Hamiltonian hermitian
CondensedMatter.TightBindingChain.hamiltonian_hermitian
Plain-language statement
The hamiltonian of the tight binding chain is hermitian.
Exact Lean statement
lemma hamiltonian_hermitian (ψ φ : T.HilbertSpace) :
⟪T.hamiltonian ψ, φ⟫_ℂ = ⟪ψ, T.hamiltonian φ⟫_ℂFormal artifact
Lean source
lemma hamiltonian_hermitian (ψ φ : T.HilbertSpace) : ⟪T.hamiltonian ψ, φ⟫_ℂ = ⟪ψ, T.hamiltonian φ⟫_ℂ := by simp only [hamiltonian, LinearMap.sub_apply, LinearMap.smul_apply, LinearMap.coe_sum, Finset.sum_apply, LinearMap.add_apply, inner_sub_left, inner_sub_right] congr 1 · -- E0 term simp only [Finset.smul_sum, sum_inner, inner_sum, inner_smul_left_eq_smul, inner_smul_right_eq_smul, localizedComp_adjoint] · -- t term simp only [Finset.smul_sum, smul_add, sum_inner, inner_sum, inner_add_left, inner_add_right, inner_smul_left_eq_smul, inner_smul_right_eq_smul, localizedComp_adjoint] exact Finset.sum_congr rfl fun n _ => by ring- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/CondensedMatter/TightBindingChain/Basic.lean:224-235
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