File size: 2,911 Bytes
9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import OAI.Probability.ClassicalON.AnnulusSmall
namespace OAI
noncomputable section
open scoped Real
namespace ClassicalON
theorem axisC_cube : axisC^3 = -axisC := by
rw [axisC,← map_pow,← map_neg]
congr 1
ext i j
fin_cases i <;> fin_cases j <;>
norm_num [axisMatrixC,pow_succ,Matrix.mul_apply,Fin.sum_univ_succ]
theorem axisC_sq_mul : axisC^2*axisC = -axisC := by
rw [← pow_succ,axisC_cube]
theorem axis_smul_mul (r : ℝ) (A B : SpinOperator 3) : (r • A)*B = r • (A*B) :=
Algebra.smul_mul_assoc r A B
theorem axis_mul_smul (r : ℝ) (A B : SpinOperator 3) : A*(r • B) = r • (A*B) :=
Algebra.mul_smul_comm r A B
def axisPolynomial (t : ℝ) : SpinOperator 3 :=
1 + (Real.sin t) • axisC + (1-Real.cos t) • axisC^2
theorem axisPolynomial_derivative (t : ℝ) :
HasDerivAt axisPolynomial (axisPolynomial t*axisC) t := by
have h := ((hasDerivAt_const t (1:SpinOperator 3)).add ((Real.hasDerivAt_sin t).smul_const axisC)).add
(((hasDerivAt_const t (1:ℝ)).sub (Real.hasDerivAt_cos t)).smul_const (axisC^2))
apply h.congr_deriv
change 0 + (Real.cos t) • axisC + (0 - -Real.sin t) • axisC^2 = axisPolynomial t*axisC
simp only [axisPolynomial,add_mul,one_mul,axis_smul_mul,axisC_sq_mul,← pow_two]
module
theorem axisPolynomial_commute (t : ℝ) : Commute (axisPolynomial t) axisC := by
change axisPolynomial t*axisC = axisC*axisPolynomial t
simp only [axisPolynomial,add_mul,mul_add,one_mul,mul_one,axis_smul_mul,
axis_mul_smul,← pow_succ,← pow_succ']
theorem exp_axisC (t : ℝ) : NormedSpace.exp (t • axisC)=axisPolynomial t := by
let f := fun t : ℝ => NormedSpace.exp (t • (-axisC))*axisPolynomial t
have hd (t : ℝ) : HasDerivAt f 0 t := by
apply ((hasDerivAt_exp_smul_const (-axisC) t).mul (axisPolynomial_derivative t)).congr_deriv
rw [mul_assoc,neg_mul axisC (axisPolynomial t),← (axisPolynomial_commute t).eq]
simp
have he : f t = f 0 := is_const_of_deriv_eq_zero (fun t => (hd t).differentiableAt)
(fun t => (hd t).deriv) t 0
have h0 : f 0 = 1 := by simp [f,axisPolynomial,zero_smul ℝ axisC,zero_smul ℝ (axisC^2)]
rw [h0] at he
have hm := congrArg (fun Q : SpinOperator 3 => NormedSpace.exp (t • axisC)*Q) he
dsimp only [f] at hm
rw [← mul_assoc,smul_neg,← NormedSpace.exp_add_of_commute (Commute.refl (t • axisC)).neg_right,
add_neg_cancel,NormedSpace.exp_zero,mul_one,one_mul] at hm
exact hm.symm
theorem exp_axisC_periodic : Function.Periodic (fun t : ℝ => NormedSpace.exp (t • axisC)) (2*Real.pi) := by
intro t
simp [exp_axisC,axisPolynomial,Real.sin_add_two_pi,Real.cos_add_two_pi]
theorem exp_axisC_pi_apply (v : EuclideanSpace ℝ (Fin 3)) :
NormedSpace.exp (Real.pi • axisC) v = WithLp.toLp 2 ![-v 0,-v 1,v 2] := by
rw [exp_axisC]
ext i
fin_cases i <;> simp [axisPolynomial,axisC_apply,pow_two] <;> ring
end ClassicalON
end
end OAI
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