Ground state energy of particle in space with minimal length and momentum

This paper derives a rigorous lower bound for the ground-state energy of quantum systems in deformed space with minimal coordinate and momentum uncertainties, applying the results to harmonic and anharmonic oscillators to obtain general expressions and determine solution domains.

The Gist

Imagine the universe as a giant, cosmic game of pool. In the standard rules of quantum mechanics (the physics of the very small), you can theoretically hit a ball with infinite precision. You can know exactly where it is and exactly how fast it's going at the same time. However, modern theories like string theory suggest that at the tiniest scales, the universe has a "pixel size." There is a limit to how small a space can be, and a limit to how precisely you can measure momentum. It's like trying to measure a distance with a ruler that has a smallest possible mark; you can't measure anything smaller than that mark.

This paper by Arsen Panas and Volodymyr Tkachuk explores what happens to the energy of a particle when we accept these "pixelated" rules of the universe.

The Setup: A Bouncy Ball in a Box

To understand this, the authors start with a classic physics problem: a harmonic oscillator. Think of this as a ball attached to a spring, bouncing back and forth. In normal physics, even at its lowest possible energy state (the "ground state"), the ball still jiggles a little bit due to quantum uncertainty.

The authors ask: If the universe has a minimum size and a minimum fuzziness for momentum, how much energy does this bouncing ball need to exist?

They use a mathematical tool called the Lagrange multiplier. You can think of this as a strict referee in a game. The referee says, "You want to find the lowest energy possible, but you must obey the new rules of the universe (the uncertainty principle)." The authors use this referee to calculate the absolute minimum energy the ball can have without breaking the new rules.

The Results: A Perfect Match

When they did the math for the simple spring-and-ball system, they found a specific formula for the lowest energy. They then compared their result to a different, more complex method (solving the Schrödinger equation, which is like solving the entire game board at once). Their "referee" method gave the exact same answer. This confirmed that their approach is accurate and reliable.

Going Deeper: Any Shape of Potential

Next, they asked, "What if the ball isn't on a spring, but is in a weirdly shaped valley or a complex bowl?" (In physics terms, this is an "arbitrary potential").

They developed a general recipe to find the minimum energy for any shape of valley, as long as the valley gets steeper as you go out (it doesn't have weird holes or spikes).

• The Recipe: They created a step-by-step method to find the "sweet spot" where the particle's position and momentum uncertainties balance out to give the lowest energy.
• The Shortcut: Since solving the full math for every shape is hard, they used a "linear approximation." Imagine drawing a straight line tangent to a curved hill to estimate its height. They did this with the "deformation" parameters (the rules of the pixelated universe).
• The Surprise: They found that for any shape of the valley, the minimum energy depends on the "momentum fuzziness" (one type of deformation) in a specific way, but it doesn't depend on the "position fuzziness" (the other type) in the first step of their calculation. It's as if the size of the universe's pixels matters more for the energy than the fuzziness of the ball's location, at least in this specific approximation.

The Limits: When the Game Breaks

The most interesting part of the paper is checking when this game is even possible to play.

They looked at a specific type of valley that gets steeper and steeper, eventually looking like a box with infinite walls (a "particle in a box"). In normal physics, a particle can always exist in a box. But in this "pixelated" universe, they found a catch:

• If the "pixels" of the universe are too big (meaning the deformation parameter $\beta$ is too large), the particle cannot exist in the box at all. The box becomes too small for the particle to fit inside the rules of the universe.
• They mapped out a "safe zone" for the parameters. If you pick a combination of "position fuzziness" and "momentum fuzziness" that falls outside this safe zone, the particle simply cannot form a stable state. It's like trying to fit a square peg in a round hole, but the hole is actually made of the laws of physics themselves.

They also found that the "strength" of the valley (how deep or steep it is) changes this safe zone. A deeper, stronger valley allows the particle to survive in a more "pixelated" universe than a weak valley would.

