**5. Applications to physical systems**

#### **5.1 Harmonic oscillator**

The harmonic oscillator has dynamical group *H*<sup>4</sup> and is a single-degree of freedom system [13–15, 23]. To the dynamical group corresponds the algebra *h*<sup>4</sup> defined by the set *<sup>a</sup>*†, *<sup>a</sup>*, *<sup>a</sup>*† f g *<sup>a</sup>*,*<sup>I</sup>* with Hilbert space the Fock space *<sup>V</sup><sup>F</sup>* : f g j*n*i, *n* ¼ 1, 2, … , so the fixed state is the ground state ∣0i, and elementary excitation operator *a*†. The quantum phase space is constructed from the unitary exponential mapping of the subspace *<sup>μ</sup>* : *<sup>a</sup>*† f g , *<sup>a</sup>* of *<sup>h</sup>*4,

$$\Omega(\mathbf{z}) = \exp\left(\mathbf{z}a^\dagger - \overline{\mathbf{z}}a\right) \in H\_4/U(\mathbf{1}) \otimes U(\mathbf{1}).\tag{52}$$

## *Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos DOI: http://dx.doi.org/10.5772/intechopen.94491*

With generators *<sup>a</sup>*†*<sup>a</sup>* and *<sup>I</sup>*, *<sup>U</sup>*ð Þ<sup>1</sup> <sup>⊗</sup> *<sup>U</sup>*ð Þ<sup>1</sup> in (52) is the maximal stability subgroup of ∣0i. As *H*4*=U*ð Þ1 ⊗ *U*ð Þ1 is isomorphic to the one-dimensional complex plane, the quantum phase space has metric *gij* ¼ *δij* and *dν*ð Þ¼ *z dzdz=π*. It is noncompact due to the infiniteness of the Fock space. There is a well-known symplectic structure on the complex plane with Poisson bracket of two functions *F*~1, *F*~<sup>2</sup> defined by

$$\left\{\tilde{F}\_1, \tilde{F}\_2\right\} = \frac{1}{i\hbar} \left(\frac{\partial \bar{F}\_1}{\partial \mathbf{z}} \frac{\partial \bar{F}\_2}{\partial \overline{\mathbf{z}}} - \frac{\partial \bar{F}\_1}{\partial \overline{\mathbf{z}}} \frac{\partial \bar{F}\_2}{\partial \mathbf{z}}\right). \tag{53}$$

It is useful to introduce the standard canonical position and momentum coordinates

$$z = \frac{1}{\sqrt{2\hbar}}(q + ip), \qquad \overline{z} = \frac{1}{\sqrt{2\hbar}}(q - ip). \tag{54}$$

The Glauber coherent states can be realized by the states ∣*z*i with the set of these states isomorphic to *H*4*=U*ð Þ1 ⊗ *U*ð Þ1 and given as

$$|z\rangle \equiv \Omega(z)|0\rangle = \exp\left(za^\dagger - \overline{z}a\right)|0\rangle = e^{-|z|^2/2} \exp\left(za^\dagger\right)|0\rangle. \tag{55}$$

The normalization constant in (55) is the Bargmann kernel

$$\mathcal{K}(z,\overline{z}) = e^{\left|x\right|^2}.\tag{56}$$

The phase space representation of the wavefunction <sup>∣</sup>Ψi∈*V<sup>F</sup>* is

$$f(\mathbf{z}) = \langle \Psi || \mathbf{z} \rangle = \sum\_{n=0}^{\infty} f\_n \frac{\mathbf{z}^n}{\sqrt{n!}}.\tag{57}$$

By Wick's Theorem, it is always possible to write an operator *A* in normal product form

$$A = A\left(a^{\dagger}, a\right) = \sum\_{k,l} A\_{k,l}^{n} \left(a^{\dagger}\right)^{k} (a)^{l}. \tag{58}$$

The phase space representation of *A* is just

$$
\tilde{U}(z,\overline{z}) = \langle z|A|z\rangle = \sum\_{k,l} A^n\_{k,l} \overline{z}^k z^l. \tag{59}
$$

