Foundations of Chemical Kinetics Lecture 4: Time dependence in - - PowerPoint PPT Presentation

Foundations of Chemical Kinetics Lecture 4: Time dependence in quantum mechanics

Marc R. Roussel Department of Chemistry and Biochemistry

Complex numbers

Complex numbers are of the form a + ib where i has the property i2 = −1. Euler’s formula: eiθ = cos θ + i sin θ Complex conjugate: If z = a + ib, the complex conjugate of z is z∗ = a − ib. Consequence: (eiθ)∗ = e−iθ

The time-dependent Schr¨

• dinger equation for a single

particle in one dimension

ˆ HΨ(x, t) = i∂ ∂t Ψ(x, t)

Relationship between the time-dependent and time-independent equations

ˆ HΨ(x, t) = i∂ ∂t Ψ(x, t)

◮ If ˆ

H is independent of time, this equation is separable, i.e. Ψ(x, t) = ψ(x)f (t)

◮ Substitute this form into the Schr¨

• dinger equation:

ˆ H [ψ(x)f (t)] = i∂ ∂t [ψ(x)f (t)] ∴ f (t)ˆ Hψ(x) = iψ(x) df (t) dt

Relationship between the time-dependent and time-independent equations (continued)

∴ f (t) ψ(x)f (t) ˆ Hψ(x) = i ψ(x) ψ(x)f (t) df (t) dt ∴ 1 ψ(x) ˆ Hψ(x) = i f (t) df (t) dt

◮ The left-hand side only depends on x, while the right-hand

side only depends on t. This can only be the case if each side is equal to a constant. Call this constant E: 1 ψ(x) ˆ Hψ(x) = E = i f (t) df (t) dt

Relationship between the time-dependent and time-independent equations (continued)

∴ ˆ Hψ(x) = Eψ(x) and df (t) dt = E if (t) = −iE f (t)

◮ The first equation is just the time-independent equation.

Relationship between the time-dependent and time-independent equations (continued)

◮ The second equation is easily solved by separation of variables:

df f = −iE dt ∴ df f = −iE

• dt

∴ ln f = −iE t

• r

f = exp

• −iE

t

Relationship between the time-dependent and time-independent equations (continued)

◮ Now putting the pieces back together, we get

Ψ(x, t) = ψ(x) exp

• −iE

t

• (1)

◮ For simplicity, define ω = E/, so that

Ψ(x, t) = ψ(x)e−iωt

◮ Note: Ψ∗Ψ = ψ∗ψ, i.e. the probability density is

time-independent and identical to that obtained from the time-independent Schr¨

• dinger equation.

◮ We call a separable solution like (1) a stationary solution.

Superposition solutions

◮ The time-dependent equation admits solutions that are

constructed by superposition of stationary solutions.

◮ Suppose that Ψ1(x, t) = ψ1(x)e−iω1t and

Ψ2(x, t) = ψ2(x)e−iω2t with ˆ Hψi = Eiψi, ωi = Ei/, and E1 = E2. Define Φ(x, t) = Ψ1(x, t) + Ψ2(x, t).

◮ We will now substitute this trial solution into Schr¨

• dinger’s

equation and verify that it satisfies the equation.

Superposition solutions (continued)

LHS RHS ˆ HΦ i ∂Φ

∂t

ˆ H

• ψ1e−iω1t + ψ2e−iω2t

i ∂

∂t

• ψ1e−iω1t + ψ2e−iω2t

e−iω1t ˆ Hψ1 + e−iω2t ˆ Hψ2 iψ1 ∂

∂t e−iω1t + iψ2 ∂ ∂t e−iω2t

E1ψ1e−iω1t + E2ψ2e−iω2t − i2ω1ψ1e−iω1t − i2ω2ψ2e−iω2t E1ψ1e−iω1t + E2ψ2e−iω2t QED