Summary

In short, this paper provides a new, rigorous way to calculate the lowest possible energy of particles in a universe that has a minimum size.

1. They proved their method works perfectly for simple springs.
2. They created a general formula that works for complex shapes.
3. They discovered that in a universe with minimum size limits, there are certain conditions where a particle simply cannot exist in a potential well. If the "fuzziness" of the universe is too high relative to the size of the container, the particle has nowhere to go.

The authors conclude that their method is a powerful and simple tool for understanding how quantum particles behave when the fabric of space itself has a fundamental limit.

Technical Summary

1. Problem Statement

The paper addresses the calculation of the ground-state energy for quantum particles in a deformed space characterized by both a minimal coordinate uncertainty (minimal length) and a minimal momentum uncertainty. This deformation arises from Generalized Uncertainty Relations (GUR) motivated by string theory and quantum gravity, which suggest that the standard Heisenberg commutation relations are modified at high energy scales.

Specifically, the authors investigate systems governed by the commutation relation:
$$[\hat{x}, \hat{p}] = i\hbar(1 + \beta' \hat{p}^2 + \alpha' \hat{x}^2)$$
This leads to the uncertainty relation:
$$\Delta x \Delta p \geq \frac{\hbar}{2}(1 + \beta' \Delta p^2 + \alpha' \Delta x^2)$$
where $\alpha'$ and $\beta'$ are deformation parameters. The primary goal is to derive a rigorous lower bound for the ground-state energy of a harmonic oscillator and arbitrary potentials within this framework, without necessarily solving the full Schrödinger equation.

2. Methodology

The authors employ a variational approach based on the method of Lagrange multipliers to find the conditional extremum of the energy functional subject to the generalized uncertainty constraint.

• Dimensionless Formulation: The problem is first non-dimensionalized using characteristic scales for length ($\Delta x_0$), momentum ($\Delta p_0$), and energy ($E_0$). For the harmonic oscillator, these are derived from the standard oscillator parameters ($\hbar, m, \omega$). For arbitrary potentials, they are derived from the potential's characteristic length $a$ and intensity $U_0$.
• Constraint Handling: The energy expectation value $\langle H \rangle$ is expressed in terms of uncertainties $\Delta x$ and $\Delta p$. The authors argue that since the unconstrained minimum of the energy function (at $\Delta x = \Delta p = 0$) lies outside the physically allowed region defined by the GUR, the true ground state energy must lie on the boundary of the allowed region. Thus, the inequality constraint is treated as an equality.
• Lagrange Multiplier System: A Lagrangian function $L(\xi, q, \lambda)$ is constructed where $\xi$ and $q$ are dimensionless uncertainties. The system of equations is solved to find the critical points ($\xi_{min}, q_{min}$).
• Linear Approximation: For arbitrary potentials, the resulting algebraic equation for $\xi_{min}$ is non-trivial. The authors solve this using a linear approximation with respect to the small deformation parameters $\alpha$ and $\beta$. They expand the solution as $\xi_{min} = \xi_0 + \xi_1 \beta + \xi_2 \alpha$.
• Numerical Verification: For the anharmonic oscillator potential $V(x) = U_0(x/a)^{2n}$, the existence of solutions is analyzed numerically using a sign-change method to determine the domain of existence for the deformation parameters.

3. Key Contributions

A. Exact Solution for the Harmonic Oscillator

The authors derive an exact analytical expression for the ground-state energy of a harmonic oscillator in this deformed space (Equation 21).

• Validation: This result is shown to exactly match the energy derived in previous literature [19] by solving the Schrödinger equation directly. This confirms the validity of the uncertainty-based variational approach for this specific case.
• Result: The energy is expressed in terms of the deformation parameters $\alpha'$ and $\beta'$, showing how minimal length and momentum modify the zero-point energy.

