Schrödinger Equation- Example:
• The use of the Schrödinger equation can be
illustrated through a further example – a particle,
of mass m, moving in a restricted space - initially
in one dimension (along the x-axis).
• There is commonly a three step procedure
involved in tackling such a problem.
• Step 1: Write and examine the relevant/related
classical expression for energy, etc.
Schrödinger Equation- Example:
Particle in a Box (Quantization):
• Quantized energies are manifested when a
particle is confined to a particular region
(usually a small region!) of space. Movement
of a particle is restricted – one, two and three
dimensional cases are encountered.
• A useful and particularly simple example is the
particle in the box (an approximate but
important model for a range of chemically
interesting problems).
PIAB – One Dimensional Case:
PIAB – One Dimension:
No Particle !
Particle “Permitted”
No Particle !
Potential
Energy
“Well”
V(x) = 0
X=0
X=a
PIAB – Boundary Conditions:
• The allowed values of potential energy place
constraints (boundary conditions) on the PIAB.
The particle cannot have infinite energy and
thus cannot exist (be found) either outside the
box or where x = 0 or where x = a.
• We could summarize the boundary conditions
conditions as
• Ψ(x) = 0 for x = 0 and for x = a.
PIAB – Schrödinger Equation:
PIAB – Schrödinger Equation:
PIAB – Schrödinger Equation:
PIAB Wave Functions:
• The solution of the Schrödinger equation
given on the previous slide has the general
form
• Ψ(x) = c1 sin(kx) + c2 cos(kx)
• The first boundary condition, Ψ(x) = 0 when x
= 0, tells us that c2 = 0. (Why?). Our wave
function thus simplifies to
• Ψ(x) = c1 sin(kx)
PIAB Wave Functions:
PIAB Wave Functions:
Normalization of PIAB Wave Functions:
Allowed Energy Eigenvalues:
PIAB – Allowed Energies:
PIAB - Class Examples:
• Consideration of plots of ψn(x) vs x.
• Consideration of plots of ψn(x)ψn(x) vs x.
• Calculation of energies as a function of
quantum number.
• PIAB energy is never zero! (We will see this
again in, for example, a consideration of “zeropoint” vibrational energies.
Class Examples:
• Normalization and probability calculations
through the evaluation of appropriate
integrals.
• Crude prediction of electronic spectra of
organic molecules containing delocalized
electrons. We probably won’t get to Vitamin
A!