Heisenberg uncertainty relations for the non-Hermitian resonance-state solutions to the Schrödinger equation

Resonance (quasinormal) states correspond to non-Hermitian solutions to the Schrödinger equation obeying outgoing boundary conditions which lead to complex energy eigenvalues and momenta. Following the normalization rule for resonance states obtained from the residue at a complex pole of the outgoing Green's function to the problem, we propose a definition of expectation value for these states and use it to investigate the extent of validity of the Heisenberg uncertainty relations for potentials that vanish after a distance. We derive analytical expressions for the expectation values involving the momentum and the position for a given resonance state and find in model calculations that the Heisenberg uncertainty relations are satisfied for a broad range of potential parameters. A comparison of our approach with that based on the regularization method by Zel'dovich yields very similar results except for resonance energies very close to the energy threshold. Our work shows that the validity of the Heisenberg uncertainty relations may be extended to the non-Hermitian resonance-state solutions to the Schrödinger equation.

Figure 1

Plot of the first three poles $k_n={\alpha }_n-i{\beta }_n$ on the fourth quadrant of the complex $k$ plane as a function of the intensity $\lambda$ of the $\delta$-shell potential given by Eq. (51) with $a=1$. For $\lambda \to \infty$ the poles $k_n$ go, as described by Eq. (54), into the infinite box model solutions and as $\lambda \to 0$ the corresponding imaginary parts ${\beta }_n$ go to $-\infty$, as described by Eq. (56). See text.

Figure 2

Plot of the uncertainty $\mathrm{\Delta }r\mathrm{\Delta }p$ for a $\delta$-shell potential as a function of the intensity $\lambda$, using Eqs. (41) (blue solid line) and (42) (orange dashed line) for the case $n=1$. The uncertainty $\mathrm{\Delta }r\mathrm{\Delta }p$ for the infinite well ($\lambda \to \infty$) (red solid square) [Eq. (63) with $n=1]$ is also included for comparison. See text.

Figure 3

Plot of the first four poles $k_n={\alpha }_n-i{\beta }_n$ on the fourth quadrant of the complex $k$ plane for a fixed barrier height $V_0=10$ as a function of the barrier width $L$ in the range from $L=100$, where they are located just above the barrier height $k_0\approx 3.16$, down to $L=0.42$, which shows that they migrate to higher values of ${\alpha }_n$ and ${\beta }_n$, except the first pole. See text.

Figure 4

Plot of the uncertainty $\mathrm{\Delta }x\mathrm{\Delta }p$ using Eqs. (41) (blue solid line) and (42) (orange dashed line) for a one-dimensional potential barrier: (a) for a fixed value of the barrier width $L=100$ as a function of the barrier height $V_0$ and (b) for a fixed value of the barrier height $V_0=10$ as a function of the barrier width $L$. Both cases refer to the resonance state with $n=1$. See text.