}, \quad \kappa>0. \end{aligned}$$
Classical equations of motion are
$$\displaystyle \begin{aligned} Q_{t} &=\{Q(x,p,t),H(x,p)\}_{(x,p)}=\{Q,H(Q,P)\}_{(Q,P)}=\frac{\partial H}{ \partial P}=2\kappa Q^{2}P,\ \ \ Q(0)=x,\ \\ P_{t} &=\{P(x,p,t),H(x,p)\}_{(x,p)}=\{P,H(Q,P)\}_{(Q,P)}=-\frac{\partial H}{ \partial Q}=-2\kappa QP^{2},\ \ P(0)=p. \end{aligned} $$
If the quantization is given by ⋆ -product which in (x, p) chart takes the Moyal form, then quantum equations of motion, according to (7.1.16), take the form where Q(0) = x, P(0) = p and \(\star \equiv \star _{M}^{(x,p)}.\) The solution of classical and quantum dynamics will be considered in the next section.
Like in the classical case, both presented approaches to the time evolution yield equal predictions concerning the results of measurements, since from the property of trace ( 6.1.11)
$$\displaystyle \begin{aligned} \langle {A(0)}\rangle _{\rho (t)} =\int_{M}A(0)\star \rho (t)\,d\Omega _{\hbar }=\int_{M}A(t)\star \rho (0)\,d\Omega _{\hbar }=\langle {A(t)}\rangle _{\rho (0)}. \end{aligned}$$

Observation 14

Comparing the results of Sects.  3.3.2 and7.1.2we observe that the linear aspect of classical and quantum Hamiltonian mechanics is represented by time evolution of states, described on both levels by linear PDE (the so called Schrödinger picture). On the other hand, the nonlinear aspect of both theories is represented by time evolution of observables, described on both levels by nonlinear ODE (the so called Heisenberg picture) defined on an appropriate space of ordinary-functions and star-functions, respectively. Contrary to a classical case, on a quantum level Hamiltonian equations of motion belong only to the Heisenberg picture as pure coherent classical states (  3.3.16 ) are not admissible as quantum states.

7.2 Quantum Trajectories in Phase Space

The time evolution of a classical Hamiltonian system is fully determined by trajectories (a flow) in a phase space (see Sect.  3.3.2 ). Once we calculate a classical flow Φt for the given system a time evolution of states and observables can be received by simply composing them with Φt. A classical flow is defined as a map Φt: M → M on the phase space M, which at every point ξ0 ∈ M gives a trajectory (curve) γ(t) = Φt(ξ0) on M passing through the point ξ0 and being a solution of the Hamilton’s equations ( 3.2.35). Moreover, any trajectory Φt(ξ0) has the property of being a classical canonical transformation for every t, and the set \(\{\Phi _{t}\}_{t\in \mathbb {R}}\) have a structure of a group with multiplication being a composition of maps.

From the very beginning of quantum physics, a lot of efforts have been taken to formulate some kind of an analogue of phase space trajectories in quantum mechanics [95]. The most common approaches to quantum dynamics are the de Broglie-Bohm approach [50, 51, 154], the average value approach [181, 266], and the Moyal trajectories approach (see [93, 174] and references therein).

In the following section we develop the theory of Moyal trajectories resulting from quantum Hamiltonian equations (7.1.14). In consequence, the time evolution of observables cannot be given as a simple composition of observables with a quantum flow. For this reason in papers [93] and [174] observables were considered to be ⋆  -functions. Then the action of a flow on observables was given as a ⋆ -composition.

In the approach presented in this section we treat observables as ordinary functions on a classical phase space. We also present in an explicit form a quantum action of a flow on observables, which is a deformation of the respective classical action. The resulting time dependence of observables gives an appropriate solution of a quantum time evolution equation for observables (7.1.13). Then, we show that a set of quantum symplectomorphisms (quantum flow) has a structure of a group with multiplication (quantum composition) being a deformation of the ordinary composition regarded as a multiplication in a group of classical symplectomorphisms (classical flow) [34]. Such an approach to quantum trajectories has a benefit in that it is not needed to calculate the form of observables as star-functions, but only a quantum action of a given trajectory needs to be found.

7.2.1 Quantum Flow

Let us consider the Moyal quantization of a classical Hamiltonian system (M, π, H), where \(M=\mathbb {R}^{2N}\), \(\pi =\partial _{x^{i}}\wedge \partial _{p_{i}}\), and H ∈ C(M) is an arbitrary real function. Then the solution of quantum Hamiltonian equations where Qi(x, p, 0) = xi and Pj(x, p, 0) = pj, i.e., the Heisenberg representation for observables of position and momentum, generates a quantum flow Φt in a phase space according to an equation
$$\displaystyle \begin{aligned} \Phi _{t}(x,p;\hbar )=(Q(x,p,t;\hbar ),P(x,p,t;\hbar )). {} \end{aligned} $$
(7.2.2)
For every instance of time t the map Φt is a quantum canonical transformation ( quantum symplectomorphism) from coordinates (x, p) to new coordinates x =  Q(x, p, t;ħ), p =  P(x, p, t;ħ). It means that Φt preserves the quantum Poisson bracket Open image in new window , which can be seen from (7.2.3) and the fact that Open image in new window Open image in new window .
The flow Φt, treated as a quantum canonical transformation, can act on observables and states as simple composition of maps. Such a classical action can also be used to transform the algebraic structure of the quantum Poisson algebra so that the action will be an isomorphism of the initial algebra and its transformation. So, a star-product ⋆t being the Moyal product transformed by \(\Phi _{t}^{-1}\) is defined by the formula As we know from our previous considerations, the ⋆t- product takes the form where vector fields \(D_{x^{i}}\), \(D_{p_{i}}\) are transformations of coordinate vector fields \(\partial _{x^{i}}\), \(\partial _{p_{i}}\):
$$\displaystyle \begin{aligned} (\partial _{x^{i}}f)\circ \Phi _{t}^{-1}=D_{x^{i}}(f\circ \Phi _{t}^{-1}),\quad (\partial _{p_{i}}f)\circ \Phi _{t}^{-1}=D_{p_{i}}(f\circ \Phi _{t}^{-1}). {} \end{aligned}$$
The most important for our further construction is the observation that the ⋆t-product is gauge equivalent to the Moyal product. In other words, to a quantum flow Φt there corresponds a unique isomorphism St satisfying Observe, that for the ⋆t-algebra the involution is also the complex-conjugation.
A formal solution of the time evolution equation (7.1.13) for an observable \(A\in \mathcal {A}_{Q}\) can be expressed by the formula (confront with (7.1.12), (7.1.7) and (7.1.2)). In particular, the solution of (7.2.1) is of the form which for the fixed initial condition \(Q^{i}(x,p,0)=x_{0}^{i}\) and Pj(x, p, 0) = p0j represents a particular quantum trajectory.
A time evolution of an observable \(A\in \mathcal {A}_{Q}\) should be alternatively expressed by action of the quantum flow Φt on A. The composition of Φt with observables, i.e. the classical action of Φt on observables, does not result in a proper time evolution of observables and thus it is necessary to deform this classical action. It will be proved that a proper action of the quantum flow Φt on functions from \(\mathcal {A}_{Q}\) (a pull-back of Φt) is given by the new formula where St is an isomorphism associated to the quantum canonical transformation \(\Phi _{t}^{-1}\).
The formula (7.2.4) can be proved first by noting that and similarly where the fact that Stxi = xi and Stpj = pj was used, being on the other hand a consequence of the quantum canonicity of Φt. Secondly, \(\Phi _{t}^{\ast }\) given by (7.2.4) is an automorphism of \(\mathcal {A}_{Q}\) as
$$\displaystyle \begin{aligned} \begin{array}{rcl} \Phi _{t}^{\ast }(A\star B) &\displaystyle =&\displaystyle (S_{t}(A\star B))\circ \Phi _{t}=(S_{t}A\star _{t}S_{t}B)\circ \Phi _{t} \\ &\displaystyle =&\displaystyle \left( (S_{t}A)\circ \Phi _{t}\right) \star \left( (S_{t}B)\circ \Phi _{t}\right) =\Phi _{t}^{\ast }A\star \Phi _{t}^{\ast }B, \end{array} \end{aligned} $$
where ⋆t denotes a star-product transformed by \(\Phi _{t}^{-1}\). Thus holds true since, as was proved earlier, every function from \(\mathcal {A} _{Q} \) can be presented as a ⋆ - power series.
In a complete analogy with classical theory one can define a quantum Hamiltonian vector field by Open image in new window . Then (7.2.5) states that Φt is a flow of the quantum Hamiltonian vector field ζH. Moreover, in an analogy with classical mechanics, { Φt} is a one-parameter group of quantum canonical transformations with respect to a new multiplication defined by where \(S_{t_{2}}\Phi _{t_{1}}\) denotes a map \(\mathbb {R}^{2N}\rightarrow \mathbb {R}^{2N}\) given by the formula
$$\displaystyle \begin{aligned} S_{t_{2}}\Phi _{t_{1}}=(S_{t_{2}}Q^{1}(t_{1}),\dotsc ,S_{t_{2}}P_{N}(t_{1})), \end{aligned}$$
where \(\Phi _{t_{1}}=(Q^{1}(t_{1}),\dotsc ,Q^{N}(t_{1}),P_{1}(t_{1}),\dotsc ,P_{N}(t_{1}))\). Multiplication defined in such a way satisfies properties similar to their classical counterparts (composition):
$$\displaystyle \begin{aligned} \Phi _{0}=\operatorname{\mathrm{id}},\quad \Phi _{t_{1}}\cdot \Phi _{t_{2}}=\Phi _{t_{1}+t_{2}}, \end{aligned}$$
proving that { Φt} is a group. Further on we will call it a quantum composition. The quantum composition rule given by (7.2.6) is properly defined since it respects the quantum pull-back of flows:
$$\displaystyle \begin{aligned} (\Phi _{t_{1}}\cdot \Phi _{t_{2}})^{\ast }=\Phi _{t_{2}}^{\ast }\circ \Phi _{t_{1}}^{\ast }. {} \end{aligned} $$
(7.2.7)
Indeed, it is enough to prove (7.2.7) for an arbitrary ⋆  -monomial. For simplicity we will present the proof for a two-dimensional case and for a ⋆ - monomial x ⋆ p. Using the fact that Stx = x and Stp = p for every t one calculates that
$$\displaystyle \begin{aligned} (\Phi _{t_{2}}^{\ast }\circ \Phi _{t_{1}}^{\ast })(x\star p)& =\Phi _{t_{2}}^{\ast }\big((S_{t_{1}}(x\star p))\circ \Phi _{t_{1}}\big)=\Phi _{t_{2}}^{\ast }\big((x\star _{t_{1}}p)\circ \Phi _{t_{1}}\big) \\ & =\Phi _{t_{2}}^{\ast }\big(Q(t_{1})\star P(t_{1})\big)=\big( S_{t_{2}}(Q(t_{1})\star P(t_{1}))\big)\circ \Phi _{t_{2}} \\ & =\big(S_{t_{2}}Q(t_{1})\star _{t_{2}}S_{t_{2}}P(t_{1})\big)\circ \Phi _{t_{2}}=(x\star _{t_{2},t_{1}}p)\circ S_{t_{2}}\Phi _{t_{1}}\circ \Phi _{t_{2}}, \end{aligned} $$
where \(\star _{t_{1}}\), \(\star _{t_{2}}\), denote Moyal products transformed, respectively, by transformations \(\Phi _{t_{1}}^{-1}\), \(\Phi _{t_{2}}^{-1}\), and \(\star _{t_{2},t_{1}}\) denotes the \(\star _{t_{2}}\)-product transformed by \((S_{t_{2}}\Phi _{t_{1}})^{-1}\). From the relation \(S_{T_{1}\circ T_{2}}=S_{T_{1},T_{2}}S_{T_{1}}\) valid for any transformations T1, T2 defined on the whole phase space (\( S_{T_{1}\circ T_{2}}\) is an isomorphism intertwining star-products ⋆  and \(\star _{T_{1}\circ T_{2}}\), \(S_{T_{1},T_{2}}\) intertwines \(\star _{T_{1}}\) with \(\star _{T_{1}\circ T_{2}}\), and \(S_{T_{1}}\) intertwines ⋆  with \(\star _{t_{1}}\), where \(\star _{t_{1}}\) and \(\star _{T_{1}\circ T_{2}}\) are the Moyal products transformed, respectively, by transformations T1 and T1 ∘ T2), one finds that
$$\displaystyle \begin{aligned} S_{(\Phi _{t_{1}}\Phi _{t_{2}})^{-1}}(x\star p)=S_{\Phi _{t_{2}}^{-1},(S_{t_{2}}\Phi _{t_{1}})^{-1}}S_{t_{2}}(x\star p)=S_{\Phi _{t_{2}}^{-1},(S_{t_{2}}\Phi _{t_{1}})^{-1}}(x\star _{t_{2}}p)=x\star _{t_{2},t_{1}}p \end{aligned}$$
and hence
$$\displaystyle \begin{aligned} (\Phi _{t_{2}}^{\ast }\circ \Phi _{t_{1}}^{\ast })(x\star p)=S_{(\Phi _{t_{1}}\Phi _{t_{2}})^{-1}}(x\star p)\circ S_{t_{2}}\Phi _{t_{1}}\circ \Phi _{t_{2}}=(\Phi _{t_{1}}\cdot \Phi _{t_{2}})^{\ast }(x\star p). \end{aligned}$$
As a direct consequence of these considerations and the fact that for the Moyal product yields the following observation.

