Quantum Mechanics Postulates

Quantum Chemistry

Postulates of Quantum Mechanics

• The state of a physical system is fully described by a wavefunction Ψ(r₁, r₂,..., t), dependent on each particle's coordinates and time.

• For every observable in classical mechanics, there exists a corresponding linear Hermitian operator in quantum mechanics.

  Observable	Operator
  Position	𝑋̂ = 𝑥 ×
  Momentum	𝑃̂ₓ = -𝑖ℏ ∂/∂𝑥
  Kinetic Energy	𝑇̂ = -ℏ²/2𝑚 ∇²
  Potential Energy	𝑉̂ = 𝑉(𝑥, 𝑦, 𝑧) ×
  Total Energy	𝐻̂ = 𝑇̂ + 𝑉̂

• In any measurement of the property linked to operator Â, observed values will be eigenvalues aᵢ, satisfying the eigenvalue equation: ÂΨᵢ = aᵢΨᵢ

• For a system with a normalized wavefunction Ψ, the average value of the observable corresponding to  is given by: ⟨A⟩ = ∫ Ψ* Â Ψ dτ

• Wavefunction evolution over time follows the time-dependent Schrödinger equation: 𝑖ℏ ∂Ψ/∂t = Ĥ Ψ

• Time-independent Hamiltonians yield wavefunctions of the form: Ψ(x, t) = Ψ(x) e^(-iEt/ℏ)

• Ψ(x) represents a solution to the time-independent Schrödinger equation: Ĥ Ψ(x) = E Ψ(x)

Operators

• An operator is a rule that transforms a function into another function, represented by  f(x) = g(x).
• A linear operator follows the principle of superposition: Â [c₁ f₁(x) + c₂ f₂(x)] = c₁ Â f₁(x) + c₂ Â f₂(x)
• A Hermitian operator satisfies: ∫ f* Â f dτ = ∫ (Â f)* f dτ
• Hermitian operators have real eigenvalues.
• Eigenfunctions of a Hermitian operator corresponding to different eigenvalues are orthogonal.

Chemistry : Acids and Base

Acids

• Substances that release H⁺ ions (hydrogen ion or proton) in aqueous solution.
• Example: $HCl_{(g)} \xrightarrow[]{H_2O} H^+{(aq)} + Cl^-{(aq)}$

Bases

• Substances that release OH⁻ ions (hydroxide or hydroxyl ion) in aqueous solution.
• Example: $NaOH_{(s)} \xrightarrow[]{H_2O} Na^+{(aq)} + OH^-{(aq)}$

Brønsted-Lowry Theory

Acids

• Chemical species that donates protons (H⁺).

Bases

• Chemical species that accepts protons (H⁺).

Conjugated Acid-Base Pair

• A set of two chemical species that differ by only one proton (H⁺).
• Example: $HCl_{(aq)} + H_2O_{(l)} \rightleftharpoons H_3O^+{(aq)} + Cl^-{(aq)}$, where HCl/Cl⁻ and H₃O⁺/H₂O are conjugate acid-base pairs.

Ampholyte

• Chemical species that can behave as either an acid or a base, depending on the species it reacts with.
• Example: H₂O, which can act as an acid or a base.

Relative Strength of Acids and Bases

• In a conjugate acid-base pair, the stronger the acid, the weaker its conjugate base, and vice versa.
• Strong acids result in weak conjugate bases.
• Weak acids result in strong conjugate bases.

Water Autoionization

• Water behaves as a very weak electrolyte because it undergoes autoionization.
• $H_2O_{(l)} + H_2O_{(l)} \rightleftharpoons H_3O^+{(aq)} + OH^-{(aq)}$, with $K_w = [H_3O^+] [OH^-] = 1.0 \times 10^{-14}$ at 25°C.
• In pure water: $[H_3O^+] = [OH^-] = 1.0 \times 10^{-7} mol dm^{-3}$