Some measurable physical quantity, if measured numerically, may lead to an uncertainty which is expr — Physical Chemistry — Kinetics Chemistry Question
Standard Deviation in One-Dimensional Quantum Mechanics
Some measurable physical quantity, if measured numerically, may lead to an uncertainty which is expressed by a standard deviation, σ. Such a standard deviation is defined as σ = √(⟨G²⟩ – ⟨G⟩²), where G is a measurable physical property; ⟨G⟩ is the average value of G; ⟨G²⟩ is the average value of G². The average values, ⟨G⟩ and ⟨G²⟩, can be obtained by integrating the corresponding physical quantity multiplied by its probability distribution over all the values of G. This definition may be applied to both classical and quantum mechanical worlds. Two examples related to the estimate of σ, one for the kinetic property of gaseous molecules and the other for the particle motion in one dimension, are given in the following.
Some useful integrals are given below:
∫₀∞ x²ⁿ exp(–ax²) dx = [1 × 3 × 5 … (2n – 1) / (2ⁿ⁺¹ aⁿ)] × √(π / a)
∫₀∞ x⁻²ⁿ⁺¹ exp(–ax²) dx = n! / (2 aⁿ⁺¹)
where n = 0, 1, 2, 3…
The distribution of speeds of gaseous molecules at a fixed temperature can be described by the following probability density, called the Maxwell–Boltzmann distribution:
F(v) = 4π (M / (2πRT))^(3/2) v² exp(–Mv² / (2RT))
where v is the speed of molecule, M is the mass of molecule, T is the temperature in Kelvin, and R is the gas constant. Calculate the average speed, ⟨v⟩, and the standard deviation, σ_v, of the distribution of speeds of the O2 molecules at 300 K. (M(O2) = 32 g mol⁻¹, R = 8.31 J K⁻¹ mol⁻¹)
Model Answer
Average speed ⟨v⟩:
⟨v⟩ = ∫₀∞ v F(v) dv
= ∫₀∞ 4π (M / (2πRT))^(3/2) v³ exp(–Mv² / (2RT)) dv
= √(8RT / (πM)) = √(8 × 8.31 × 300 / (3.14 × 0.032)) = 4.45 × 10² m s⁻¹
Standard deviation σ_v:
⟨v²⟩ = ∫₀∞ v² F(v) dv
= ∫₀∞ 4π (M / (2πRT))^(3/2) v⁴ exp(–Mv² / (2RT)) dv
= 3RT / M = 3 × 8.31 × 300 / 0.032 = 2.33 × 10⁵ m² s⁻²
σ_v = √(⟨v²⟩ – ⟨v⟩²) = √(2.33 × 10⁵ – (4.45 × 10²)²) = 1.87 × 10² m s⁻¹
Suppose a particle moving in the x direction has a normalized wave function,
ψ(x) = [(1 / (2πσ²))^(1/4)] exp(–x² / (4πσ²))
Calculate the average position, ⟨x⟩, and the standard deviation, σ_x, of the position distribution of the particle after a large number of measurements of x.
Model Answer
Average position ⟨x⟩:
⟨x⟩ = ∫‑∞∞ ψ*(x) x ψ(x) dx = 0
Standard deviation σ_x:
⟨x²⟩ = ∫‑∞∞ ψ*(x) x² ψ(x) dx = σ²
σ_x = √(⟨x²⟩ – ⟨x⟩²) = σ
In quantum mechanics, momentum for one dimension can be expressed by an operator, i.e.,
p_hat = -i (h / 2π) d/dx, where h is the Planck's constant. Calculate the average momentum, ⟨p⟩, and the standard deviation, σ_p, for the particle with the same wave function described in part 2.
Model Answer
Average momentum ⟨p⟩:
⟨p⟩ = ∫‑∞∞ ψ*(x) [-i (h / 2π) d/dx] ψ(x) dx = 0
Average of p², ⟨p²⟩:
⟨p²⟩ = ∫‑∞∞ ψ*(x) [- (h / 2πσ)²] d²/dx² ψ(x) dx = h² / (16π²σ²)
Standard deviation σ_p:
σ_p = √(⟨p²⟩ – ⟨p⟩²) = h / (4πσ)
Calculate the uncertainty product of position and momentum, σ_x σ_p, for the above quantum mechanical example.
Model Answer
σ_x σ_p = σ × (h / (4πσ)) = h / (4π)