[Paper Review] Exact diagonalization of the $d-$dimensional confined quantum harmonic oscillator

Kunle Adegoke, Adenike Olatinwo+3 more|arXiv (Cornell University)|Apr 5, 2016

Quantum Mechanics and Non-Hermitian Physics2 references3 citations

TL;DR

This paper presents an exact matrix representation for the Hamiltonian of the d-dimensional confined quantum harmonic oscillator using eigenstates of the kinetic energy operator as basis functions. By constructing explicit N^d × N^d matrices for the kinetic and potential energy operators, the method enables direct diagonalization without root-finding or numerical integration, yielding accurate energy spectra. The key contribution is a closed-form matrix expression that facilitates efficient computation and enables a sixth-order analytic approximation for the 1D case.

ABSTRACT

In the existing literature various numerical techniques have been developed to quantize the confined harmonic oscillator in higher dimensions. In obtaining the energy eigenvalues, such methods often involve indirect approaches such as searching for the roots of hypergeometric functions or numerically solving a differential equation. In this paper, however, we derive an explicit matrix representation for the Hamiltonian of a confined quantum harmonic oscillator in higher dimensions, thus facilitating direct diagonalization.

Motivation & Objective

• To develop a direct, exact matrix representation of the d-dimensional confined harmonic oscillator Hamiltonian.
• To eliminate indirect numerical methods such as root-finding in hypergeometric functions or solving differential equations.
• To enable efficient and accurate energy spectrum computation via direct diagonalization.
• To derive an approximate analytic expression for the 1D confined harmonic oscillator energy levels up to sixth order in ω.
• To provide a computationally tractable framework for higher-dimensional quantum systems with hard-wall confinement.

Proposed method

• Basis states are constructed from eigenfunctions of the kinetic energy operator in 1D, given by φ_r(x_i) = √(1/L) cos[π/2 sin²(rπ/2) - (r+1)πx_i/(2L)], forming a complete orthonormal set.
• The full d-dimensional Hilbert space is spanned by direct product states ψ_s(x) = ∏_{i=1}^d φ_{s_i}(x_i), with s indexing N^d states.
• The Hamiltonian matrix elements are derived as H_st = ∑_{i=1}^d c_{i_st} (T_i)_{s_i t_i} + ∑_{i=1}^d c_{i_st} (V_i)_{s_i t_i}, where c_i are binary matrices enforcing index matching across dimensions.
• The kinetic energy matrix T is diagonal in the basis, with eigenvalues ε_s = ε ∑_{i=1}^d (s_i + 1)^2, where ε = π²ℏ²/(8mL²).
• The potential energy matrix V is constructed via matrix elements (V_i)_{s_i t_i} = ⟨φ_{s_i}|(1/2)mω²x_i²|φ_{t_i}⟩, computed using trigonometric integrals.
• The full Hamiltonian is assembled as H = T + V, with matrix elements expressed in terms of Kronecker delta structures and cosine terms to capture off-diagonal couplings.

Experimental results

Research questions

• RQ1 Can an exact matrix representation of the d-dimensional confined harmonic oscillator be derived without relying on indirect numerical methods?
• RQ2 How can the kinetic and potential energy operators be explicitly represented in a finite-dimensional basis of confined states?
• RQ3 What is the structure of the Hamiltonian matrix in the basis of kinetic energy eigenstates for d-dimensional systems?
• RQ4 Can a high-order analytic approximation for the 1D confined harmonic oscillator energy spectrum be derived from the matrix formulation?
• RQ5 What is the role of the binary coupling matrices c_i in enabling the decomposition of the full Hamiltonian into decoupled 1D contributions?

Key findings

• The Hamiltonian is expressed as an explicit N^d × N^d matrix with closed-form elements involving δ-functions, cosine terms, and inverse-square terms, enabling direct diagonalization.
• For the 1D case, the matrix reduces to an N×N form with H_st = εδ_st(s+1)^2 + (λ²ε/8)δ_st(π²/6 - 1/(s+1)^2) + (λ²ε/2)(1/(s−t)^2 + δ_st − 1/(s+t+2)^2)cos²((s−t)π/2) for s≠t.
• An approximate analytic energy spectrum is derived up to sixth order in ω, with E_r ≈ ∑_{m=0}^3 {λ^{2m}ε / 2^{4m−1} ∑_{n=0}^m [(-1)^n ζ(2m−2n)c_n^{(m)} / (r+1)^{2(m+n−1)}]}
• The second-order energy correction is E_r^{(2)} = (λ⁴ε/128)[ζ(4)/(r+1)^2 − 5ζ(2)/(r+1)^4 + 7/(r+1)^6], with ζ(2)=π²/6 and ζ(4)=π⁴/90.
• The method avoids solving differential equations or root-finding in special functions, offering a numerically stable and efficient alternative for confined quantum systems.
• The approach is generalizable to arbitrary d and N, with the matrix structure preserving the Kronecker sum form of the original Hamiltonian.