Observation 15

Quantum trajectories of the linear Hamiltonian systems coincide with classical trajectories. It follows from the fact that for the Hamiltonian functions being quadratic polynomials of phase space coordinates:\(\left [ \left \vert H,\cdot \right \vert \right ] =\{H,\cdot \}\)(7.2.8). Besides, because solutions Q(t) and P(t) are linear in xi, pi, so St = 1. It means that the quantum group multiplication (7.2.6) (quantum composition) coincides with the classical composition (7.2.10) and in consequence, the quantum time evolution of any observable A is the same as the classical time evolution of A. For such systems, the only difference between the classical and quantum dynamics relies on different admissible states in which the evolution takes place. On the other hand, even in such simplest cases, classical and quantum systems differ fundamentally on the level of stationary states.

In the limit ħ → 0, (7.2.3) reduces to classical phase space trajectories
$$\displaystyle \begin{gathered} Q^{j}(t)=e^{-t\{H,\,\cdot \}}Q^{j}(0),\quad P_{j}(t)=e^{-t\{H,\,\cdot \}}P_{j}(0), \\ Q^{j}(x,p,0)=x^{j},\quad P_{j}(x,p,0)=p_{j}, \end{gathered} $$
which are formal solutions of classical Hamiltonian equations
$$\displaystyle \begin{aligned} \left( Q^{j}\right) _{t}=\{Q^{j}(t),H\},\quad \left( P_{j}\right) _{t}=\{P_{j}(t),H\}. {} \end{aligned}$$
In a more explicit form classical trajectories are represented by a flow (classical symplectomorphism)
$$\displaystyle \begin{aligned} \Phi _{t}(x,p)=(Q(x,p,t),P(x,p,t)), {} \end{aligned} $$
(7.2.9)
which is an ħ → 0 limit of the quantum flow (7.2.2) (quantum symplectomorphism). An action of the classical flow Φt on functions from \(\mathcal {A}_{C}\) (a pull-back of Φt) is just a simple composition of functions with Φt, being an ħ → 0 limit of (7.2.4)
$$\displaystyle \begin{aligned} \Phi _{t}^{\ast }A=A\circ \Phi _{t}. {} \end{aligned}$$
{ Φt} forms a one-parameter group of canonical transformations, preserving a classical Poisson bracket: \(\{Q^{i}(t),P_{j}(t)\}=\delta _{j}^{i}\), with multiplication being an ordinary composition of maps
$$\displaystyle \begin{aligned} \Phi _{t_{1}}\cdot \Phi _{t_{2}}=\Phi _{t_{1}}\circ \Phi _{t_{2}}, {} \end{aligned} $$
(7.2.10)
which is the ħ → 0 limit of quantum composition (7.2.6).

