Secular Equation and Product of Eigenvalues for Spin $S=2$ System in Non-Axisymmetric Crystal Field

Problem Statement

For a quantum mechanical system with spin quantum number $S = 2$S=2, the spin Hamiltonian $\mathcal{H}$H in a non-axisymmetric crystal field is given by:

$$\mathcal{H} = D\hat{S}_z^2 + E(\hat{S}_x^2 - \hat{S}_y^2)$$H=DS^z2​+E(S^x2​−S^y2​)

where $\hat{S}_x, \hat{S}_y, \hat{S}_z$S^x​,S^y​,S^z​ are angular momentum operators for spin $S = 2$S=2 in units with $\hbar = 1$ℏ=1, and $D$D, $E$E are positive constants.

This Hamiltonian is represented as a $5\times5$5×5 Hermitian matrix in the basis of eigenstates $|m\rangle$∣m⟩ ($m = 2, 1, 0, -1, -2$m=2,1,0,−1,−2) of $\hat{S}_z$S^z​.

The secular equation determining the energy eigenvalues, with $I$I as the identity matrix:

$$\det(\lambda I - \mathcal{H}) = 0$$det(λI−H)=0

has five eigenvalues $\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5$λ1​,λ2​,λ3​,λ4​,λ5​ (with multiplicity).

For the values of $D$D and $E$E given in the constraints, find the product of the five eigenvalues $\prod_{i=1}^5\lambda_i$∏i=15​λi​.

Constraints

• Axially symmetric splitting parameter: $D = 1$D=1
• Non-axisymmetric (rhombic) splitting parameter: $E = 4$E=4
• $D$D, $E$E, and eigenvalues $\lambda$λ are treated as dimensionless values.

Input Format

Give the value of $\lambda_1\lambda_2\lambda_3\lambda_4\lambda_5$λ1​λ2​λ3​λ4​λ5​ as a positive integer.