Singular Limits in Quantum Mechanics

Abstract:

When we apply singular limits to certain parameters in a differential equation, it reduces to another equation. Here, we investigate whether the solutions of these equations are related by the same singular limits.

Calogero-Sutherland model

The Schrödinger equation can be exactly solved for a quantum mechanical system consisting of N identical particles on a line that interact via an inverse square potential and are confined by a harmonic well. Bill Sutherland showed that the corresponding problem on a circle can also be exactly solved.

Before the works of Calogero and Sutherland, an example of an exactly solvable quantum N-particle system comprised N particles on a line interacting pairwise via a delta function potential.

The Hamiltonian for the Calogero model is given by:

H = -¹/₂∇² + ¹/₂ω²Σ_n=1^Nx_n² + ^g2/₂Σ_m,n=1;m≠n^N(x_n − x_m)^-2

Where ∇² is the Laplacian operator in the N-dimensional space spanned by the N coordinates x_n, and ψ is the eigenfunction corresponding to the energy eigenvalue E.

Get quality help now

Sweet V

Verified writer

Proficient in: Physics

4.9 (984)

“ Ok, let me say I’m extremely satisfy with the result while it was a last minute thing. I really enjoy the effort put in. ”

+84 relevant experts are online

Hire writer

The energy eigenvalues for the Calogero model are given by the formula:

E_k = ω(k + N/2 + N(N - 1)/2(a + 1/2)), k = 0, 1, 2, ...

Where a = ¹/₂√(1 + 2g²), with g² = l(l+1) being a positive coupling constant determining the strength of the repulsive part of the two-body interaction, and ω² determines the strength of the attractive part of the interaction. The Hamiltonian is nonrelativistic and consists of a repulsive part that is singular at the origin and an attractive part consisting of an external harmonic well.

For the Sutherland potential, we consider a finite system of N particles of mass m on a ring of length L.

The Hamiltonian (for ω = 0) is then given by:

H = -ħ²/_2mΣ_i=1^N(d²/dx_i²) + l(l + 1)π²/L²Σ_i≠j^Nsin²(π(x_i − x_j)/L)

The interactive term is inversely proportional to the square of the chord distance between the pairs of particles. In the thermodynamic limit (L→∞), it mimics the inverse square potential, i.e., the Calogero potential model.

Eigenstates of the Sutherland potential in one dimension

The Hamiltonian for a one-body system in a Sutherland potential in one dimension is given by:

H = -ħ²/_2m(d²/dx²) + l(l + 1)π²/L²sin²(πx/L)

For the ground state, we take the solution as:

ψ = sin((l+1)πx/L)

Substituting it into the Schrödinger equation, we obtain the ground state energy as:

E₀ = ((l + 1)²ħ²π²)/(2mL²)

For the excited states, we assume a solution of the form:

ψ = f(cos(πx/L))sin((l+1)πx/L)

Substituting it into the Schrödinger equation, we obtain the energy eigenstates as:

E_n = ((n+ l + 1)²ħ²π²)/(2mL²)

Where the function f(cos(πx/L)) is the Gegenbauer Polynomials C_l+1ⁿ(cos(πx/L)).

Eigenstates of the Calogero potential in one dimension

The Hamiltonian for a one-body system in a Calogero potential (with ω = 0) in one dimension is given by:

H = -ħ²/_2m(d²/dx²) + l(l + 1)/x²

This is a scattering problem and involves no bound states. Let us assume a solution of the form:

ψ = f(x)x^l+1

Substituting it into the Schrödinger equation, we get:

x(d²f/dx²) + 2(l + 1)(df/dx) + Ef(x) = 0

This equation can be reduced to Bessel's equation using the following transformation:

f(x) = xⁿz

Writing 2(l + 1) = a and E = k², the equation reduces to:

x²(d²z/dx²) + x(dz/dx) + (k²x² - n²)z = 0

Where z = J_n(kx) is the Bessel function. The final wavefunction is given by:

ψ = √xJ_n(kx)

And the energy is given by:

E = (k²ħ²)/(2m)

Conclusion

In this paper, we have explored the Calogero-Sutherland model in quantum mechanics, specifically focusing on the eigenstates of the Sutherland potential in one dimension and the eigenstates of the Calogero potential in one dimension. These models provide valuable insights into the behavior of quantum systems with different potentials and interactions.

Our analysis has revealed the energy eigenvalues and wavefunctions for these systems, shedding light on their quantum mechanical properties. The Calogero-Sutherland model offers a rich framework for understanding the behavior of particles interacting in various ways, from inverse square potentials to harmonic wells.

Overall, this paper contributes to the broader understanding of quantum mechanics and the behavior of particles in different potential landscapes. Further research can extend these findings to more complex systems and explore the implications for various physical phenomena.

Bibliography

1. F. Calogero, "Solution of a three-body problem in one dimension," Journal of Mathematical Physics, vol. 10, no. 12, pp. 2191–2196, 1969.
2. F. Calogero, "Calogero-moser system," Scholarpedia, vol. 3, no. 8, p. 7216, 2008.