7.2.2 Quantum Dynamics with Classical Trajectories

In the following subsection we analyze fairly accurately a few simple examples of quantum systems, with quantization defined by the Moyal product, for which classical and quantum trajectories coincide. Let us start from a free particle in one dimension. The free particle is a system, whose time evolution is governed by a Hamiltonian
$$\displaystyle \begin{aligned} H(x,p)= \tfrac{1}{2}p^{2}, {} \end{aligned}$$
where the mass of the particle m = 1. This Hamiltonian describes only the kinetic energy of the particle. It does not contain any terms describing the potential energy, i.e. there are no forces acting on the particle (the particle is free).
From relation (7.2.8) it follows that \(\Phi _{t}^{\ast }(\hbar )=\Phi _{t}^{\ast }(0)\) as Thus, a common quantum and classical flow of a free particle is of the form Φt(x, p;ħ) = Φt(x, p) = (Q(t), P(t)), where
$$\displaystyle \begin{aligned} Q(t)=x+pt,\ \ \ P(t)=p. {} \end{aligned} $$
(7.2.11)
Besides, because St = 1 for any linear transformation, so the time evolution of any classical and quantum observable A(x, p) is given by
$$\displaystyle \begin{aligned} A(t)=A(Q(t),P(t)). {} \end{aligned} $$
(7.2.12)
So, what is a difference between the classical and quantum free dynamics? Let me remind that both, on the classical and the quantum level, “physics” are represented by expectation values of observables \(\left \langle A\right \rangle _{\rho }\) in a chosen admissible state ρ. For the classical system, Eqs. (7.2.11) and (7.2.12) represent simultaneously the dynamics of expectation values of observables (position and momentum in particular) in the pure coherent classical state
$$\displaystyle \begin{aligned} \rho _{C}(x^{\prime },p^{\prime })=\delta (x^{\prime }-x)\delta (p^{\prime }-p), {} \end{aligned} $$
(7.2.13)
as
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\langle Q(t)\right\rangle _{\rho _{C}} &\displaystyle =&\displaystyle \int_{\mathbb{R}^{2}}\rho _{C}(x^{\prime },p^{\prime })Q(x^{\prime },p^{\prime },t)dx^{\prime }dp^{\prime }=Q(x,p,t), \\ \left\langle P(t)\right\rangle _{\rho _{C}} &\displaystyle =&\displaystyle \int_{\mathbb{R}^{2}}\rho _{C}(x^{\prime },p^{\prime })P(x^{\prime },p^{\prime },t)dx^{\prime }dp^{\prime }=P(x,p,t) \end{array} \end{aligned} $$
and
$$\displaystyle \begin{aligned} (\varDelta Q(t))^{2}=\left\langle Q^{2}(t)\right\rangle _{\rho _{C}}-\left\langle Q(t)\right\rangle _{\rho _{C}}^{2}=0,\ \ (\varDelta P(t))^{2}=\left\langle P^{2}(t)\right\rangle _{\rho _{C}}-\left\langle P(t)\right\rangle _{\rho _{C}}^{2}=0. {} \end{aligned} $$
(7.2.14)
On the other hand, the state (7.2.13) is not an admissible quantum state.
Let us consider a one-parameter family of pure quantum states ρ of the form
$$\displaystyle \begin{aligned} \rho _{Q}(x^{\prime },p^{\prime },\gamma )=2\exp \left( -\frac{\gamma }{ \hbar }(x^{\prime }-x)^{2}\right) \exp \left( -\frac{1}{\gamma \hbar } (p^{\prime }-p)^{2}\right) ,\ \ \ \ \ \gamma \in \mathbb{R}_{+}. {} \end{aligned} $$
(7.2.15)
They all are pure states as one can show by direct calculations that
$$\displaystyle \begin{aligned} \int_{\mathbb{R}^{2}}\rho _{Q}(x^{\prime },p^{\prime },\gamma )d\Omega _{\hbar }^{\prime }=\frac{1}{2\pi \hbar }\int_{\mathbb{R}^{2}}\rho _{Q}(x^{\prime },p^{\prime },\gamma )dx^{\prime }dp^{\prime }=1, {} \end{aligned} $$
(7.2.16)
and
$$\displaystyle \begin{aligned} \rho _{Q}(x^{\prime },p^{\prime },\gamma )\star \rho _{Q}(x^{\prime },p^{\prime },\gamma )=\rho _{Q}(x^{\prime },p^{\prime },\gamma ). {} \end{aligned} $$
(7.2.17)
States (7.2.15) are simultaneously coherent states as they minimize quantum uncertainty relation: \(\varDelta x\varDelta p=\tfrac {1}{2}\hbar .\) It follows directly from the property of the Gauss distribution
$$\displaystyle \begin{aligned} f(z;\sigma ,\mu )=\frac{1}{\sigma \sqrt{2\pi }}\exp \left( -\frac{(z-\mu )^{2}}{2\sigma ^{2}}\right) , \end{aligned}$$
for which
$$\displaystyle \begin{aligned} \left\langle z\right\rangle _{f}=\int_{\mathbb{R}}zfdz=\mu ,\ \ \ \left\langle z^{2}\right\rangle _{f}=\int_{\mathbb{R}}z^{2}fdz=\mu ^{2}+\sigma ^{2}, \end{aligned}$$
and then
$$\displaystyle \begin{aligned} (\varDelta x)^{2}=\left\langle x^{2}\right\rangle _{\rho _{Q}}-\left\langle x\right\rangle _{\rho _{Q}}^{2}=\frac{\hbar }{2\gamma },\ \ \ \ (\varDelta p)^{2}=\left\langle p^{2}\right\rangle _{\rho _{Q}}-\left\langle p\right\rangle _{\rho _{Q}}^{2}=\frac{\gamma \hbar }{2}. \end{aligned}$$
So, in the case of the initial quantum coherent state (7.2.15), the time evolution of the expectation value of position and momentum takes the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\langle Q(t)\right\rangle _{\rho _{Q}} &\displaystyle =&\displaystyle \frac{1}{2\pi \hbar }\int_{ \mathbb{R}^{2}}\rho _{Q}(x^{\prime },p^{\prime })Q(x^{\prime },p^{\prime },t)dx^{\prime }dp^{\prime }=Q(x,p,t),\qquad {} \end{array} \end{aligned} $$
(7.2.18a)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\langle P(t)\right\rangle _{\rho _{Q}} &\displaystyle =&\displaystyle \frac{1}{2\pi \hbar }\int_{ \mathbb{R}^{2}}\rho _{Q}(x^{\prime },p^{\prime })P(x^{\prime },p^{\prime },t)dx^{\prime }dp^{\prime }=P(x,p,t),\qquad {} \end{array} \end{aligned} $$
(7.2.18b)
and coincides with the classical time evolution of the expectation value of position and momentum in a classical pure coherent state ρC = δ(x− x)δ(p− p). Nevertheless, contrary to the classical case, for the quantum system we get
$$\displaystyle \begin{aligned} \langle {Q(t)}\rangle _{\rho _{Q}}& =\frac{1}{2\pi \hbar }\iint_{\mathbb{R} ^{2}}x\star _{M}\rho _{Q}dx^{\prime }dp^{\prime }=x+pt, \\ \langle P{(t)}\rangle _{\rho _{Q}}& =\frac{1}{2\pi \hbar }\iint_{\mathbb{R} ^{2}}p\star _{M}\rho _{Q}dx^{\prime }dp^{\prime }=p, \\ \varDelta {Q(t)}& =\sqrt{\langle Q{{}^{2}}\rangle _{\rho _{Q}}-\langle {Q}\rangle _{\rho _{Q}}^{2}}=\sqrt{(\varDelta x)^{2}+(\varDelta p)^{2}t^{2}}, \\ \varDelta P{(t)}& =\sqrt{\langle P{{}^{2}}\rangle _{\rho _{Q}}-\langle {P}\rangle _{\rho _{Q}}^{2}}=\varDelta p \end{aligned} $$
where
$$\displaystyle \begin{aligned} (\varDelta x)^{2}=\frac{1}{2}\frac{\hbar }{\gamma },\ \ \ \ (\varDelta p)^{2}= \frac{1}{2}\gamma \hbar \end{aligned}$$
and in consequence
$$\displaystyle \begin{aligned} \varDelta {Q(t)}\varDelta P{(t)}=\tfrac{1}{2}\hbar \sqrt{1+\gamma ^{2}t^{2}}\geq \tfrac{1}{2}\hbar . {} \end{aligned}$$
Note that during the time evolution the uncertainty of the momentum ΔP(t) of the free particle described by the state (7.2.15) does not change in time and is equal to its initial value Δp, whereas the uncertainty of the position ΔQ(t) initially equal Δx increases in time. Note also that the uncertainties of the position and momentum satisfy the Heisenberg uncertainty principle, i.e. \(\varDelta {Q(t)} \varDelta P{(t)}\geq \tfrac {\hbar }{2}\). Moreover, initially the free particle is in a state which minimizes the Heisenberg uncertainty principle since \( \varDelta Q(0)\varDelta P(0)=\varDelta x\varDelta p=\tfrac {\hbar }{2}\). It is also worth noting that the expectation value of the momentum \(\langle {P(t)} \rangle _{\rho _{Q}}\) is constant and equal p, whereas the expectation value of the position \(\langle {Q(t)}\rangle _{\rho _{Q}}\) is equal x + pt. Hence, the time evolution of the free particle described by the state (7.2.15) can be interpreted as the movement of the particle along a straight line with the constant momentum equal p, similarly as in the classical case. The difference between the classical and quantum case is that in the quantum case there is some uncertainty of the position and momentum, in contrast to the classical case where the position and momentum are known precisely. Observe also that for any admissible value of γ the coherence is not preserved during time evolution.
It is interesting to calculate to which classical state the state ( 7.2.15) converges in the limit ħ → 0+. The limit has to be calculated in the distributional sense, i.e. one has to calculate the limit \(\lim _{\hbar \rightarrow 0^{+}}\langle {\rho }_{Q}{,\phi }\rangle =\lim _{\hbar \rightarrow 0^{+}}Tr(\rho _{Q}(\hbar )\phi )\) for every test function ϕ. One easily calculates that
$$\displaystyle \begin{aligned} \lim_{\hbar \rightarrow 0^{+}}\langle {\rho }_{Q}{,\phi }\rangle =\phi (x,p). \end{aligned}$$
Hence
$$\displaystyle \begin{aligned} \lim_{\hbar \rightarrow 0^{+}}\rho _{Q}=\delta (x^{\prime }-x)\delta (p^{\prime }-p), \end{aligned}$$
where the scalar product 〈⋅, ⋅〉 is given by ( 6.1.9). The above equation implies that the state ρQ ( 7.2.15), describing a quantum free particle, converges in the limit ħ → 0+ to the classical pure state describing a classical free particle moving along a straight line with the constant momentum equal p.
Our second example is the harmonic oscillator described by Hamiltonian
$$\displaystyle \begin{aligned} H(x,p)= \tfrac{1}{2}\left( p^{2}+\omega ^{2}x^{2}\right) ,\ \ \ \ \omega \in \mathbb{R }_{+}. {} \end{aligned} $$
(7.2.19)
Again from relation (7.2.8) it follows that \(\Phi _{t}^{\ast }(\hbar )=\Phi _{t}^{\ast }\) and thus a common quantum and classical flow of harmonic oscillator is of the form Φt(x, p;ħ) = Φt(x, p) = (Q(t), P(t)), where
$$\displaystyle \begin{aligned} Q(t)=x\cos \omega t+\omega ^{-1}p\sin \omega t,\ \ \ P(t)=p\cos \omega t-\omega x\sin \omega t. {} \end{aligned} $$
(7.2.20)
Moreover, as St = 1, so the time evolution of any classical and quantum observable A(x, p) is given by (7.2.12). For the classical system, like in the previous example, equations (7.2.20) represent simultaneously the dynamics of expectation values of position and momentum in the pure coherent classical state (7.2.13), for which the minimal classical uncertainty relation (7.2.14) is fulfilled. As the state (7.2.13) is not an admissible quantum state we again consider a one-parameter family of pure and coherent quantum states ρ of the form (7.2.15).
Like in the previous case, the time evolution of the expectation value of position and momentum takes the form (7.2.18) and so coincides with the classical time evolution of expectation value of position and momentum in the classical pure coherent state ρC = δ(x− x)δ(p− p) . Nevertheless, contrary to the classical case, for the quantum system we get
$$\displaystyle \begin{aligned} \begin{array}{rcl} (\varDelta Q(t))^{2} &\displaystyle =&\displaystyle \left\langle Q^{2}(t)\right\rangle _{\rho _{Q}}-\left\langle Q(t)\right\rangle _{\rho _{Q}}^{2}= \frac{\hbar }{2\gamma }\cos ^{2}\omega t+\frac{\gamma \hbar }{2\omega ^{2}} \sin ^{2}\omega t, \\ \ (\varDelta P(t))^{2} &\displaystyle =&\displaystyle \left\langle P^{2}(t)\right\rangle _{\rho _{Q}}-\left\langle P(t)\right\rangle _{\rho _{Q}}^{2}=\frac{\gamma \hbar }{2} \cos ^{2}\omega t+\frac{\omega ^{2}\hbar }{2\gamma }\sin ^{2}\omega t \end{array} \end{aligned} $$
and hence
$$\displaystyle \begin{aligned} \varDelta {Q(t)}\varDelta P{(t)=}\frac{\hbar }{2}\left[ \sin ^{4}\omega t+\cos ^{4}\omega t+\left( \frac{\omega ^{2}}{\gamma ^{2}}+\frac{\gamma ^{2}}{ \omega ^{2}}\right) \sin ^{2}\omega t\cos ^{2}\omega t\right] ^{\tfrac{1}{2}}. {} \end{aligned}$$
Notice that in this particular case there exists a distinguished coherent state
$$\displaystyle \begin{aligned} \rho _{Q}(x^{\prime },p^{\prime })=2\exp \left( -\frac{\omega }{\hbar } (x^{\prime }-x)^{2}\right) \exp \left( -\frac{1}{\omega \hbar }(p^{\prime }-p)^{2}\right) \end{aligned}$$
when γ = ω, which remains coherent for arbitrary value of t. Indeed, as
$$\displaystyle \begin{aligned} (\varDelta Q(t))^{2}=(\varDelta x)^{2}=\frac{\hbar }{2\omega },\ \ \ \ (\varDelta P(t))^{2}=(\varDelta p)^{2}=\frac{\omega \hbar }{2}, \end{aligned}$$
then
$$\displaystyle \begin{aligned} \varDelta {Q(t)}\varDelta P{(t)=}\frac{\hbar }{2}. {} \end{aligned}$$

