Page 234 - Instant notes

G4
THE WAVE NATURE OF MATTER

Key Notes
A wavefunction describes the region of space in which the
particle it represents is located. The square of the wavefunction is
proportional to the probability of finding the particle at that
location.
The Schrödinger equation, Hψ i =E i ψ i is the basic equation for
calculating the wavefunctions ψ i of a quantum mechanical system
described by a Hamiltonian operator H (usually the sum of the
kinetic and potential energy). Each wavefunction is associated
with a specific energy E i of the system.
The imposition of boundary conditions on the solutions of the
Schrödinger equation restricts a system to a set of physically
allowable wavefunctions (and energies) and is the origin of
quantization. An example boundary condition requires the value
of the wavefunction to be zero at the wall of an infinitely deep
potential well.
The magnitude of the uncertainty which must co-exist between
the position and momentum of a particle is∆p∆x≥ћ/2.

Several general features of quantum mechanical systems are
illustrated by the solution of the Schrödinger equation for a
particle (mass m) constrained to one-dimensional motion between
walls of infinite potential a distance L apart (the box): the energy
2 2
2
is quantized, E n =n h /8mL ; energy levels are more closely
spaced for a larger box; the probability of finding the particle at
different positions within the box is not uniform; the system
possesses intrinsic zero point energy.
The zero point energy is the minimum energy a system can
possess. It is frequently non-zero as a consequence of
Heisenberg’s uncertainty principle.
The energy of a particle undergoing rotational motion with
2
2 2
moment of inertia I=mr is quantized, E n =n ћ /2I Both positive
and negative values of n are allowed because the particle can
rotate with the same energy in either direction.
Two or more states of a system are degenerate if they possess the
same energy.