feat(QuantumMechanics): basic properties/defs for HarmonicOscillator

Physlib.lean

@@ -280,6 +280,7 @@ public import Physlib.QFT.QED.AnomalyCancellation.VectorLike
 public import Physlib.QuantumMechanics.FiniteTarget
 public import Physlib.QuantumMechanics.FreeParticle.Basic
 public import Physlib.QuantumMechanics.HarmonicOscillator.Basic
+public import Physlib.QuantumMechanics.HarmonicOscillator.LadderOperators
 public import Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.Basic
 public import Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.Completeness
 public import Physlib.QuantumMechanics.HarmonicOscillator.OneDimension.Eigenfunction

Physlib/QuantumMechanics/HarmonicOscillator/Basic.lean

@@ -5,31 +5,252 @@ Authors: Gregory J. Loges
 -/
 module
 
-public import Physlib.Meta.TODO.Basic
+public import Physlib.Meta.Informal.Basic
+public import Physlib.QuantumMechanics.Operators.Momentum
+public import Physlib.QuantumMechanics.Operators.Multiplication
+public import Physlib.QuantumMechanics.QuantumSystem.Basic
 /-!
 
 # The quantum harmonic oscillator
 
--/
+## i. Overview
 
-@[expose] public section
+The harmonic oscillator is one of the most important examples in non-relativistic quantum mechanics.
+It describes a particle of mass `m` subject to a positive-definite quadratic potential
+in `d` dimensions.
+
+- `Basic.lean` : Properties of the potential, definition of isotropic oscillators,
+    kinetic, potential and Hamiltonian operators.
+- `LadderOperators.lean` : Definitions of the raising/lowering/number operators
+    and their algebraic properties.
+
+## ii. Key results
 
-TODO "Define `HarmonicOscillator` as a structure extending `SpaceDQuantumSystem`
-  (c.f. `Hydrogen.Basic.lean` for an example). In general the potential is determined by
-  a positive-definite, real symmetric matrix `V = ½m(xᵗ·A·x)`.
-  Note that such matrices can always be diagonalized so perhaps it suffices to take `A` diagonal.
-  A special case with enhanced symmetry is the isotropic harmonic oscillator with `A = ω²·𝕀`."
+## iii. Table of contents
 
-TODO "Define the raising/lowering/number operators for the quantum harmonic oscillator."
+- A. Basic properties
+  - A.1. Positive mass
+  - A.2. Positive natural frequencies
+- B. Characteristic lengths
+- C. The quadratic potential function
+  - C.1. Positive-definite matrix
+  - C.2. Quadratic form
+  - C.3. Potential function
+- D. Isotropic oscillators
+- E. Hilbert space
+- F. Operators
+  - E.1. Kinetic energy
+  - E.2. Potential energy
+  - E.3. Hamiltonian
+- G. As a quantum system
 
-TODO "Prove the commutation relations for the raising/lowering/number/Hamiltonian operators
-  of the quantum harmonic oscillator."
+## iv. References
 
-TODO "Determine the spectrum of the quantum harmonic oscillator in terms of the eigenvalues
-  of the matrix `A ≻ 0` appearing in the potential."
+-/
+
+@[expose] public section
+
+TODO "Determine the spectrum of the quantum harmonic oscillator in terms of
+  the natural frequencies and integer quantum numbers."
 
 TODO "Determine the energy eigenstates of the quantum harmonic oscillator
   in the 'Cartesian basis' in terms of Hermite polynomials."
 
 TODO "Determine the energy eigenstates of the isotropic quantum harmonic oscillator
   in the 'spherical basis' in terms of spherical harmonics."
+
+noncomputable section
+namespace QuantumMechanics
+
+/-- The `d`-dimensional quantum harmonic oscillator. -/
+structure HarmonicOscillator (d : ℕ) where
+  /-- The mass (positive). -/
+  m : ℝ
+  hm : 0 < m
+  /-- The natural frequencies (positive). -/
+  ω : Fin d → ℝ
+  hω : ∀ i, 0 < ω i
+
+variable {d : ℕ} (Q : HarmonicOscillator d) (i : Fin d)
+
+namespace HarmonicOscillator
+
+open Constants SpaceDHilbertSpace MeasureTheory
+
+/-!
+## A. Basic properties
+-/
+
+/-!
+### A.1. Positive mass
+-/
+
+@[simp]
+lemma m_pos : 0 < Q.m := Q.hm
+
+@[simp]
+lemma m_nonneg : 0 ≤ Q.m := Q.hm.le
+
+@[simp]
+lemma m_ne_zero : Q.m ≠ 0 := Q.hm.ne'
+
+/-!
+### A.2. Positive natural frequencies
+-/
+
+@[simp]
+lemma ω_pos : 0 < Q.ω i := Q.hω i
+
+@[simp]
+lemma ω_nonneg : 0 ≤ Q.ω i := (Q.hω i).le
+
+@[simp]
+lemma ω_ne_zero : Q.ω i ≠ 0 := (Q.hω i).ne'
+
+/-!
+## B. Characteristic lengths
+-/
+
+/-- The characteristic length `ξ i ≔ √ℏ / (√Q.m * √(Q.ω i))`. -/
+def ξ : ℝ := √ℏ / (√Q.m * √(Q.ω i))
+
+lemma ξ_eq : Q.ξ i = √ℏ / (√Q.m * √(Q.ω i)) := rfl
+
+@[simp]
+lemma ξ_pos : 0 < Q.ξ i := by simp [ξ_eq]
+
+@[simp]
+lemma ξ_nonneg : 0 ≤ Q.ξ i := (Q.ξ_pos i).le
+
+@[simp]
+lemma ξ_ne_zero : Q.ξ i ≠ 0 := (Q.ξ_pos i).ne'
+
+lemma ξ_sq : (Q.ξ i) ^ 2 = ℏ / (Q.m * Q.ω i) := by rw [Q.ξ_eq]; field_simp; simp [← mul_rotate]
+
+lemma ξ_inv : (Q.ξ i)⁻¹ = √Q.m * √(Q.ω i) / √ℏ := by simp [ξ_eq]
+
+lemma ξ_inv' : (Q.ξ i)⁻¹ = Q.m * Q.ω i * Q.ξ i / ℏ := by field_simp; simp [ξ_sq, mul_assoc]
+
+/-!
+## C. The quadratic potential function
+-/
+
+section
+
+open Matrix
+
+/-!
+### C.1. Positive-definite matrix
+-/
+
+/-- The positive-definite matrix defining the quadratic potential function. -/
+def potentialMatrix : Matrix (Fin d) (Fin d) ℝ := diagonal ((2⁻¹ * Q.m) • Q.ω ^ 2)
+
+lemma potentialMatrix_eq : Q.potentialMatrix = diagonal ((2⁻¹ * Q.m) • Q.ω ^ 2) := rfl
+
+lemma potentialMatrix_isHermitian : Q.potentialMatrix.IsHermitian := by simp [potentialMatrix_eq]
+
+@[simp]
+lemma potentialMatrix_mulVec (v : Fin d → ℝ) :
+    Q.potentialMatrix *ᵥ v = (2⁻¹ * Q.m) • (Q.ω ^ 2 * v) := by
+  ext
+  simp [potentialMatrix_eq, smul_mulVec, mulVec_diagonal]
+
+/-!
+### C.2. Quadratic form
+-/
+
+/-- The positive-definite quadratic form associated to the potential matrix. -/
+def potentialQuadraticForm : QuadraticForm ℝ (Fin d → ℝ) := Q.potentialMatrix.toQuadraticForm'
+
+/-!
+### C.3. Potential function
+-/
+
+/-- The quadratic potential function, `½m · ∑ i, ωᵢ²·xᵢ²`. -/
+def potentialFunction : Space d → ℝ := Q.potentialQuadraticForm ∘ Space.val
+
+lemma potentialFunction_eq : Q.potentialFunction = Q.potentialQuadraticForm ∘ Space.val := rfl
+
+/-- The potential function for the harmonic oscillator is a.e. strongly measurable. -/
+informal_lemma potentialFunction_aestronglyMeasurable where
+  deps := [``HarmonicOscillator]
+  tag := "QM-HO-potAESM"
+
+end
+
+/-!
+## D. Isotropic oscillators
+-/
+
+/-- A Harmonic oscillator is isotropic if all natural frequencies are equal. -/
+def IsIsotropic : Prop := ∀ i j, Q.ω i = Q.ω j
+
+lemma isIsotropic_def : Q.IsIsotropic ↔ ∀ i j, Q.ω i = Q.ω j := Iff.rfl
+
+lemma isIsotropic_of_one (Q : HarmonicOscillator 1) : Q.IsIsotropic := by simp [isIsotropic_def]
+
+/-!
+## E. Hilbert space
+-/
+
+/-- The Hilbert space for the quantum harmonic oscillator. -/
+@[nolint unusedArguments]
+abbrev HS (_ : HarmonicOscillator d) : Type _ := SpaceDHilbertSpace d
+
+/-!
+## F. Operators
+-/
+
+/-!
+### F.1. Kinetic energy
+-/
+
+/-- The kinetic energy operator, `p²/2m`. -/
+def kineticOperator : Q.HS →ₗ.[ℂ] Q.HS := (2 * Q.m)⁻¹ • momentumSqOperator
+
+/-!
+### F.2. Potential energy
+-/
+
+section
+
+open MeasureTheory Complex
+
+/-- The potential operator which maps `ψ` to `Q.potentialFunction • ψ`. -/
+def potentialOperator : Q.HS →ₗ.[ℂ] Q.HS := 𝓜 volume (ofReal ∘ Q.potentialFunction)
+
+/-- The potential operators for the harmonic oscillator is self-adjoint. -/
+informal_lemma potentialOperator_isSelfAdjoint where
+  deps := [``HarmonicOscillator.potentialFunction_aestronglyMeasurable]
+  tag := "QM-HO-potSA"
+
+end
+
+/-!
+### F.3. Hamiltonian
+-/
+
+/-- The Hamiltonian for the harmonic oscillator. -/
+def hamiltonian : Q.HS →ₗ.[ℂ] Q.HS := Q.kineticOperator + Q.potentialOperator
+
+lemma hamiltonain_eq : Q.hamiltonian = Q.kineticOperator + Q.potentialOperator := rfl
+
+/-- The Hamiltonian for the harmonic oscillator is essentially self-adjoint. -/
+informal_lemma hamiltonian_essentially_self_adjoint where
+  deps := [``HarmonicOscillator.hamiltonian]
+  tag := "QM-HO-hamESA"
+
+/-!
+## G. As a quantum system
+-/
+
+/-- The `d`-dimensional harmonic oscillator as a quantum system
+  (self-adjoint Hamiltonian acting on a Hilbert space). -/
+informal_definition toQuantumSystem where
+  deps := [``HarmonicOscillator.hamiltonian_essentially_self_adjoint]
+  tag := "QM-HO-sys"
+
+end HarmonicOscillator
+end QuantumMechanics
+end

Physlib/QuantumMechanics/HarmonicOscillator/LadderOperators.lean

@@ -0,0 +1,54 @@
+/-
+Copyright (c) 2026 Gregory J. Loges. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Gregory J. Loges
+-/
+module
+
+public import Physlib.QuantumMechanics.HarmonicOscillator.Basic
+/-!
+
+# Ladder operators
+
+This is currently a stub, intended to house the ladder operators of the quantum harmonic oscillator.
+
+-/
+
+@[expose] public section
+
+/-!
+## A. Ladder operators
+-/
+
+TODO "Define the raising/lowering operators for the quantum harmonic oscillator."
+
+TODO "Prove that the raising/lowering operators are adjoints of one another (tag as simp?)."
+
+TODO "Prove the commutation relations for the ladder operators, `[a, a†] = δ`."
+
+/-!
+## B. Number operators
+-/
+
+TODO "Define the number operators for the quantum harmonic oscillator
+  in terms of the ladder operators."
+
+TODO "Prove that the number operators are symmetric/self-adjoint."
+
+TODO "Prove the commutation relations for the number operators, `[N i, N j] = 0`."
+
+TODO "Prove the commutation relations between number and ladder operators,
+  `[N, a] = -δ a` and `[N, a†] = δ a†`."
+
+/-!
+## C. Hamiltonian
+-/
+
+TODO "Define a Hamiltonian in terms of the number operators."
+
+TODO "Prove the commutation relations between the Hamiltonian and ladder/number operators."
+
+TODO "Relate the 'number operator' Hamiltonian to the 'K + T' Hamiltonian
+  (=/≤/≥ depending on their domains)."
+
+TODO "Prove that the two Hamiltonians define the same quantum system."

