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.
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