Page 238 - Instant notes

Physical chemistry 224

Fig. 1. (a) The wavefunction of a
particle with a well-defined position.
(b) The superposition of a number of
wavefunctions of different
wavelengths. The superposition of an
infinite number of wavefunctions of
different wavelength is required to
produce the spike wavefunction of a
particle with a well-defined position.

The magnitude of the uncertainty which must coexist between position and momentum is
quantitatively given by:

where ∆p and ∆x are the uncertainties in momentum and position, respectively. The value
of ћ/2 is very small so the phenomenon is not directly observable at the scale of everyday
macroscopic objects. For example, the uncertainty in position of an object of mass 1.0 kg
−3
−1
travelling with a velocity known to be better than 1.0×10 m s precision is 5.3×10 −26
m. This uncertainty is many orders of magnitude smaller than the size of an atomic
nucleus. However, the same uncertainty in velocity for an electron of mass 9.11×10 −31 kg
implies an uncertainty in electron position far larger than the size of an atom.

Particle in a box

The application of Schrödinger’s equation to a particle undergoing one-dimensional
translational motion between confined limits demonstrates how imposition of boundary
conditions gives rise to one of the fundamental principles of quantum mechanics,
quantization. The two walls of the box are at positions x=0 and x=L along the x-axis.
Inside the box the particle (mass m) moves freely in the x-direction, and the potential
energy V=0. The potential energy rises abruptly to infinity at the walls.
The Schrödinger equation for the particle in the box is: