Atkins’ Physical ChemistryEighth Edition

Chapter 8Quantum Theory:

Introduction and Principles

Copyright © 2006 by Peter Atkins and Julio de Paula

Peter Atkins • Julio de Paula

Born interpretation of the wavefunctionBorn interpretation of the wavefunction

• The value of |The value of |ΨΨ||22 (or (or ΨΨ**ΨΨ if complex) if complex)

||ΨΨ||22 ∝∝ probability of finding particle at that point 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 If the wavefunction of a particle has the value ΨΨ at at

some point some point xx, the probability of finding the , the probability of finding the particleparticle

between between xx and and x x + + dxdx is proportional to | is proportional to |ΨΨ||22 . .

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 dτ

dτ = 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 Ψ1 can constructively/destructively interfere with the regions of Ψ2.

Based on the Born interpretation, an acceptablewavefunction must be:

1) Continuous

2) Single-valued

3) Finite

To ensure that the particleis in the system, thewavefunction must benormalized:

Fig 8.24

|Ψ|2 ∝ probability|NΨ|2 = probability

Normalization |Ψ|2 ∝ probability|NΨ|2 = probability

ΨE)x(Vdx

Ψd

m2 2

22

Time-independent Schrodinger equation for particleof mass m moving in one dimension, x:

Sum of all probabilities must equal 1 or:

1dxΨΨN *2So: 2/1

* dxΨΨ

1N

Normalized ψ in three dimensions:

1dxdydzΨΨ*

1dΨΨ* τ

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:

dτ = r2 sin θ dr dθ dφ

Fig 8.23 Spherical polar coordinates for systems

of spherical symmetry

r = 0 - ∞

θ = 0 – π

φ = 0 - 2π

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 foundanywhere!

Edxd

m2 2

22

)ikxexp(B)ikxexp(A m2

kE

22

22A

Fig 8.25 Square of the modulus of a wavefunctionfor a free particle of mass, m.

22A Assume B = 0, then probability:

• Assume A = B, then probability

using:

• Now position is quantized!

kxcosA4 222

kx sin ikx cose

kx sin ikx coseikx-

ikx

Fig 8.25 Square of the modulus of a wavefunctionfor a free particle of mass, m.

kxcosA4 222 Assume A = B, then probability:

Operators, Eigenfunctions, and EigenvaluesOperators, Eigenfunctions, and Eigenvalues

• Systematic method to extract info from wavefunction

• Operator for an observable is applied to wavefunctionto obtain the value of the observable

• (Operator)(function) = (constant)(same function)

• (Operator)(Eigenfunction) = (Eigenvalue)(Eigenfunction)

e.g.,

EH

where )x(Vdxd

m2H

2

22

Operators, Eigenfunctions, and EigenvaluesOperators, Eigenfunctions, and Eigenvalues

• e.g., is the position operator for one dimension

is the momentum operator

xx

dxd

ipx

What is the linear momentum of a particle described by thewavefunction: )ikxexp(AΨ

Ψk)ikxexp(kAdx

)ikxexp(dAik

i

dx

)ikxexp(dA

iΨ

dx

d

iΨp

Operators, Eigenfunctions, and EigenvaluesOperators, Eigenfunctions, and Eigenvalues

is the position operator for one dimension

is the momentum operator

xx

dxd

ipx

Suppose we want operator for potential energy, V = ½ kx2:

22

1 kxV

Likewise the operator for kinetic energy, EK = px2/2m:

2

22

dxd

m2dxd

idxd

im21

KE

Fig 8.26 Kinetic energy of a particle witha non-periodic wavefunction.

• 2nd derivative gives measure

of curvature of function

• The larger the 2nd derivative the

greater the curvature.

• The greater the curvature the

greater the EK.

Fig 8.27 Observed kinetic energy of a particle isan average over the entire spacecovered by the wavefunction.