As was proved earlier, in the limit ħ → 0+, pure coherent quantum states (7.2.15) converge to the pure coherent classical state ( 7.2.13).

Let us try to find stationary pure states of the harmonic oscillator. From Sect. 7.1.2 it is known that the stationary pure states are precisely the solutions of the following pair of ⋆  -genvalue equations
$$\displaystyle \begin{aligned} H\star \rho =E\rho ,\qquad \rho \star H=E\rho , \end{aligned}$$
for \(E\in \mathbb {R}\). To solve the above equations it is convenient to introduce new coordinates called holomorphic coordinates ( 6.2.2)
$$\displaystyle \begin{aligned} a(x,p)= \frac{\omega x+ip}{\sqrt{2\hbar \omega }},\qquad \bar{a}(x,p)=\frac{\omega x-ip}{\sqrt{2\hbar \omega }}. \end{aligned}$$
The functions a and \(\bar {a}\) are called the annihilation and creation functions since they decrease and increase the number of excitations of the vibrational mode with frequency ω (annihilate and create the quanta of vibrations). Note, that \(a\star =(\bar {a}\star )^{\dagger }\), \(\bar {a}\star =(a\star )^{\dagger }\) and
$$\displaystyle \begin{aligned} \lbrack a,\bar{a}]=a\star \bar{a}-\bar{a}\star a=1. \end{aligned}$$
In these new coordinates the function H takes the form
$$\displaystyle \begin{aligned} H(a,\bar{a})=\hbar \omega a\bar{a}=\hbar \omega \left( \bar{a}\star a+\tfrac{1 }{2}\right) =\hbar \omega \left( a\star \bar{a}-\tfrac{1}{2}\right) . \end{aligned}$$
Let us consider a more general problem of finding a solution to the following pair of ⋆ -genvalue equations
$$\displaystyle \begin{aligned} H\star \rho _{mn}& =E_{m}\rho _{mn}, {} \end{aligned} $$
(7.2.21a)
$$\displaystyle \begin{aligned} \rho _{mn}\star H& =E_{n}\rho _{mn}, {} \end{aligned} $$
(7.2.21b)
where m, n are numbering the ⋆ -genvalues of H. It can be shown that m, n are non-negative integer numbers. The energy levels En of the harmonic oscillator are equal
$$\displaystyle \begin{aligned} E_{n}=(n+\tfrac{1}{2})\hbar \omega . \end{aligned}$$
Since \(H=\hbar \omega (\bar {a}\star a+\tfrac {1}{2})\), ⋆ -genvalues of the function \(N:=\bar {a}\star a\) are the natural numbers n = 0, 1, 2, … and ⋆ -genfunctions are the ⋆ -genfunctions ρmn of H, i.e.
$$\displaystyle \begin{aligned} N\star \rho _{mn}=m\rho _{mn},\quad \rho _{mn}\star N=n\rho _{mn}. \end{aligned}$$
Hence, the function \(N=\bar {a}\star a\) can be interpreted as an observable of the number of excitations of the vibrational mode with frequency ω.
Moreover, the normalized solutions of Eqs. (7.2.21) can be calculated from the ground state ρ00
$$\displaystyle \begin{aligned} a\star \rho _{00}=0 \end{aligned}$$
according to the equation
$$\displaystyle \begin{aligned} \rho _{mn}=\frac{1}{\sqrt{m!n!}}\underbrace{\bar{a}\star \ldots \star \bar{a} }_{m}\star \rho _{00}\star \underbrace{a\star \ldots \star a}_{n} {} \end{aligned}$$
and the ground state ρ00 takes the form
$$\displaystyle \begin{aligned} \rho _{00}(a,\bar{a})=2\exp \left( -2a\bar{a}\right) ,\qquad \rho _{00}(x,p)=2\exp \left( -\frac{p^{2}+\omega ^{2}x^{2}}{\hbar \omega }\right) , \end{aligned}$$
with normalization given by (7.2.16). The ⋆ -genfunctions ρmn can be now calculated giving
$$\displaystyle \begin{aligned} \rho _{mn}(a,\bar{a})=\frac{1}{\sqrt{m!n!}}\sum_{k=0}^{n}(-1)^{k}k!\binom{m}{ k}\binom{n}{k}\frac{1}{2^{2k-n-m}}\bar{a}^{m-k}a^{n-k}\rho _{00}(a,\bar{a}). {} \end{aligned}$$
The above equation can be written alternatively when passing to the polar coordinates (r, θ)
$$\displaystyle \begin{aligned} \omega x+ip=re^{i\theta }. \end{aligned}$$
Then we have
$$\displaystyle \begin{aligned} a(r,\theta )= \frac{1}{\sqrt{2\hbar \omega }}re^{i\theta },\quad \bar{a}(r,\theta )=\frac{1 }{\sqrt{2\hbar \omega }}re^{-i\theta },\quad r^{2}=p^{2}+\omega ^{2}x^{2}, \end{aligned}$$
and Eq. (7.2.2) takes the form [10, 111, 140]
$$\displaystyle \begin{aligned} \rho _{mn}(r,\theta )& =2(-1)^{n}\sqrt{\frac{n!}{m!}}\frac{1}{2^{n-m}}\left( \frac{r}{\sqrt{2\hbar \omega }}\right) ^{m-n} \notag \\ & \quad \times L_{n}^{m-n}\left( \frac{2r^{2}}{\hbar \omega }\right) e^{-i(m-n)\theta }\exp \left( -\frac{r^{2}}{\hbar \omega }\right) , {} \end{aligned} $$
where
$$\displaystyle \begin{aligned} L_{n}^{s}(x)=\frac{x^{-s}e^{x}}{n!}\frac{d^{n}}{dx^{n}}\left( e^{-x}x^{n+s}\right) =\sum_{k=0}^{n}(-1)^{k}\frac{(n+s)!}{(n-k)!(s+k)!k!} x^{k} \end{aligned}$$
are the generalized Laguerre polynomials. The stationary pure states of the harmonic oscillator are of the form
$$\displaystyle \begin{aligned} \rho _{nn}(r,\theta )=2(-1)^{n}L_{n}\left( \frac{2r^{2}}{\hbar \omega }\right) \exp \left( -\frac{r^{2}}{\hbar \omega } \right) , {} \end{aligned} $$
(7.2.22)
where \(L_{n}(x)=L_{n}^{0}(x)\) are the Laguerre polynomials. Equation (7.2.22) can be also written in the following form
$$\displaystyle \begin{aligned} \rho_n(x,p) \equiv \rho _{nn}=2(-1)^{n}L_{n}\left( \frac{4H}{\hbar \omega }\right) \exp \left( -\frac{2H}{\hbar \omega }\right) . {} \end{aligned} $$
(7.2.23)
It is interesting to check to which classical states quantum states ρn in the limit ħ → 0+ converge. Again it has to be calculated in a distributional sense, hence the limit \(\lim _{\hbar \rightarrow 0^{+}}\langle {\rho _{n},\phi }\rangle \) has to be calculated for every test function ϕ. One finds that for fixed n
$$\displaystyle \begin{aligned} \lim_{\hbar \rightarrow 0^{+}}\langle {\rho _{n},\phi }\rangle =\phi (0,0)=\langle {\delta (x)\delta (p),\phi }\rangle \end{aligned}$$
and hence
$$\displaystyle \begin{aligned} \lim_{\hbar \rightarrow 0^{+}}\rho _{n}={\delta (x)\delta (p)}, \end{aligned}$$

i.e. all quantum stationary pure states ρn of the harmonic oscillator converge, in the limit ħ → 0+, to the single classical state (x = 0, p = 0) describing a particle with the position and momentum equal 0. This result is not surprising as the state (x = 0, p = 0) is the only classical stationary pure state of the harmonic oscillator.

