Foundations of Chemical Kinetics Lecture 2: The single-particle - - PowerPoint PPT Presentation

Foundations of Chemical Kinetics Lecture 2: The single-particle time-independent Schr¨

• dinger

equation

Marc R. Roussel Department of Chemistry and Biochemistry

Quantum mechanics of single particles

◮ De Broglie’s theoretical arguments (1924) and their

experimental verification by Davisson and Germer (1927) showed that particles have wave properties.

◮ Debye commented after a presentation by Schr¨

• dinger on de

Broglie’s work at a seminar in Zurich in the Fall of 1925 that, if particles behaved like waves, they should have a wave equation.

◮ That wave equation was provided by Schr¨

• dinger a few

months later.

◮ Born (1926) proposed that the wavefunction was connected to

probability.

The wavefunction

◮ For a single particle, the wavefunction, ψ, is a function of the

spatial coordinates, i.e. ψ = ψ(x, y, z).

◮ In general, the wavefunction may be complex-valued, i.e. it

has real and imaginary parts. However, in many simple cases, we can get away with real-valued wavefunctions.

◮ The square of the wavefunction is a probability density, i.e. ◮ |ψ(x, y, z)|2 is the probability of locating a particle near the

coordinates (x, y, z) per unit volume; or

◮ |ψ(x, y, z)|2 dx dy dz is the probability that the particle will be

found in a volume of dimensions dx × dy × dz centered on the coordinates (x, y, z); or

◮

R |ψ(x, y, z)|2 dx dy dz is the probability that the particle

will be found in the region R.

The wavefunction and particle positions

◮ The wavefunction contains all the information that can be

known about a particle.

◮ Particles do not have precise positions in quantum mechanics. ◮ A measurement of the position ◮ is limited in precision by the uncertainty principle; and ◮ “localizes” the particle through wavefunction collapse

(transformation of the wavefunction to a new wavefunction that is peaked near the position returned in the measurement); and

◮ the localization is random and governed by the probability

density implied by the original wavefunction.

The time-independent wave equation for a single particle in one dimension

a.k.a. Schr¨

• dinger’s equation

− 2 2m d2ψ dx2 + V (x)ψ = Eψ V (x) is the potential energy of the particle. E is the total energy of the particle. Constants: = h/2π, m is the particle mass. Alternative way of writing the equation: ˆ Hψ = Eψ where ˆ H is the Hamiltonian operator defined by ˆ H = − 2 2m d2 dx2 + V (x)

The particle in a box

◮ A simple model of a particle held inside a box with

impenetrable walls V (x) = for 0 < x < L ∞ for x ≤ 0

• r

x ≥ L

◮ Inside the box, we get the Schr¨

• dinger equation

− 2 2m d2ψ dx2 = Eψ

◮ Boundary conditions: The wavefunction has to go to zero at

the boundaries (x = 0 and x = L)

The particle in a box (continued)

◮ Solution:

ψn(x) = A sin nπx L

• , n = 1, 2, 3, . . .

with En = n2h2 8mL2

◮ The constant A is obtained by requiring that ψ2

n be a

probability density. Then L ψ2

n dx = 1 =

⇒ A =

• 2/L

Particle-in-a-box energy spectrum

En = n2h2 8mL2 Key observations:

◮ The energy levels are quantized: Only certain

energies are allowed.

◮ There is a minimum amount of energy which the

system must have known as the zero-point energy. Definitions: Ground state: lowest possible energy state 1st excited state: 2nd lowest energy state 2nd excited state: . . .

Particle-in-a-box wavefunction

• 1.5
• 1
• 0.5

0.5 1 1.5 0.2 0.4 0.6 0.8 1 L1/2 ψ2(x) x/L node

The harmonic oscillator

m m2 m1

In Schr¨

• dinger’s equation:

◮ Coordinate x is the displacement from the equilibrium

extension of the spring.

◮ V (x) ≈ 1

2kx2

◮ For the two masses connected by a spring, instead of m, use

the reduced mass µ = m1m2/(m1 + m2).

Harmonic oscillator (continued)

Schr¨

• dinger equation for the “diatomic” case:

− 2 2µ d2ψ dx2 + 1 2kx2ψ = Eψ

◮ Energies: Ev = ω0

• v + 1

2

• , v = 0, 1, 2, . . ., with ω0 =
• k/µ

◮ Again note quantization, zero-point energy ◮ Energy levels are equally spaced

Harmonic oscillator energy levels

1 2 3 4 5 6

• 3
• 2
• 1

1 2 3 E/− hω0 α1/2 x classical turning points

Harmonic oscillator ground-state wavefunction

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

• 4
• 2

2 4 α-1/4 ψ0 α1/2 x

New phenomenon: tunneling = ⇒ non-zero wavefunction in classically forbidden region

Relevance to kinetics

x (x) V