variables for distance. We also will need to determine the number of degrees
of freedom, whether the object is constrained to move only in one dimen-
sion or in three dimensions. For example, for the first problem, we will
consider an object trapped in a one-dimensional box with a potential of zero
inside of the box. As a result, only one variable, x, is used in Schrödinger’s
equation and the potential term is always zero. This is followed by invest-
igating how the problem is changed when the object is free to move in
two dimensions. For the subsequent applications, a different potential term
is used and, consequently, different differential equations are solved. For
the harmonic oscillator, the problem is again one-dimensional but the
potential term is that for a spring. For the hydrogen atom, we will need
to use three dimensions with an electrostatic potential.
Once the potential is substituted and the number of dimensions is chosen,
we will determine the wavefunction by solving Schrödinger’s’ equation.
In each case, a solution to the equation will be presented to have a better
understanding of what gives rise to the specific forms of the wavefunctions.
As we will see, each of these equations is a well-known second-order dif-
ferential equation that was studied and solved by mathematicians in the
1800s. Indeed, at that point in time, many of the differential equations with
solutions relevant to chemistry and physics were solved, although their
application was not known until many years later. For the initial applica-
tions, the solutions will be functions that have no meaning in themselves.
However, for the hydrogen atom, the wavefunctions will be shown to be
related to the well-known orbitals, identified as s, p, d, etc.
Once the wavefunction has been determined, then it is possible to pre-
dict all of the properties of the system, within the constraints of quantum
mechanics. For example, the wavefunction can be used to determine the
probability of finding the object at a certain position in space, the average
momentum of the object, or the average position. Thus in principle, all
of the functional properties of the object can be predicted, at least as an
average. For the hydrogen atom, this provides a means to predict what
happens in spectroscopy when light strikes an atom.
Interpretation of quantum mechanics
Quantum mechanics has been accepted because, in each case, the calculated
values have been found to be correct when compared with experimental
data, while classical mechanics has failed in many cases. This led to the
idea that classical physics was correct only for those situations involving
the physical world, whereas for small particles, namely electrons, protons,
and neutrons, quantum mechanics applied. Unlike classical mechanics,
the object in quantum mechanics is never at a specific location but rather
only has a certain probability of being at a certain location. Likewise, the
object has only average values rather than absolute values of position,