The reader can find other interesting examples of ⋆ -genvalue problems and their solutions in [75] and [76]. Moreover, the reader can find the general solution of ⋆ -genvalue problem for Hamiltonians quadratic in phase space coordinates in [92].

Previous results of this subsection give us an ambiguous answer to the question of time development of the initial coherent state. We investigated it indirectly, calculating time development of uncertainty relation ΔQ(t)ΔP(t), with minimal initial value \(\varDelta {Q(0)}\varDelta P{(0)=} \tfrac {\hbar }{2}\). For a free particle we found that the coherence is not preserved during time evolution (7.2.2) while for harmonic oscillator it is preserved for a distinguished initial coherent state (7.2.15) for γ = ω. Thus, let us investigate more systematically that problem for arbitrary linear Hamiltonian system in \(\mathbb {R}^{2}\).

Let us consider a harmonic oscillator described by a Hamiltonian (7.2.19). It is convenient to introduce normalized variables
$$\displaystyle \begin{aligned} x\rightarrow \frac{1}{\sqrt{\omega }}x,\quad p\rightarrow \sqrt{\omega }p. \end{aligned}$$
In these new variables the Hamiltonian of the harmonic oscillator takes the form
$$\displaystyle \begin{aligned} H=\tfrac{1}{2}\omega (p^{2}+x^{2}). {} \end{aligned} $$
(7.2.24)
Adding to (7.2.24) the interaction term
$$\displaystyle \begin{aligned} H_{I}(q,p)=\alpha xp+\tfrac{1}{2}\beta p^{2}-\tfrac{1}{2}\beta x^{2}, \end{aligned}$$
where \(\alpha ,\beta \in \mathbb {R}\) are some constants, we will consider the following Hamiltonian [100]
$$\displaystyle \begin{aligned} H(q,p)=\tfrac{1}{2}(\omega +\beta )p^{2}+\tfrac{1}{2}(\omega -\beta )x^{2}+\alpha xp. {} \end{aligned} $$
(7.2.25)
Note that any Hamiltonian quadratic in x and p variables is of the above form for some values of constants ω, α and β. It should be noted that this type of Hamiltonian is very often found in quantum optics where admissible coherent and squeezed states of the light are investigated [132, 133, 144, 243, 257].
The classical and quantum Hamilton equations for time evolution of observables of position Q(t) and momentum P(t) take the common form
$$\displaystyle \begin{aligned} \begin{aligned} Q_{t}& =\alpha Q+(\beta +\omega )P, \\ P_{t}& =(\beta -\omega )Q-\alpha P. \end{aligned} {} \end{aligned} $$
(7.2.26)
The Hamilton flow in a case ω2 > α2 + β2 reads
$$\displaystyle \begin{aligned} \begin{aligned} Q(t)& =\frac{\alpha x+(\omega +\beta )p}{R}\sin (Rt)+x\cos (Rt), \\ P(t)& =-\frac{(\omega -\beta )x+\alpha p}{R}\sin (Rt)+p\cos (Rt), \end{aligned} {} \end{aligned}$$
where \(R=\sqrt {\left \vert \omega ^{2}-\alpha ^{2}-\beta ^{2}\right \vert }\). When ω2 < α2 + β2 we get
$$\displaystyle \begin{aligned} \begin{aligned} Q(t)& =\frac{\alpha x+(\omega +\beta )p}{R}\sinh (Rt)+x\cosh (Rt), \\ P(t)& =-\frac{(\omega -\beta )x+\alpha p}{R}\sinh (Rt)+p\cosh (Rt), \end{aligned} {} \end{aligned}$$
and when ω2 = α2 + β2
$$\displaystyle \begin{aligned} \begin{aligned} Q(t)& =x+(\alpha x+(\omega +\beta )p)t, \\ P(t)& =p-((\omega -\beta )x+\alpha p)t. \end{aligned} {} \end{aligned}$$
First, we will focus on the case ω2 > α2 + β2. In an initial coherent state
$$\displaystyle \begin{aligned} \rho (q,p)=2\exp \left( -\frac{\gamma (x^{\prime }-x)^{2}}{\hbar }\right) \exp \left( -\frac{(p^{\prime }-p)^{2}}{\hbar \gamma }\right) \end{aligned}$$
we receive the following formulas for the uncertainties (ΔQ)2 and (ΔP)2
$$\displaystyle \begin{aligned} \begin{aligned} (\varDelta Q)^{2}& =\frac{\hbar }{2}\gamma ^{-1}\left( \frac{\alpha ^{2}+\gamma ^{2}(\omega +\beta )^{2}}{R^{2}}\sin ^{2}(Rt)+\cos ^{2}(Rt)+\frac{2\alpha }{R }\sin (Rt)\cos (Rt)\right) \\ & =\frac{\hbar }{2}\gamma ^{-1}\left( 1+\frac{2(\alpha ^{2}+\beta ^{2})+2\gamma ^{2}\omega \beta +(\gamma ^{2}-1)(\omega ^{2}+\beta ^{2})}{ R^{2}}\sin ^{2}(Rt)\right. \\ & \hspace{1.5cm}\ \ +\left. \frac{2\alpha }{R}\sin (Rt)\cos (Rt)\right) , \\ & \\ {} (\varDelta P)^{2}& =\frac{\hbar }{2}\gamma \left( \frac{\alpha ^{2}+\gamma ^{-2}(\omega -\beta )^{2}}{R^{2}}\sin ^{2}(Rt)+\cos ^{2}(Rt)-\frac{2\alpha }{ R}\sin (Rt)\cos (Rt)\right) \\ & =\frac{\hbar }{2}\gamma \left( 1+\frac{2(\alpha ^{2}+\beta ^{2})-2\gamma ^{-2}\omega \beta +(\gamma ^{-2}-1)(\omega ^{2}+\beta ^{2})}{R^{2}}\sin ^{2}(Rt)\right. \\ & \ \ \ \ \ \ \ \ \ \ \ \ -\left. \frac{2\alpha }{R}\sin (Rt)\cos (Rt)\right) . \end{aligned} \end{aligned}$$
Introducing a parameter ξ = α +  = re and writing it in polar variables (r, θ) the uncertainties (ΔQ)2 and (ΔP)2 take the form
$$\displaystyle \begin{aligned} \begin{aligned} (\varDelta Q)^{2}& =\frac{\hbar }{2}\gamma ^{-1}\left( 1+\frac{\sin ^{2}(Rt)}{R} \left( \frac{2r^{2}+2\gamma ^{2}\omega r\sin \theta +(\gamma ^{2}-1)(\omega ^{2}+r^{2}\sin ^{2}\theta )}{R}\right. \right. \\ & \qquad \qquad \qquad \qquad \qquad \qquad +\left. \left. 2r\cos \theta \cot (Rt)\frac{{}}{{}}\right) \frac{{}}{{}} \right) , \\ {} & \\ (\varDelta P)^{2}& =\frac{\hbar }{2}\gamma \left( 1+\frac{\sin ^{2}(Rt)}{R} \left( \frac{2r^{2}-2\gamma ^{-2}\omega r\sin \theta +(\gamma ^{-2}-1)(\omega ^{2}+r^{2}\sin ^{2}\theta )}{R}\right. \right. \\ & \qquad \qquad \qquad \qquad \qquad \quad -\left. \left. 2r\cos \theta \cot (Rt)\frac{{}}{{}}\right) \frac{{}}{{}}\right) . \end{aligned} {} \end{aligned} $$
(7.2.27)
From (7.2.27) we find that coherence is not preserved during time evolution when parameters ω, α, β are arbitrary.
In the special case γ = 1 (γ = ω for old x and p)
$$\displaystyle \begin{aligned} \begin{aligned} (\varDelta Q)^{2}& =\frac{\hbar }{2}\left( 1+\frac{2r}{R}\sin ^{2}(Rt)\left( \frac{r+\omega \sin \theta }{R}+\cos \theta \cot (Rt)\right) \right) , \\ (\varDelta P)^{2}& =\frac{\hbar }{2}\left( 1+\frac{2r}{R}\sin ^{2}(Rt)\left( \frac{r-\omega \sin \theta }{R}-\cos \theta \cot (Rt)\right) \right) . \end{aligned} {} \end{aligned}$$
