Quanta Wave Number exp N
CondensedMatter.TightBindingChain.quantaWaveNumber_exp_N
Project documentation
The quantized wavenumbers form a subset of the BrillouinZone. -/ lemma quantaWaveNumber_subset_brillouinZone : T.QuantaWaveNumber ⊆ T.BrillouinZone := by rintro _ ⟨n, rfl⟩ have hT := T.a_pos have hNpos : 0 < T.N := lt_of_le_of_lt (Nat.zero_le _) n.isLt simp only [BrillouinZone, Set.mem_Ico] generalize T.N = x at * generalize T.a = a at * have hx : (0 :...
Exact Lean statement
lemma quantaWaveNumber_exp_N (n : ℕ) (k : T.QuantaWaveNumber) :
Complex.exp (Complex.I * k * n * T.N * T.a) = 1Formal artifact
Lean source
lemma quantaWaveNumber_exp_N (n : ℕ) (k : T.QuantaWaveNumber) : Complex.exp (Complex.I * k * n * T.N * T.a) = 1 := by refine Complex.exp_eq_one_iff.mpr ?_ obtain ⟨_, m, rfl⟩ := k use ((m : Int) - (T.N / 2 : ℕ)) * (n : ℤ) have hpp : (T.N : ℂ) ≠ 0 := by simp [Ne.symm (NeZero.ne' T.N)] have hT' : (T.a : ℂ) ≠ 0 := Complex.ne_zero_of_re_pos T.a_pos simp only [Complex.ofReal_mul, Complex.ofReal_div, Complex.ofReal_ofNat, Complex.ofReal_natCast, Complex.ofReal_sub, Int.cast_mul, Int.cast_sub, Int.cast_natCast] field_simp- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/CondensedMatter/TightBindingChain/Basic.lean:329-338
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