Irreversible evolution of a wave packet in the rigged-Hilbert-space quantum mechanics

It is well known that a state with complex energy cannot be the eigenstate of a self-adjoint operator, such as the Hamiltonian. Resonances, i.e., states with exponentially decaying observables, are not vectors belonging to the conventional Hilbert space. One can describe these resonances in an unusual mathematical formalism based on the so-called rigged Hilbert space (RHS). In the RHS, the states with complex energy are denoted as Gamow vectors (GVs), and they model decay processes. We study the GVs of the reversed harmonic oscillator, and we analytically and numerically investigate the unstable evolution of wave packets. We introduce the background function to study initial data that are not composed only by a summation of GVs, and we analyze different wave packets belonging to specific function spaces. Our work furnishes support for the idea that irreversible wave propagation can be investigated using rigged-Hilbert-space quantum mechanics and provides insight for the experimental investigation of irreversible dynamics.

Figure 1

Functions ${\varphi }_{\epsilon }$ with compact support defined in Eq. (28) for several $\epsilon$ values.

Figure 2

One-dimensional evolution of $|{\varphi }_1(u,t)|$ [Eq. (28) with $\epsilon =1]$.

Figure 3

Evolution of $|{\varphi }_1(u,t)|$ [Eq. (28) with $\epsilon =1]$.

Figure 4

Numerically calculated projections $C_N\left(t\right):={\langle U\left(t\right){\varphi }_1|f_N^{+}\rangle }^{*}$ on the $N$-order resonances of a function with compact support [Eq. (28) with $\epsilon =1]$ in the ($u,v$) representation, in a semilogarithmic scale.

Figure 5

Transformed ${\varphi }_{\epsilon }$ [Eq. (30) for various $\epsilon ]$.

Figure 6

One-dimensional evolution of a transformed function with compact support [Eq. (30), $\epsilon =1/2]$ with a RHO potential.

Figure 7

Evolution of a transformed function with compact support [Eq. (30), $\epsilon =1/2]$ with a RHO potential.

Figure 8

Pictorial representation of Gelfand triplet defined in Eq. (7). Here ${\mathrm{\Phi }}_{+}\equiv \mathcal{D}, \mathrm{\Phi }\equiv S\left(\mathbb{R}\right),$ and $\mathcal{H}\equiv {\mathcal{L}}^2\left(\mathbb{R}\right)$. One can get a Euler-Venn diagram also for the triplet in Eq. (6) by replacing ${\mathrm{\Phi }}_{+}$ with ${\mathrm{\Phi }}_{-}$ and $\mathcal{D}$ with $\mathcal{Z}$.

Figure 9

Projections ${\langle {\varphi }_1|f_n^{+}\rangle }^{*}$ of a function with compact support [defined in Eq. (28), $\epsilon =1]$.

Figure 10

Projections ${\langle \varphi |f_n^{+}\rangle }^{*}$ of the Gaussian function.

Figure 11

One-dimensional evolution of the Gaussian wave $\varphi \left(u\right)=\frac{e^{-\frac{u^2}{2}}}{\sqrt[4]{\pi }}$.

Figure 12

Evolution of the Gaussian wave $\varphi \left(u\right)=\frac{e^{-\frac{u^2}{2}}}{\sqrt[4]{\pi }}$.

Figure 13

Evolution of the projections $C_N\left(t\right):={\langle U\left(t\right)\varphi |f_N^{+}\rangle }^{*}$ on the $N$-order resonances of the Gaussian wave $\varphi \left(u\right)=\frac{e^{-\frac{u^2}{2}}}{\sqrt[4]{\pi }}$ in the ($u,v$) representation, in a linear scale.

Figure 14

Evolution of the projections on the $N$-order resonances $C_N\left(t\right):={\langle U\left(t\right)\varphi |f_N^{+}\rangle }^{*}$ of the Gaussian wave $\varphi \left(u\right)=\frac{e^{-\frac{u^2}{2}}}{\sqrt[4]{\pi }}$ in the ($u,v$) representation, in a semilogarithmic scale.

Figure 15

Fit of the Gaussian function $\varphi \left(u\right)=\frac{e^{-\frac{u^2}{2}}}{\sqrt[4]{\pi }}$ by the function ${\varphi }_{{\epsilon }_0}\left(u\right)$ (${\epsilon }_0=1.802425$).

Figure 16

Comparison of the Gaussian evolution $\varphi (u,t)$ with ${\varphi }_{{\epsilon }_0}(u,t)$. ${\varphi }_{{\epsilon }_0}(u,t)$ focalizes without any loss or dispersion of energy, while the Gaussian presents a dispersive background (see also Fig. 17).

Figure 17

We want to analyze if a region exists where ${\varphi }_N^{\text{BG}}(u,t)\simeq \varphi (u,t)-{\varphi }_{{\epsilon }_0}(u,t)$. A comparison between ${\varphi }_{20}^{\text{BG}}(u,t)$ and $\varphi (u,t)-{\varphi }_{{\epsilon }_0}(u,t)$ is here reported in semilogarithmic scale. These two wave packets are well overlapped on their borders.

Figure 18

One-dimensional evolution of the Gaussian wave $\varphi \left(x\right)=\frac{e^{-\frac{x^2}{2}}}{\sqrt[4]{\pi }}$ under a RHO potential.

Figure 19

Evolution of the wave $\left|\varphi \right|$ with initial condition $\varphi \left(x\right)=\frac{e^{-\frac{x^2}{2}}}{\sqrt[4]{\pi }}$.