New approach to nonperturbative quantum mechanics with minimal length uncertainty

The existence of a minimal measurable length is a common feature of various approaches to quantum gravity such as string theory, loop quantum gravity, and black-hole physics. In this scenario, all commutation relations are modified and the Heisenberg uncertainty principle is changed to the so-called Generalized (Gravitational) Uncertainty Principle (GUP). Here, we present a one-dimensional nonperturbative approach to quantum mechanics with minimal length uncertainty relation which implies $X=x$ to all orders and $P=p+\frac{1}{3}\beta p^3$ to first order of GUP parameter $\beta$, where $X$ and $P$ are the generalized position and momentum operators and $[x,p]=i\hslash$. We show that this formalism is an equivalent representation of the seminal proposal by Kempf, Mangano, and Mann and predicts the same physics. However, this proposal reveals many significant aspects of the generalized uncertainty principle in a simple and comprehensive form and the existence of a maximal canonical momentum is manifest through this representation. The problems of the free particle and the harmonic oscillator are exactly solved in this GUP framework and the effects of GUP on the thermodynamics of these systems are also presented. Although $X$, $P$, and the Hamiltonian of the harmonic oscillator all are formally self-adjoint, the careful study of the domains of these operators shows that only the momentum operator remains self-adjoint in the presence of the minimal length uncertainty. We finally discuss the difficulties with the definition of potentials with infinitely sharp boundaries.

Figure 1

The temporal behavior of $x$ and $p$, and the phase space of the harmonic oscillator for $\beta =0$ (blue line) and $\beta =1$ (red line). We set $m=\omega =1$ and $a=5$ ($\overline{\omega }=\sqrt{26}$).

Figure 2

Comparing $E_n/\hslash \omega$ (red line) and $E_n^{(\mathrm{SC})}/\hslash \omega$ (blue line) for $\gamma =2$ with the ordinary harmonic oscillator spectrum (green line).

Figure 3

The classical mean energy of $N$ harmonic oscillators in thermal equilibrium versus temperature for $\beta =1$ (red line) and $\beta =0$ (blue line). We set $m=1$.

Figure 4

The quantum mechanical mean energy of the harmonic oscillator $E_n/\hslash \omega$ versus $k_{B}T/\hslash \omega$ for $\gamma =1$ (red line) and $\gamma =0$ (blue line).

Figure 5

The classical (dashed line) and quantum mechanical (solid line) mean energy of the harmonic oscillator for $\gamma =0.2$ (red line) and $\gamma =0$ (blue line).