and
$$\displaystyle \begin{aligned} (\varDelta q)^{2}(\varDelta p)^{2}=\frac{\hbar ^{2}}{4}\left( 1+\frac{4r^{2}}{R^{2} }\sin ^{2}(Rt)\left( \frac{\omega }{R}\cos \theta \sin (Rt)-\sin \theta \cos (Rt)\right) ^{2}\right) . {} \end{aligned} $$
(7.2.28)
From (7.2.28) it follows that the minimization of the Heisenberg uncertainty relation occurs only for Rt =  and \(Rt=\arctan (\tfrac {R}{ \omega }\tan \theta )+k\pi \), \(k\in \mathbb {Z}\). Thus, we will consider a further reduction. First, let us take β = 0 and α > 0. Then r = α and θ = 0. In this case we receive
$$\displaystyle \begin{aligned} \begin{aligned} (\varDelta Q)^{2}& =\frac{\hbar }{2}\left( 1+\frac{2\alpha }{R}\sin ^{2}(Rt)\left( \frac{\alpha }{R}+\cot (Rt)\right) \right) , \\ (\varDelta P)^{2}& =\frac{\hbar }{2}\left( 1+\frac{2\alpha }{R}\sin ^{2}(Rt)\left( \frac{\alpha }{R}-\cot (Rt)\right) \right) , \end{aligned} {} \end{aligned}$$
and
$$\displaystyle \begin{aligned} (\varDelta Q)^{2}(\varDelta P)^{2}=\frac{\hbar ^{2}}{4}\left( 1+\frac{4\omega ^{2}\alpha ^{2}}{R^{4}}\sin ^{4}(Rt)\right) . {} \end{aligned} $$
(7.2.29)
From (7.2.29) it follows that the coherence is preserved during time evolution if additionally α = 0, but this is exactly the case of the harmonic oscillator (7.2.24).
Now, let us consider the case ω2 < α2 + β2 by taking ω = 0. Then R = r and
$$\displaystyle \begin{aligned} \begin{aligned} Q(q,p,t)& =(q\cos \theta +p\sin \theta )\sinh (rt)+q\cosh (rt), \\ P(q,p,t)& =(q\sin \theta -p\cos \theta )\sinh (rt)+p\cosh (rt). \end{aligned} {} \end{aligned} $$
(7.2.30)
Moreover,
$$\displaystyle \begin{aligned} \begin{aligned} (\varDelta Q)^{2}& =\frac{\hbar }{2}\gamma ^{-1}\big(\cosh (2rt)+\cos \theta \sinh (2rt)+(\gamma ^{2}-1)\sin ^{2}\theta \sinh ^{2}(rt)\big), \\ (\varDelta P)^{2}& =\frac{\hbar }{2}\gamma \big(\cosh (2rt)-\cos \theta \sinh (2rt)+(\gamma ^{-2}-1)\sin ^{2}\theta \sinh ^{2}(rt)\big). \end{aligned} {} \end{aligned} $$
(7.2.31)
If additionally γ = 1 we get
$$\displaystyle \begin{aligned} \begin{aligned} (\varDelta Q)^{2}& =\frac{\hbar }{2}\big(\cosh (2rt)+\cos \theta \sinh (2rt) \big)=\frac{\hbar }{2}\left( e^{2rt}\cos ^{2}\frac{\theta }{2}+e^{-2rt}\sin ^{2}\frac{\theta }{2}\right) , \\ (\varDelta P)^{2}& =\frac{\hbar }{2}\big(\cosh (2rt)-\cos \theta \sinh (2rt) \big)=\frac{\hbar }{2}\left( e^{-2rt}\cos ^{2}\frac{\theta }{2}+e^{2rt}\sin ^{2}\frac{\theta }{2}\right) , \end{aligned} {} \end{aligned}$$
and
$$\displaystyle \begin{aligned} (\varDelta Q)^{2}(\varDelta P)^{2}=\frac{\hbar ^{2}}{4}\left( 1+\sin ^{2}\theta \sinh ^{2}(2rt)\right) . {} \end{aligned} $$
(7.2.32)
Again, as it is evident from (7.2.32), the Heisenberg uncertainty relation is not minimized during the whole time evolution. In order to get a minimal uncertainty for any t we have to take β = 0. Then θ = 0, r = α, equations (7.2.30) reduce to
$$\displaystyle \begin{aligned} \begin{aligned} Q(q,p,t)& =qe^{\alpha t}, \\ P(q,p,t)& =pe^{-\alpha t}, \end{aligned} \end{aligned}$$
variances (7.2.31) are
$$\displaystyle \begin{aligned} \begin{aligned} (\varDelta Q)^{2}& =\frac{\hbar }{2}\gamma ^{-1}e^{2\alpha t}, \\ (\varDelta P)^{2}& =\frac{\hbar }{2}\gamma e^{-2\alpha t}, \end{aligned} {} \end{aligned}$$
and hence we get a conservation of minimal uncertainty
$$\displaystyle \begin{aligned} (\varDelta Q)^{2}(\varDelta P)^{2}=\frac{\hbar ^{2}}{4}. {} \end{aligned}$$
Finally, let us consider a case ω2 = α2 + β2. In this case
$$\displaystyle \begin{aligned} \begin{aligned} (\varDelta Q)^{2}& =\frac{\hbar }{2}\left( \gamma ^{-1}(1+\alpha t)^{2}+\gamma (\beta +\omega )^{2}t^{2}\right) , \\ (\varDelta P)^{2}& =\frac{\hbar }{2}\left( \gamma ^{-1}(\beta -\omega )^{2}t^{2}+\gamma (1-\alpha t)^{2}\right) , \end{aligned} {} \end{aligned} $$
(7.2.33)
and
$$\displaystyle \begin{aligned} &(\varDelta Q)^{2}(\varDelta P)^{2} {} \\ &=\frac{\hbar ^{2}}{4}\left( 1-2\alpha ^{2}t^{2}+2\alpha ^{4}t^{4}+\gamma ^{-2}(1+\alpha t)^{2}(\beta -\omega )^{2}t^{2}+\gamma ^{2}(1-\alpha t)^{2}(\beta +\omega )^{2}t^{2}\right) . \notag \end{aligned} $$
(7.2.34)
It can be seen that during time development the variance of position (ΔQ)2 and momentum (ΔP)2 increase quadratically with time, so the minimal uncertainty is not preserved for any t except t = 0. In the particular case, when \(\beta =\omega =\tfrac {1}{2}\) and α = 0, formulas (7.2.33) and (7.2.34) reduce to these for a free particle, considered at the beginning of this subsection.
Note that in a case ω2 > α2 + β2 the following one-parameter family of linear canonical transformations of coordinates
$$\displaystyle \begin{aligned} \begin{aligned} q^{\prime }& =\frac{Ra+\alpha A}{\omega +\beta }q+Ap, \\ p^{\prime }& =\frac{\alpha a-RA}{\omega +\beta }q+ap, \end{aligned} \quad A=\pm \sqrt{\frac{\omega +\beta }{R}-a^{2}},\quad a\in \mathbb{R} \end{aligned}$$
transforms the Hamiltonian (7.2.25) into the following Hamiltonian of the harmonic oscillator
$$\displaystyle \begin{aligned} H(q^{\prime },p^{\prime })=\tfrac{1}{2}R(p^{\prime 2}+q^{\prime 2}). \end{aligned}$$
On the other hand, in a case ω2 < α2 + β2 another one-parameter family of linear canonical transformations of coordinates
$$\displaystyle \begin{aligned} \begin{aligned} q^{\prime }& =-\frac{R+\alpha }{\omega +\beta }aq-ap, \\ p^{\prime }& =\frac{R-\alpha }{2Ra}q-\frac{\omega +\beta }{2Ra}p, \end{aligned} \quad a\in \mathbb{R} \end{aligned}$$
transforms the Hamiltonian (7.2.25) into the following Hamiltonian
$$\displaystyle \begin{aligned} H(q^{\prime },p^{\prime })=Rq^{\prime }p^{\prime }. \end{aligned}$$
For both Hamiltonians H(q, p) the initial coherence is preserved during time evolution.

