Atkins’ Physical Chemistry Eighth Edition Chapter 8 Quantum Theory: Introduction and Principles Copyright © 2006 by Peter Atkins and Julio de Paula Peter Atkins Julio de Paula

Born interpretation of the wavefunction The value of |Ψ| 2 (or Ψ * Ψ if complex) |Ψ| 2 ∝ probability of finding particle at that point For a 1-D system: For a 1-D system: If the wavefunction of a particle has the value Ψ at some point x, the probability of finding the particle between x and x + dx is proportional to |Ψ| 2.

Fig 8.19 Probability of finding a particle in some region of a 1-D system, Ψ(x)

Fig 8.20 Born Interpretation: Probability of finding a particle in some volume of a 3-D system, Ψ(r) Probability density = |Ψ| 2 Probability = |Ψ| 2 d τ d τ = dx dy dz

Fig 8.21 Sign of wavefunction has no direct physical significance |Ψ| 2 (or Ψ * Ψ if complex) > 0 However, the positive and negative regions of Ψ 1 can constructively/destructively interfere with the regions of Ψ 2.

Based on the Born interpretation, an acceptable wavefunction must be: 1) Continuous 2) Single-valued 3) Finite To ensure that the particle is in the system, the wavefunction must be normalized: Fig 8.24 |Ψ| 2 ∝ probability |NΨ| 2 = probability

Normalization |Ψ| 2 ∝ probability |NΨ| 2 = probability Time-independent Schrodinger equation for particle of mass m moving in one dimension, x: Sum of all probabilities must equal 1 or: So:

Normalized ψ in three dimensions: Or: For systems of spherical symmetry (atoms) it is best to use spherical polar coordinates:

Fig 8.22 Spherical polar coordinates for systems of spherical symmetry Now: Ψ(r, θ, φ) x → r sin θ cos φ y → r sin θ sin φ z → r cos θ Volume element becomes: d τ = r 2 sin θ dr dθ dφ

Fig 8.23 Spherical polar coordinates for systems of spherical symmetry r = 0 - ∞ θ = 0 – π φ = π

Consider a free particle of mass, m, moving in 1-D. Assume V = 0 From Schrodinger equation solutions are: Assume B = 0, then probability Particle may be found anywhere!

Fig 8.25 Square of the modulus of a wavefunction for a free particle of mass, m. Assume B = 0, then probability:

Assume A = B, then probability using: Now position is quantized!

Fig 8.25 Square of the modulus of a wavefunction for a free particle of mass, m. Assume A = B, then probability:

Operators, Eigenfunctions, and Eigenvalues Systematic method to extract info from wavefunction Operator for an observable is applied to wavefunction to obtain the value of the observable (Operator)(function) = (constant)(same function) (Operator)(Eigenfunction) = (Eigenvalue)(Eigenfunction) e.g., where

Operators, Eigenfunctions, and Eigenvalues e.g., is the position operator for one dimension is the momentum operator What is the linear momentum of a particle described by the wavefunction:

Operators, Eigenfunctions, and Eigenvalues is the position operator for one dimension is the momentum operator Suppose we want operator for potential energy, V = ½ kx 2 : Likewise the operator for kinetic energy, E K = p x 2 /2m:

Fig 8.26 Kinetic energy of a particle with a non-periodic wavefunction. 2nd derivative gives measure of curvature of function The larger the 2 nd derivative the greater the curvature. The greater the curvature the greater the E K.

Fig 8.27 Observed kinetic energy of a particle is an average over the entire space covered by the wavefunction.