bc6f58b2c4
- Implemented `2_exact_casimir_projectors.py` to construct exact SO(3) isotypic projectors using the Casimir operator, replacing Monte-Carlo methods. - Created `3_apply_exact_projectors.py` to apply the exact projectors to example states, calculating ||A_j||_* estimates with improved precision. - Developed `4_spherical_basis.py` to build a Condon-Shortley-consistent spherical basis for a single spin-1 leg using ladder operators. - Introduced `6_six_j_recoupling_proof.py` to provide a complete proof of the cut-recoupling formula for full collective SU(2) symmetry, verifying the relationship between reduced blocks A_j^(1) and A_p^(2). - Added a README file to document the execution order and purpose of each script in the symmetric states project.
65 lines
2.3 KiB
Python
65 lines
2.3 KiB
Python
"""
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Step 1: build a Condon-Shortley-consistent spherical basis {|1,-1>,|1,0>,|1,+1>}
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for a single spin-1 leg, starting from the EXACT Cartesian generators
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(J_a)_{bc} = -i eps_{abc} (already verified symbolically in
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exact_casimir_projectors.py), and using the ladder-operator construction
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so we do not have to trust a memorized phase convention.
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"""
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import numpy as np
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def J_component(a):
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eps = np.zeros((3,3,3))
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eps[0,1,2]=eps[1,2,0]=eps[2,0,1]=1
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eps[0,2,1]=eps[2,1,0]=eps[1,0,2]=-1
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M = np.zeros((3,3), dtype=complex)
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for b in range(3):
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for c in range(3):
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M[b,c] = -1j*eps[a,b,c]
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return M
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Jx, Jy, Jz = J_component(0), J_component(1), J_component(2)
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Jp = Jx + 1j*Jy # raising
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Jm = Jx - 1j*Jy # lowering
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# sanity
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print("[Jx,Jy]-i Jz max err:", np.abs(Jx@Jy-Jy@Jx - 1j*Jz).max())
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print("J^2 (single leg), should be 2*I:")
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print(np.round(Jx@Jx+Jy@Jy+Jz@Jz,6))
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# eigenvectors of Jz
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evals, evecs = np.linalg.eigh(Jz) # Jz Hermitian? check
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print("Jz Hermitian check:", np.abs(Jz - Jz.conj().T).max())
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print("Jz eigenvalues:", np.round(evals,6))
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# pick |1,-1> = eigenvector with eigenvalue closest to -1, fix phase: first
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# nonzero component real positive
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idx_m1 = np.argmin(np.abs(evals - (-1)))
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v_m1 = evecs[:, idx_m1]
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# fix global phase
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k = np.argmax(np.abs(v_m1))
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v_m1 = v_m1 * np.exp(-1j*np.angle(v_m1[k]))
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if v_m1[k].real < 0: v_m1 = -v_m1
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print("\n|1,-1> (Cartesian components x,y,z):", np.round(v_m1,4))
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# ladder up: |1,0> = Jp|1,-1> / ||...|| (standard CS convention: J+|j,m>=sqrt((j-m)(j+m+1))|j,m+1>, positive real coefficient)
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v0_raw = Jp @ v_m1
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n0 = np.linalg.norm(v0_raw)
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v_0 = v0_raw / n0
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print("|1,0> raw ladder norm (expect sqrt((1-(-1))*(1+(-1)+1))=sqrt(2)):", n0)
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v_p1_raw = Jp @ v_0
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n_p1 = np.linalg.norm(v_p1_raw)
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v_p1 = v_p1_raw / n_p1
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print("|1,+1> raw ladder norm (expect sqrt((1-0)*(1+0+1))=sqrt(2)):", n_p1)
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# check orthonormality and Jz eigenvalues
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basis = np.stack([v_m1, v_0, v_p1], axis=1) # columns
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print("\northonormality check (should be I_3):")
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print(np.round(basis.conj().T @ basis, 6))
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for name, v, m in [("|1,-1>", v_m1, -1), ("|1,0>", v_0, 0), ("|1,+1>", v_p1, 1)]:
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Jzv = Jz @ v
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print(f"{name}: Jz|.> - {m}|.> max err = {np.abs(Jzv - m*v).max():.2e}")
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np.save("spherical_basis_single_leg.npy", basis) # columns m=-1,0,+1
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print("\nsaved spherical_basis_single_leg.npy (columns ordered m=-1,0,+1)")
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