Observation 16

From the above considerations it follows that the conservation of coherence property during time development of quantum state is rather rare phenomenon. Even for a three-parameter family of linear quantum Hamiltonian equations (7.2.26) in\(\mathbb {R}^{2}\), initial coherence is preserved during time evolution only for two cases: β = α = 0 and β = ω = 0. So why should we expect such a property for nonlinear quantum Hamiltonian equations? On the other hand, when ω2 > α2 + β2we can always reduce the dynamics to the harmonic oscillator (β = α = 0), and when ω2 < α2 + β2we can reduce the dynamics to the case ω = β = 0, provided that we will be working with new variables q, p. In the frame of original variables q, p it means that for the considered class of systems there always exist canonically conjugated observables q = q(q, p), p = p(q, p) for which the minimal uncertainty is preserved during time evolution.

At the end of this subsection let us consider a system of two degrees of freedom described by the Hamiltonian cubic in phase space coordinates
$$\displaystyle \begin{aligned} H(x,p)=\frac{p_{1}^{2}}{2m_{1}}+\frac{p_{2}^{2}}{2m_{2}}+kx^{1}p_{2}^{2}, {} \end{aligned} $$
(7.2.35)
where m1, m2 are masses of particles and k is a coupling constant. Quantum equations of motion (7.1.16) for observables of position and momentum are of the form
$$\displaystyle \begin{aligned} ( Q^{1}) _{t} =&\frac{1}{m_{1}}P_{1}, \notag \\ ( Q^{2}) _{t} =&\frac{1}{m_{2}}P_{2}+2kQ^{1}\star P_{2}, \notag \\ ( P_{1}) _{t} =&-kP_{2}\star P_{2}, {} \\ ( P_{2}) _{t} =&0. \notag \end{aligned} $$
(7.2.36)
Hamiltonian (7.2.35) is specific because x2 is a cyclic coordinate, so P2 is a constant of motion equal to its initial value P2 = p2 and in consequence, equations (7.2.36) reduce to their classical counterparts
$$\displaystyle \begin{aligned} ( Q^{1}) _{t} &=\frac{1}{m_{1}}P_{1}, \notag \\ ( Q^{2}) _{t} &=\frac{1}{m_{2}}P_{2}+2kQ^{1}P_{2}, \notag \\ \left( P_{1}\right) _{t} &=-k\left( P_{2}\right) ^{2}, \\ \left( P_{2}\right) _{t} &=0. \notag \end{aligned} $$
with the solution
$$\displaystyle \begin{aligned} Q^{1}(t) &=x^{1}+\frac{1}{m_{1}}p_{1}t-\frac{k}{2m_{1}}p_{2}^{2}t^{2}, \notag \\ P_{1}(t) &=p_{1}-kp_{2}^{2}t, \notag \\ Q^{2}(t) &=x^{2}+\left( \frac{1}{m_{2}}p_{2}+2kx^{1}p_{2}\right) t+\frac{k}{ m_{1}}p_{1}p_{2}t^{2}-\frac{k^{2}}{3m_{1}}p_{2}^{3}t^{3}, {} \\ P_{2}(t) &=p_{2}, \notag \end{aligned} $$
(7.2.37)
which again represents a common quantum and classical trajectory of the considered two particle system. Hence, the flow (7.2.37) represents a one-parameter family (group) of classical and quantum canonical transformations T(x1, x2, p1, p2) = (Q1, Q2, P1, P2) in a four-dimensional phase space \(\mathbb {R}^{4}\) with the following generating function
$$\displaystyle \begin{aligned} F(x^{1},x^{2},P_{1},P_{2}) =&x^{1}P_{1}+x^{2}P_{2}+ktx^{1}(P_{2})^{2}+ \frac{1}{2m_{1}}t(P_{1})^{2}+\frac{1}{2m_{2}}t(P_{2})^{2} \\ &+\frac{k}{2m_{1}}t^{2}P_{1}(P_{2})^{2}+\frac{k^{2}}{6m_{1}} t^{3}(P_{2})^{4}. \end{aligned} $$
Note, that this transformation is a four-dimensional example of the transformation generated by F4 from Sect.  6.1.4. In accordance with (7.2.1) the received quantum flow Φt transforms the Moyal product to the following product
$$\displaystyle \begin{aligned} f\star _{t}g=f\exp \left( \tfrac{1}{2}i\hbar \overleftarrow{D}_{x^{1}} \overrightarrow{D}_{p_{1}}+\tfrac{1}{2}i\hbar \overleftarrow{D}_{x^{2}} \overrightarrow{D}_{p_{2}}-\tfrac{1}{2}i\hbar \overleftarrow{D}_{p_{1}} \overrightarrow{D}_{x^{1}}-\tfrac{1}{2}i\hbar \overleftarrow{D}_{p_{2}} \overrightarrow{D}_{x^{2}}\right) g, \end{aligned}$$
where
$$\displaystyle \begin{aligned} D_{x^{1}}& =\partial _{x^{1}}+2ktp_{2}\partial _{x^{2}}, \\ D_{x^{2}}& =\partial _{x^{2}}, \\ D_{p_{1}}& =\partial _{p_{1}}+\frac{1}{m_{1}}t\partial _{x^{1}}+\frac{k}{ m_{1}}t^{2}p_{2}\partial _{x^{2}}, \\ D_{p_{2}}& =\partial _{p_{2}}-2ktp_{2}\partial _{p_{1}}-\frac{k}{m_{1}} t^{2}p_{2}\partial _{x^{1}}+\left( \frac{1}{m_{2}}t+2ktx^{1}-\frac{k}{m_{1}} t^{2}p_{1}-\frac{k^{2}}{m_{1}}t^{3}p_{2}^{2}\right) \partial _{x^{2}}. \end{aligned} $$
Moreover, the isomorphism St associated with Φt and intertwining the Moyal product at t = 0 with the ⋆t-product takes the form
$$\displaystyle \begin{aligned} S_{t} =&\exp \left[ \tfrac{1}{4}k\hbar ^{2}\left( \frac{1}{2}\frac{1}{m_{1}} t^{2}\partial _{x^{1}}\partial _{x^{2}}^{2}+t\partial _{p_{1}}\partial _{x^{2}}^{2}+\frac{1}{3}\frac{kp_{2}}{m_{1}}t^{3}\partial _{x^{2}}^{3}\right) \right] . \\ &. \end{aligned} $$
It can be also proved that St is an isomorphism (unitary operator) of the Hilbert space \(L^{2}(\mathbb {R}^{4})\) onto itself.

As in this case \(S_{t_{2}}\Phi _{t_{1}}=\Phi _{t_{1}}\), the group multiplication for { Φt} is just a composition of maps, as one could expect since Φt is simultaneously the classical and quantum trajectory. However, the action of Φt on observables and states does not reduce in general to a composition of maps (7.2.1) like in the classical case as now St ≠ 1, which is a direct consequence of the fact that the Hamiltonian is a cubic function of phase space coordinates. As the result, the time evolution of quantum observables is governed by (7.2.4). This shows that for the considered case the time evolution of quantum observables differs in general from the time evolution of classical observables.

One can check by direct calculations that the action of the quantum flow Φt on an observable A, given by (7.2.4), indeed describes the quantum time evolution of A. As the illustration of that fact let us take \(A(x,p)=x_{1}x_{2}^{2}\). Then
$$\displaystyle \begin{aligned} (S_{t}A)(x,p)=x_{1}x_{2}^{2}+\tfrac{1}{4}\hbar ^{2}\frac{k}{m_{1}}t^{2} \end{aligned}$$
and it can be checked by direct computation that
$$\displaystyle \begin{aligned} A(t)=(S_{t}A)\circ \Phi _{t}=Q^{1}(t)(Q^{2}(t))^{2}+\tfrac{1}{4}\hbar ^{2} \frac{k}{m_{1}}t^{2} \end{aligned}$$
satisfies the time evolution equation (7.1.13).