Physlib/QuantumMechanics/HarmonicOscillator/OneDimension/Basic.lean

@@ -8,7 +8,6 @@ module
 public import Physlib.QuantumMechanics.Operators.OneDimension.Parity
 public import Physlib.QuantumMechanics.Operators.OneDimension.Momentum
 public import Physlib.QuantumMechanics.Operators.OneDimension.Position
-public import Physlib.Meta.TODO.Basic
 /-!
 
 # 1d Harmonic Oscillator
@@ -77,20 +76,22 @@ open MeasureTheory
 variable (Q : HarmonicOscillator)
 
 @[simp]
-lemma m_mul_ω_div_two_ℏ_pos : 0 < Q.m * Q.ω / (2 * ℏ) := by
-  apply div_pos
-  · exact mul_pos Q.hm Q.hω
-  · exact mul_pos (by norm_num) ℏ_pos
+lemma m_pos : 0 < Q.m := Q.hm
 
 @[simp]
-lemma m_mul_ω_div_ℏ_pos : 0 < Q.m * Q.ω / ℏ := by
-  apply div_pos
-  · exact mul_pos Q.hm Q.hω
-  · exact ℏ_pos
+lemma m_nonneg : 0 ≤ Q.m := Q.hm.le
+
+@[simp]
+lemma m_ne_zero : Q.m ≠ 0 := Q.hm.ne'
+
+@[simp]
+lemma ω_pos : 0 < Q.ω := Q.hω
 
-lemma m_ne_zero : Q.m ≠ 0 := by
-  have h1 := Q.hm
-  linarith
+@[simp]
+lemma ω_nonneg : 0 ≤ Q.ω := Q.hω.le
+
+@[simp]
+lemma ω_ne_zero : Q.ω ≠ 0 := Q.hω.ne'
 
 /-!
 
@@ -137,7 +138,7 @@ lemma ξ_inverse : Q.ξ⁻¹ = √(Q.m * Q.ω / ℏ) := by
 
 lemma one_over_ξ_sq : (1/Q.ξ)^2 = Q.m * Q.ω / ℏ := by
   rw [one_over_ξ]
-  refine Real.sq_sqrt (le_of_lt Q.m_mul_ω_div_ℏ_pos)
+  exact Real.sq_sqrt (by simp [div_nonneg])
 
 @[inherit_doc momentumOperator]
 scoped[QuantumMechanics.OneDimension.HarmonicOscillator] notation "Pᵒᵖ" => momentumOperator

Physlib/QuantumMechanics/PlanckConstant.lean

@@ -24,12 +24,15 @@ namespace Constants
 def ℏ : Subtype fun x : ℝ => 0 < x := ⟨1.054571817e-34, by norm_num⟩
 
 /-- Planck's constant is positive. -/
+@[simp]
 lemma ℏ_pos : 0 < (ℏ : ℝ) := ℏ.2
 
 /-- Planck's constant is non-negative. -/
+@[simp]
 lemma ℏ_nonneg : 0 ≤ (ℏ : ℝ) := le_of_lt ℏ.2
 
 /-- Planck's constant is not equal to zero. -/
+@[simp]
 lemma ℏ_ne_zero : (ℏ : ℝ) ≠ 0 := ne_of_gt ℏ.2
 
 end Constants