Download Wavefunctions and Bound Systems

Wavefunctions and Bound
Systems
Chapters Q9 (3
rd
ed)
What’s a Wavefunction?
• Looking a head Schroedinger’s
Equation:
Hy= E y
Wavefunction “psi” solves Schroedinger’s equation
and contains, in its components, all of the
information we need to determine values of
observables…
Wavefunctions are only part of the
story…
• The information in the wavefunction is “coded” in
its components. Actual values for observables
depend on “how you ask” the wavefunction
• Operators tell us what we want to know:
• Example: momentum
– Classical:
p  mv

– Quantum:
d
p  i
dx
This is not as strange as it seems –
you got used to thinking of
momentum as “em” times “vee” –
this is just another way to think
about it!
example (cont…)
Consider a free particle traveling in space…

An eigenvalue
equation
Eigenvalue for
py  py
momentum
d
i ( kx t )
i ( kx t )
i
( Ae
)  pAe
dx
Momentum operator
Wavefunction (eigenfunction)
What wavefunctions are (and are not!)
• Wavefunctions are
mathematical ideas that depict
probability distributions (Born
interpretation)
• Wavefunctions can be
described using the
mathematics of waves but are
not “real”
• Wavefunctions obey strict
mathematical rules:
– continuous, differentiable, finite
• Consider the wave function below:
This is the wavefunction
This is the probability
distribution
Applications…
• Particle in a Box
• Quantized Spring!
• Bohr Atom
Molecular Examples of Particle in a
Box…
From Carrots to Quantum Physics!
• Why are carrots
orange?
C40 H56
beta Carotene and Cyanine
• CY-5
– Bond length 0.14 nm
– # bonds 6
– Level n = 3  4
• b-Carotene ?
– Bond length 0.14 nm
– # bonds 18
– Level n = 9  10
How a quanton is “bound”
depends on the Potential
Energy Function V(x)
• V(x) is where the chemistry happens!
• The Shape of the potential sets the shape
of possible wavefunctions (energy
eigenfunctions)
• Energy Eigenfunctions are stationary
states (ie – standing waves!)
Three Potentials in Chp 7
“Box”
“Spring”
“Coulomb”
The quanton’s wavefunction must “fit” within these
potentials – that sets l and hence the energy and
momentum of the quanton
Suggested Study Problems from
Chp 7
Q7B.1, Q7B.4, Q7S.4