In the case *A* is simply a generator of *H*4, we can write (59) as

$$\begin{aligned} \tilde{a}^\dagger &= \langle z|a^\dagger|z\rangle, & \tilde{a} &= \langle z|a|z\rangle, \\ \tilde{a}^\dagger \tilde{a} &= \langle z|a^\dagger a|z\rangle = \left|z\right|^2, & \tilde{I} &= \langle z|I|z\rangle = I. \end{aligned} \tag{60}$$

The corresponding algebraic structure of *H*<sup>4</sup> in the phase-space representation is

$$i\hbar\_{\mathfrak{c}}\{\ddot{a},\ddot{a}^{\dagger}\}=\tilde{I}, \qquad i\hbar\_{\mathfrak{c}}\{\ddot{a}^{\dagger}\ddot{a}\}=-\tilde{a}, \qquad i\hbar\_{\mathfrak{c}}\{\ddot{a}^{\dagger}\ddot{a},\ddot{a}^{\dagger}\}=\tilde{a}^{\dagger}.\tag{61}$$

Here ℏ*<sup>c</sup>* is used in the classical analogy. The algebraic structure of the *H*<sup>4</sup> generators is preserved when commutators are replaced by Poisson brackets in phase space. Using (54) the Dirac quantization condition and *H*~ are given by

*A Collection of Papers on Chaos Theory and Its Applications*

$$
\dot{H}[q,p] = i\hbar \{q,p\}, \qquad \dot{H}(q,p) = \langle \mathbf{z}|H|\mathbf{z}\rangle. \tag{62}
$$

For the forced harmonic oscillator, the classical analogy of the Hamiltonian is given by

$$\begin{split} \ddot{H}(q,p) &= \frac{\alpha}{2} \left( p^2 + q^2 \right) + i\sqrt{2} \Re(\lambda(t)q) - \sqrt{2} \Re(\lambda(t)p) \\ &= \frac{\alpha}{2} \left( p^2 + q^2 \right) + \frac{1}{\sqrt{2}} \left( \lambda(t) + \overline{\lambda}(t) \right) q \rangle + \frac{1}{\sqrt{2}i} \left( (\lambda(t) - \overline{\lambda}(t))p \right). \end{split} \tag{63}$$

Hamilton's equations in (44) can be used to evaluate the *t* derivatives of *q* and *p*:

$$\frac{dq}{dt} = ap + \frac{\mathbf{1}}{\sqrt{2}i} \left(\lambda(t) - \overline{\lambda}(t)\right), \qquad \frac{dp}{dt} = -aq - \frac{\mathbf{1}}{\sqrt{2}} \left(\lambda(t) + \overline{\lambda}(t)\right). \tag{64}$$

Hence combining these two derivatives, we obtain

$$\frac{d}{dt}(q+ip) = -i\alpha(q+ip) - \sqrt{2}i\lambda(t). \tag{65}$$

Multiplying both sides by the integrating factor *ei<sup>ω</sup><sup>t</sup>* and then integrating with respect to *t*, the solution is

$$q(t) + ip(t) = e^{-i\alpha t}(q(0) + ip(0)) - i\sqrt{2}e^{-i\alpha t} \int\_0^t \lambda(\tau)e^{i\alpha \tau}d\tau = x(t)\sqrt{2\hbar\_c}.\tag{66}$$

If the initial state is ∣0i or a coherent state ∣*z*ð Þi 0 , then the exact quantum solution is

$$|\psi(t)\rangle = |z(t)\rangle e^{i\rho(t)}\tag{67}$$

and *z t*ð Þ is given by (66). The phase *φ* is a quantum effect obtained from *z t*ð Þ

$$
\rho(t) = -\frac{1}{2}at - \int\_0^t \Re[\lambda(\tau)z(\tau)]d\tau. \tag{68}
$$

This seems to imply the classical analogy provides an exact quantum solution if the Hamiltonian is a linear function of the generators of *G*.
