CHEM 3541Physical Chemistry
CHEM 3541 Physical Chemistry
Week1.Textbook::Atkins'PhysicalChemistry,7thed.,pp.304-316:Atkins'Physical Chemistry,10th ed.,pp.290, 292-305
Week1 • Textbook: • Atkins’ Physical Chemistry, 7th ed., pp. 304-316 • Atkins’ Physical Chemistry, 10th ed., pp. 290, 292-305
Schrodinger Equation (S.E.)HY=EY:H:Hamiltonianoperator. Y: Wave function, Y(ri,r2, ..,rn): H = R + V: Kinetic energy operator and potentialenergy operator
Schrödinger Equation (S.E.) • 𝐻 Ψ = 𝐸Ψ • 𝐻: Hamiltonian operator • Ψ: Wave function, Ψ 𝑟 Ԧ 1 , 𝑟 Ԧ 2 , . , 𝑟 Ԧ 𝑛 • 𝐻 = 𝐾 + 𝑉: Kinetic energy operator and potential energy operator
1-D system with one particled2n2.H:hisPlanckconstantV(x), hdx2 2m2T· In many situations, V(x) operator is simply a functionn2d2+V(x)] (x) =E(x)S.E
1-D system with one particle • 𝐻 = − ℏ 2 2𝑚 ⋅ d 2 d𝑥 2 + 𝑉 𝑥 , ℏ = ℎ 2𝜋 , ℎ is Planck constant • In many situations, 𝑉 𝑥 operator is simply a function • S.E. − ℏ 2 2𝑚 ⋅ d 2 d𝑥 2 + 𝑉 𝑥 Ψ 𝑥 = 𝐸Ψ(𝑥)
Probability density.[Y(x)/2 is called probability density. The probability to find the particle between x and x + dx is1(x)/2dx·Normalization.Total probability to find theparticle between-co and +oo should be 1· If J|Y(x)/2dx = 1, then Y(x) is normalizedy'(x): Normalization of Y'(x): Y(x) +()2dx
Probability density • Ψ 𝑥 2 is called probability density • The probability to find the particle between 𝑥 and 𝑥 + d𝑥 is Ψ 𝑥 2d𝑥 • Normalization • Total probability to find the particle between −∞ and +∞ should be 1 ∞− If• +∞ Ψ 𝑥 2d𝑥 = 1, then Ψ 𝑥 is normalized • Normalization of Ψ′ 𝑥 : Ψ 𝑥 = Ψ′ 𝑥 ∞− +∞ Ψ′ 𝑥 2d𝑥