B. Generalization to Arbitrary Potentials

The paper extends the method to arbitrary potentials of the form $V(x) = U_0 U((x/a)^2)$, provided the potential satisfies specific convexity conditions (Jensen's inequality).

• Rigorous Lower Bound: For arbitrary potentials, the derived energy $E_{min}$ constitutes a rigorous lower bound to the true ground-state energy. This is because $\langle V(x) \rangle \geq V(\Delta x)$ due to Jensen's inequality, whereas the harmonic oscillator case is exact because $\langle x^2 \rangle = (\Delta x)^2$ when $\langle x \rangle = 0$.
• Linear Approximation Formula: A general expression for the ground-state energy in the linear approximation of deformation parameters is derived (Equation 53).
  • Key Finding: The linear correction term for the coordinate uncertainty ($\xi_2$) is found to be zero for any potential. This implies that the dependence of the minimal uncertainty on the momentum deformation parameter ($\beta$) is stronger than on the coordinate deformation parameter ($\alpha$) in the linear regime.

C. Domain of Existence Analysis

The authors investigate the conditions under which bound states exist in deformed space for anharmonic oscillators $V(x) \propto x^{2n}$.

• Limiting Case (Particle in a Box): As $n \to \infty$, the potential approaches a "particle in a box." The analysis confirms that the existence of solutions requires $\beta < 1/2$ (in dimensionless units), which aligns perfectly with exact solutions for a particle in a box with minimal length.
• Convergence: Numerical results show that as $n$ increases, the domain of existence for the pair $(\alpha, \beta)$ shrinks and converges to the limit defined by the particle-in-a-box constraint.
• Physical Interpretation: If the deformation parameters $(\alpha, \beta)$ fall outside the calculated existence domain for a specific potential, it implies that no bound states exist for a particle in that specific deformed space. The particle cannot be confined by the potential well under those specific deformation conditions.

4. Results

1. Harmonic Oscillator Energy: The derived exact energy (Eq. 21) and its linear approximation (Eq. 56) are consistent with exact Schrödinger solutions and previous linear approximations.
2. Anharmonic Oscillator: The linear approximation formula (Eq. 55) for the potential $V(x) = \gamma(x/a)^{2n}$ matches the authors' previous work [23] (which considered only minimal length, i.e., $\alpha'=0$), validating the generalization to include minimal momentum.
3. Existence Domains:
   • For $n=1$ (harmonic oscillator), the existence domain is determined solely by the GUR constraint $\alpha\beta < 1/4$.
   • For $n > 1$, the existence domain is strictly smaller than the GUR constraint. There are regions where the uncertainty principle is satisfied, but the specific potential cannot support a bound state due to the deformation.
   • The limit $\beta \to 1/2$ is recovered as $n \to \infty$.

5. Significance

• Methodological Robustness: The paper demonstrates that the uncertainty principle approach combined with Lagrange multipliers is a powerful and rigorous tool for estimating ground-state energies in deformed spaces. It avoids the complexity of solving differential equations while yielding exact results for the harmonic oscillator and rigorous bounds for others.
• Physical Insight into Deformation: The work highlights that minimal length and momentum uncertainties impose physical constraints on the existence of matter. It is not just a mathematical modification of energy levels; for certain combinations of deformation parameters, quantum systems (like anharmonic oscillators) may cease to have bound states entirely.
• Universality: The derived linear approximation formulas provide a practical tool for estimating energy corrections in various quantum systems (hydrogen atom, Coulomb-like problems, etc.) without needing to solve the specific Schrödinger equation for each case.
• Future Applications: The authors suggest this method can be applied to any uncertainty relation depending only on $\Delta x$ and $\Delta p$, opening avenues for studying other quantum gravity models.

In conclusion, the paper successfully bridges the gap between exact solutions and variational approximations in deformed quantum mechanics, providing a comprehensive framework for understanding how minimal length and momentum affect the stability and energy of quantum systems.