Busch Model: Two-Particle Energy in Traps

• The Busch model is a quantization condition for two interacting particles in a 3D harmonic oscillator trap, relating short-range s-wave scattering phase shifts to discrete energy levels.
• It employs a phase-shift approximation where energy shifts follow the 'phase-shift × level-spacing' rule, yielding errors as low as 0.5% for excited states in smooth traps.
• Finite-range generalizations and comparisons with hard-wall geometries underscore the model’s sensitivity to trap conditions and its relevance to ultracold atomic systems.

The Busch model provides a rigorous quantization condition for the energy spectrum of two point-like particles interacting via short-range, s-wave interactions inside a three-dimensional (3D) isotropic harmonic oscillator (H.O.) trap. Central to the model is the relationship between the discrete trap spectrum and the free-space scattering phase shifts. The Busch formula and its generalizations serve as benchmarks for studies in few-body and many-body quantum physics, ultracold atomic systems, and for the implementation of pseudopotential models in trapped geometries (Kohler, 2011, Goswami et al., 2016).

1. Formulation of the Busch Model: Assumptions and Derivation

The system consists of two identical particles of mass $m$ interacting via a short-range (delta-function) potential, confined within a 3D isotropic H.O. trap of frequency $\omega$. The center-of-mass and relative coordinates separate, yielding the total energy in trap units ($\hbar\omega$): $E_{\rm tot} = E_{\rm rel} + E_{\rm cm} = 2n + 3,$ where $E_{\rm cm} = \tfrac{3}{2}$ for the center-of-mass and $E_{\rm rel} = E$ denotes the relative energy; $n=0, 1, 2, \dots$ labels the quantum number of relative motion.

Defining the oscillator length $a_{\rm osc} = \sqrt{\hbar / m \omega}$, the relative motion is governed by s-wave scattering. Outside the interaction range, the relative wavefunction must match a free-space s-wave form, leading to a quantization condition dependent on the free-space phase shift $\delta(k)$.

The Busch formula relates the scattering phase shift to the discrete trap spectrum: $$\tan \delta(k) = -\,\frac{\sqrt{\eta}\;\Gamma\left(\frac{1-\eta}{4}\right)}{2\;\Gamma\left(\frac{3-\eta}{4}\right)} \,, \quad \eta = 2E,$$ with $\Gamma$ denoting the Gamma function (Kohler, 2011).

2. Energy Shift and Phase-Shift Approximation

The noninteracting benchmark energies are $E^{(0)} = 2n+\frac{3}{2}$, with interacting levels shifted by $\Delta$. The level spacing is $dE=2$. Introducing the dimensionless shift $S = \Delta / dE$ yields $E = 2n+\frac{3}{2} + 2S$ and $\eta = 4n+3+4S$.

Algebraic manipulation, utilizing the Gamma-function reflection formula, recasts the Busch result: $$\tan\delta(k) = -\,A(z)\,\tan(\pi S), \quad z = n+S+\tfrac{1}{2},$$

$$A(z) = \frac{\sqrt{z-\tfrac{1}{4}}\; \Gamma(z)}{\Gamma(z+\tfrac{1}{2})}.$$

Numerically, $A(z) \to 1$ rapidly for increasing $z$, enabling the phase-shift approximation: $$\tan\delta(k) \approx -\tan(\pi S) \implies S \approx -\frac{\delta(k)}{\pi} \,.$$ This yields the "phase-shift × level-spacing" rule for the energy shift: $$\Delta \approx -\frac{\delta(k)}{\pi} \, dE, \qquad dE=2.$$ This approximation is physically significant in replacing the exact spectrum with a result determined solely by the phase shift at energy $k$ and the local level spacing (Kohler, 2011).

3. Finite-Range Generalizations and Effective Range Effects

Extending the model to include finite-range corrections, (Goswami et al., 2016) constructs model potentials $V_-(r)$ and $V_+(r)$ tailored for negative and positive scattering lengths, respectively, and parameterized by the effective range $r_0$. These potentials are formulated to satisfy the effective-range expansion: $$k\cot\delta_0(k) = -1/a_s + \tfrac{1}{2}r_0 k^2 + \cdots$$ In the $r_0 \to 0$ limit, these model interactions recover the Busch contact pseudopotential spectrum in 3D. For large $r_0$, the spectrum approaches that of noninteracting particles. However, in 1D, as $r_0 \to 0$, the spectrum does not reduce smoothly to the 1D contact limit, indicating a singularity in the 1D zero-range model—a distinctive feature not present in 3D (Goswami et al., 2016).