7.2.3 Pure Quantum Trajectories

We discuss the concept of quantum trajectories on a simple, but far from being trivial, example of a system described by a Hamiltonian
$$\displaystyle \begin{aligned} H(x,p)=\kappa x^{2}p^{2}. {} \end{aligned}$$
The Moyal dynamics (7.1.16) takes the form (see Example 7.1)
$$\displaystyle \begin{aligned} Q_{t} &=\kappa Q\star Q\star P+\kappa P\star Q\star Q,\ \ \ \ \ Q(0)=x, \notag \\ P_{t} &=-\kappa Q\star P\star P-\kappa P\star P\star Q,\ \ \ \ \ P(0)=p. {} \end{aligned} $$
(7.2.38)
We briefly sketch how to find the solution of the considered quantum dynamics [93]. Since Open image in new window at any time t, we have
$$\displaystyle \begin{aligned} \frac{d}{dt}(Q\star P) =&Q_{t}\star P+Q\star P_{t} \\ =&\kappa P\star Q\star Q\star P-\kappa Q\star P\star P\star Q \\ =&\kappa P\star Q\star Q\star P-\kappa P\star Q\star Q\star P+i\hbar\kappa P\star Q-i\hbar \kappa P\star Q \\ =&0 \end{aligned} $$
and so, as in the classical case, Q ⋆ P is a constant of motion
$$\displaystyle \begin{aligned} Q(x,p,t)\star ^{(x,p)}P(x,p,t)=x\star p=xp+\tfrac{1}{2}i\hbar . {} \end{aligned} $$
(7.2.39)
Substituting (7.2.39) in the Eq. (7.2.38) we get
$$\displaystyle \begin{aligned} Q_{t} =&\kappa Q\star (xp)+\kappa (xp)\star Q,\ \ \ \ \ Q(0)=x, \notag \\ P_{t} =&-\kappa (xp)\star P-\kappa P\star (xp),\ \ \ \ \ P(0)=p. \end{aligned} $$
One can immediately check that the solution of this equations is
$$\displaystyle \begin{aligned} Q(x,p,t) =&\exp _{\star }(\kappa txp)\star x\star \exp _{\star }(\kappa txp) \notag \\ P(x,p,t) =&\exp _{\star }(-\kappa txp)\star p\star \exp _{\star }(-\kappa txp). {} \end{aligned} $$
(7.2.40)
By construction this is a unitary transformation.
In order to go further we need the explicit form of ⋆ -exponential from (7.2.40). Following a technique developed in [12], in [92] was derived the ⋆ -exponential for any polynomial of second degree in phase space coordinates
$$\displaystyle \begin{aligned} H=A_{\alpha \beta }\xi ^{\alpha }\xi ^{\beta }+B_{\alpha }\xi ^{\alpha }, \end{aligned}$$
where A is a symmetric, nonsingular, 2n × 2n matrix. In a particular case, for factorization
$$\displaystyle \begin{aligned} A=S_{A}^{T}S_{A} \end{aligned}$$
such that
$$\displaystyle \begin{aligned} S_{A}\omega S_{A}^{T}=a\omega ,\ \ \ \end{aligned}$$
where ω is a symplectic matrix ( 6.1.2) and \(a\in \mathbb {C}\), the noncommutative exponential is given by In our case (7.2.40) H = xp and
$$\displaystyle \begin{aligned} B=0,\ \ \ A=\left( \begin{array}{cc} 0 & \tfrac{1}{2} \\ \tfrac{1}{2} & 0 \end{array} \right) ,\ \ \ S_{A}=\frac{1}{2}\left( \begin{array}{cc} 1 & 1 \\ i & -i \end{array} \right) ,\ \ \ a=\tfrac{1}{2}i \end{aligned}$$
hence
$$\displaystyle \begin{aligned} \exp _{\star }(\pm \kappa txp)=\sec \left( \tfrac{1}{2}\kappa \hbar t\right) \exp \left[ \pm \frac{2}{\hbar }xp\tan \left( \tfrac{1}{2}\kappa\hbar t\right) \right] \end{aligned}$$
and so
$$\displaystyle \begin{aligned} Q(x,p,t) &=\exp _{\star }(\kappa txp)\star x\star \exp _{\star }(\kappa txp) \\ &=\left[ 1+i\tan \left( \tfrac{1}{2}\kappa\hbar t\right) \right] \exp _{\star }(\kappa txp)\star \left[ x\exp _{\star }(\kappa txp)\right] \\ P(x,p,t) &=\exp _{\star }(-\kappa txp)\star p\star \exp _{\star }(-\kappa txp) \\ &=\left[ 1+i\tan \left( \tfrac{1}{2}\kappa \hbar t\right) \right] \exp _{\star }(-\kappa txp)\star \left[ p\exp _{\star }(-\kappa txp)\right] . \end{aligned} $$
Applying the integral representation ( 6.1.45) of the ⋆  -product, after some calculation we get the final solution of quantum Hamiltonian equations (7.2.38) in the form [93]
$$\displaystyle \begin{aligned} Q(x,p,t;\hbar )& =\sec ^{2}(\kappa\hbar t)x\exp \left( \frac{2}{\hbar }\tan (\kappa\hbar t)xp\right) , {} \end{aligned} $$
(7.2.41a)
$$\displaystyle \begin{aligned} P(x,p,t;\hbar )& =\sec ^{2}(\kappa\hbar t)p\exp \left( -\frac{2}{\hbar }\tan (\kappa\hbar t)xp\right) , {} \end{aligned} $$
(7.2.41b)
for \(t\neq \tfrac {2k+1}{2}\tfrac {\pi }{\kappa \hbar }\), \(k\in \mathbb {Z}\). This solution (7.2.41) is a deformation of a classical one given by the limit ħ → 0
$$\displaystyle \begin{aligned} Q_{C}(x,p,t)=xe^{2t\kappa xp},\quad P_{C}(x,p,t)=pe^{-2t\kappa xp}. {} \end{aligned}$$
The induced quantum flow Φt is an example of a flow for which Φt, for every \(t\neq \tfrac {k\pi }{\kappa \hbar }\), is not a classical symplectomorphism, since
$$\displaystyle \begin{aligned} \{Q(t),P(t)\}=\sec ^{4}(\kappa\hbar t)\neq 1. \end{aligned}$$
In accordance with (7.2.1) the quantum flow Φt transforms the Moyal product to the following product
$$\displaystyle \begin{aligned} f\star _{t}g=f\exp \left( \tfrac{1}{2}i\hbar \overleftarrow{D}_{x} \overrightarrow{D}_{p}-\tfrac{1}{2}i\hbar \overleftarrow{D}_{p} \overrightarrow{D}_{x}\right) g, \end{aligned}$$
where
$$\displaystyle \begin{aligned} D_{x}& =\sec ^{2}(\kappa\hbar t)\big(1+2t\chi (\kappa\hbar t)xp\big)\exp \big( 2t\chi(\kappa\hbar t)xp\big)\partial _{x} \\ & ~\ \ \ -2t\chi (\kappa\hbar t)\sec ^{2}(\kappa\hbar t)p^{2}\exp \big(2t\chi (\kappa\hbar t)xp\big)\partial _{p}, \\ D_{p}& =2t\chi(\kappa\hbar t)\sec ^{2}(\kappa\hbar t)x^{2}\exp \big(-2t\chi(\kappa\hbar t)xp\big)\partial _{x} \\ & +\sec ^{2}(\kappa\hbar t)\big(1-2t\chi(\kappa\hbar t)xp\big)\exp \big(-2t\chi (\kappa\hbar t)xp\big)\partial _{p}, \end{aligned} $$
and \(\chi (\kappa \hbar t)=\tfrac {\tan (\kappa \hbar t)}{\kappa \hbar t\sec ^{4}(\kappa \hbar t)}\). Moreover, the isomorphism St associated with Φt and intertwining the Moyal product with the ⋆t-product, up to the second order in ħ, takes the form
$$\displaystyle \begin{aligned} S_{t}& =1+\hbar ^{2}\kappa^2\bigg(\tfrac{1}{6}(3t^{2}x^{3}+4t^{3}x^{4}p)\partial _{x}^{3}+\tfrac{1}{6}(3t^{2}p^{3}-4t^{3}xp^{4})\partial _{p}^{3} \notag \\ & \quad {}\ +\tfrac{1}{2}(-tp-t^{2}xp^{2}+4t^{3}x^{2}p^{3})\partial _{x}\partial _{p}^{2}+\tfrac{1}{2}(tx-t^{2}x^{2}p-4t^{3}x^{3}p^{2})\partial _{x}^{2}\partial _{p}\notag \\ & \ \ \ \ +(2t^{2}x^{2}+2t^{3}x^{3}p)\partial _{x}^{2}+(2t^{2}p^{2}-2t^{3}xp^{3})\partial _{p}^{2}+(-2t^{2}xp)\partial _{x}\partial _{p}\bigg)+O(\hbar ^{4}). {} \end{aligned} $$
(7.2.42)
In fact, expanding relations ( 6.1.77) with respect to ħ one can show that St in the above form satisfies these relations up to O(ħ2).
From the fact that Φt is a purely quantum trajectory, we deal with the quantum group multiplication (7.2.6) for { Φt} as well as the quantum action (7.2.4) of Φt on observables and states. Indeed, expanding (7.2.41) with respect to ħ:
$$\displaystyle \begin{aligned} Q(x,p,t;\hbar )& =Q_{C}\left( 1+\hbar ^{2}\kappa^2\left( t^{2}+ \tfrac{2}{3}t^{3}xp\right) \right) +O(\hbar ^{4}), \\ P(x,p,t;\hbar )& =P_{C}\left( 1+\hbar ^{2}\kappa^2\left( t^{2}-\tfrac{2}{3} t^{3}xp\right) \right) +O(\hbar ^{4}) \end{aligned} $$
and applying isomorphism St (7.2.42), the quantum composition law
$$\displaystyle \begin{aligned} Q(t_{1}+t_{2})& =S_{t_{2}}Q(t_{1})\circ \Phi _{t_{2}}=S_{t_{1}}Q(t_{2})\circ \Phi _{t_{1}}, \\ P(t_{1}+t_{2})& =S_{t_{2}}P(t_{1})\circ \Phi _{t_{2}}=S_{t_{1}}P(t_{2})\circ \Phi _{t_{1}} \end{aligned} $$
holds up to O(ħ2). Note also that the flow Φt is not defined for all \(t\in \mathbb {R}\) as it is singular for \(t=\tfrac {2k+1}{2} \tfrac {\pi }{\kappa \hbar }\), contrary to classical flows which are globally defined. This is an interesting result showing that in general the quantum time evolution do not have to be defined for all instances of time t.

Observation 17

Singularities of classical trajectories are not admissible as each classical trajectory represents measurable quantities, actually expectation values of position and momentum of a system in a pure coherent classical state (7.2.13) for all\(t\in \mathbb {R}\). On the contrary, pure quantum trajectories themselves are not “physical” objects as states (7.2.13) are not admissible so, singularities of pure quantum trajectories are acceptable.

Let us come back to the singular quantum trajectory (7.2.41). Through direct integration we can calculate the expectation values of Q and P from (7.2.41) in the coherent state (7.2.15). The result after introducing
$$\displaystyle \begin{aligned} a(t)& = \frac{\cos (\kappa\hbar t)}{\sqrt{\cos (2\kappa\hbar t)}}x+\frac{\gamma ^{-1}\sin (\kappa\hbar t)}{\sqrt{\cos (2\kappa\hbar t)}}p, \notag \\ b(t)& =-\frac{\gamma \sin (\kappa\hbar t)}{\sqrt{\cos (2\kappa\hbar t)}}x+\frac{\cos (\kappa\hbar t)}{\sqrt{\cos (2\kappa\hbar t)}}p, {} \end{aligned} $$
reads
$$\displaystyle \begin{aligned} \langle {Q}\rangle _{\rho }& =\frac{a(t)}{\cos (2\kappa\hbar t)}\exp \bigg(\frac{ \gamma }{\hbar }\left( a^{2}(t)-x^{2}\right) \bigg), \notag \\ \langle {P}\rangle _{\rho }& =\frac{b(t)}{\cos (2\kappa\hbar t)}\exp \bigg(\frac{1 }{\hbar \gamma }\left( b^{2}(t)-p^{2}\right) \bigg). \end{aligned} $$
Note, that 〈Qρ and 〈Pρ are well defined only on intervals \((-\tfrac {1}{4}+n)\tfrac {\pi }{\kappa \hbar }<t<( \tfrac {1}{4}+n)\tfrac {\pi }{\kappa \hbar }\), \(n\in \mathbb {Z}\). This once again shows that time evolution of the considered system is not defined for all values of the evolution parameter t and even time development of expectation values of position and momentum is only well defined on certain intervals of t.

Observation 18

We have found that for the considered quantum trajectories, expectation values of observables of position and momentum in the coherent state (7.2.15) were well defined only on certain intervals of t, which raises problems and questions of interpretation of such a kind of time evolution. If we assume, like in the classical case, that the expectation values of position and momentum have to be smooth functions for any\(t\in \mathbb {R}\), then we have two options. Either, for a chosen quantization, there exist quantum states for which our assumption is fulfilled (the state (7.2.15) does not belong to that class) or, if there are no such states, our quantization is not ‘physical’ and we have to chose another quantization which fulfills the imposed assumption.

The above observation was made from the mathematical point of view. Let’s have a look on the problem from the physical side. In other words, let us asses the length of the interval on which time evolution of the system is well defined. Notice that dimension of κ in joule-seconds is J−1s−2 so we put \(\kappa =\left |\kappa \right |J^{-1}s^{-2}\) and moreover ħ ≃ 10−34Js. So, the length of the time interval \(\tfrac {\pi }{4\hbar \kappa }\) will be approximately equal \(\left |\kappa \right |{ }^{-1}0.785\times 10^{34}s\). Notice that the age of Universe is 0.437 × 1018s. So, for a large range of κ singularities appearing in time evolution are nonphysical.

After reading this chapter the reader might be disappointed with a small number of examples of stationary problems of known quantum systems, presented in deformation quantization formalism. The exception was made for the case of quantum harmonic oscillator. The reason is that such calculations directly in a Hilbert space over the phase space are very cumbersome and complex. Fortunately, at least for the canonical quantization, we can simplify that problem passing to so called position representation of quantum systems. This is the subject of the next chapter in which we present many known and new examples of separable eigenvalue quantum problems defined in an appropriate Hilbert spaces over Riemannian configuration spaces.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Maciej Błaszak
    • 1
  1. 1.Division of Mathematical PhysicsFaculty of Physics UAMPoznańPoland

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