GSO002 Problem 18
Problem Statement
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, the spin Hamiltonian H in a non-axisymmetric crystal field is given by:
H=DS^z2+E(S^x2−S^y2)where S^x,S^y,S^z are angular momentum operators for spin S=2 in units with ℏ=1, and D, E are positive constants.
This Hamiltonian is represented as a 5×5 Hermitian matrix in the basis of eigenstates ∣m⟩ (m=2,1,0,−1,−2) of S^z.
The secular equation determining the energy eigenvalues, with I as the identity matrix:
det(λI−H)=0has five eigenvalues λ1,λ2,λ3,λ4,λ5 (with multiplicity).
For the values of D and E given in the constraints, find the product of the five eigenvalues ∏i=15λi.
Constraints
- Axially symmetric splitting parameter: D=1
- Non-axisymmetric (rhombic) splitting parameter: E=4
- D, E, and eigenvalues λ are treated as dimensionless values.
Input Format
Give the value of λ1λ2λ3λ4λ5 as a positive integer.
Solution
1. Expressing the Hamiltonian Using Raising and Lowering Operators
Using S^±=S^x±iS^y:
S^x2−S^y2=21(S^+2+S^−2)The action of raising and lowering operators: S^±∣m⟩=S(S+1)−m(m±1)∣m±1⟩.
2. Matrix Representation
For S=2, S(S+1)=6. The Hamiltonian matrix in basis (∣2⟩,∣1⟩,∣0⟩,∣−1⟩,∣−2⟩)T is:
H=4D06E000D03E06E0006E03E0D0006E04D3. Block Diagonalization
The even-m states (∣2⟩,∣0⟩,∣−2⟩) and odd-m states (∣1⟩,∣−1⟩) are decoupled. Rearranging:
Block H1 (odd-m subspace):
H1=(D3E3ED)Block H2 (even-m subspace):
H2=4D6E06E06E06E4D4. Product of Eigenvalues = Determinant
By Vieta's formulas, the product of all eigenvalues equals det(H)=det(H1)×det(H2).
det(H1)=D2−9E2 det(H2)=4D(0−6E2)−6E(6E⋅4D)=−24DE2−24DE2=−48DE2 i=1∏5λi=(D2−9E2)(−48DE2)=48DE2(9E2−D2)5. Substituting D=1, E=4
=48×1×16×(9×16−1)=768×143=109824Answer: 109824