The energy levels for finite-range interactions are determined numerically via the solution of the radial Schrödinger equation using the Numerov algorithm, integrating outward from the origin with appropriate boundary conditions and locating the discrete levels by root-finding of an energy-dependent shooting function.

4. Quantitative Accuracy and Error Structure

The energy shift derived from the phase-shift approximation can be quantitatively compared to the exact solution. For the ground relative state ($n=0$) and small $|a|$, the maximum error is approximately 3%. For excited states ($n\geq 1$), the error decreases rapidly to below 0.5%. In the strongly interacting limit ($|a|\to\infty$), corresponding to unitarity, the phase shift approaches $\pm \frac{\pi}{2}$ and the approximation becomes exact, with the shift equal to one level-spacing. The overall accuracy illustrates that the H.O. geometry is particularly favorable for phase-shift-based effective interaction schemes (Kohler, 2011).

5. Comparison to Spherical Infinite Square-Well Traps

In the spherical infinite square-well geometry of radius $R$, the corresponding energy spectrum is determined by the requirement that the radial wavefunction vanishes at $r=R$, with the phase shift modifying the quantization condition: $$k^{(0)}_n R = n\pi, \qquad k_n R + \delta(k_n) = n\pi$$ The resulting shift is: $$S = \frac{\Delta}{dE} = -\frac{k_n+k^{(0)}_n}{k^{(0)}_{n+1}+k^{(0)}_n} \frac{\delta(k_n)}{\pi}$$ In the $R\to\infty$ or large $n$ limit, $S \to -\delta/\pi$ as in the H.O. case. However, for finite $R$, the error in the phase-shift approximation can be as large as 30%. The discrepancy arises from the hard-wall boundary, which modifies the wavefunction matching and level density compared to the smooth H.O. case. The error depends on both the scattering phase shift and the relative scale of the interaction range to the trap size (Kohler, 2011).

6. Physical Implications and Role of Trap Geometry

The geometry of the confining trap dramatically impacts the relationship between free-space scattering and the discrete spectrum. In 3D isotropic H.O. traps, the "phase-shift × level-spacing" rule is numerically accurate (error $<$ 0.5% for all but the ground state) due to the absence of hard walls and the linear growth of level density with energy. In contrast, hard-wall geometries such as spherical and cubic boxes can exhibit substantial deviations unless the box is taken to the infinite-size limit. The origin of this distinction can be traced to the boundary conditions imposed by the trap geometry—smooth (harmonic) versus sharp (hard wall)—an insight already noted in the work of DeWitt (1956) (Kohler, 2011).

For finite-range interactions in 3D, the Busch spectrum is precisely recovered as the range parameter $r_0 \to 0$. In highly anisotropic traps or for sufficiently large effective range, the spectrum can interpolate between various dimensional regimes or interaction limits, enabling systematic exploration relevant to ultracold atoms in tight confinements and effective field theory approaches (Goswami et al., 2016). In 1D, however, the singular nature of the contact limit precludes straightforward reproduction of the δ-function spectrum by shrinking smooth finite-range potentials.

7. Summary Table: Accuracy of Phase-Shift Approximation

Trap Geometry	Typical Error in $\Delta \approx -(\delta/\pi)dE$	Behavior for Large $n$ or Trap Size
3D Isotropic H.O.	$\lesssim 0.5\%$ (excited), $\leq 3\%$ (ground)	Error $\to 0$ as $|a| \to \infty$
Spherical Box ($R$ finite)	Up to 30%	Error $\to 0$ as $R \to \infty$

In conclusion, the Busch model and its finite-range generalization constitute foundational tools for understanding two-body quantum spectra in external traps. The validity of phase-shift-based approximations is highly geometry-dependent, being robust in 3D harmonic confinement and significantly less so in hard-wall geometries. These results underpin the design and interpretation of effective interactions in few- and many-body quantum systems in applied cold-atom and nuclear physics contexts (Kohler, 2011, Goswami et al., 2016).