Classical mechanics vs. quantum mechanics

What is quantum mechanics and what does it do?
In very general terms, the basic problem that both classical Newtonian mechanics
and quantum mechanics seek to address can be stated very simply: if the state of a
dynamic system is known initially and something is done to it, how will the state of the
system change with time in response?
In this chapter, we will give a brief overview of, first, how Newtonian mechanics
goes about solving the problem for systems in the macroscopic world and, then, how
quantummechanics does it for systems on the atomic and subatomic scale.We will see
qualitatively what the differences and similarities of the two schemes are and what the
domain of applicability of each is.
Brief overview of classical mechanics
To answer the question posed above systematically, we must first give a more rigorous
formulation of the problem and introduce the special language and terminology (in
double quotation marks) that will be used in subsequent discussions. For the macro-
scopic world, common sense tells us that, to begin with, we should identify the
‘‘system’’ that we are dealing with in terms of a set of ‘‘static properties’’ that do not
change with time in the context of the problem. For example, the mass of an object
might be a static property. The change in the ‘‘state’’ of the systemis characterized by a
set of ‘‘dynamic variables.’’ Knowing the initial state of the system means that we can
specify the ‘‘initial conditions of these dynamic variables.’’What is done to the system
is represented by the ‘‘actions’’ on the system. How the state of the system changes
under the prescribed actions is then described by how the dynamic variables change
with time. This means that there must be an ‘‘equation of motion’’ that governs the
time-dependence of the state of the system. Themathematical solution of the equation
of motion for the dynamic variables of the system will then tell us precisely the state of
the system at a later time t>0; that is to say, everything about what happens to the
system after something is done to it.
For definiteness, let us start with the simplest possible ‘‘system’’: a single particle, or
a point system, that is characterized by a single static property, its mass m.We assume
that itsmotion is limited to a one-dimensional linear space (1-D, coordinate axis x, for
example). According to Newtonian mechanics, the state of the particle at any time t is
1completely specified in terms of the numerical values of its position x(t) and velocity
vx(t), which is the rate of change of its position with respect to time, or vx(t)¼dx(t)/dt.
All the other dynamic properties, such as linearmomentumpx(t)¼mvx, kinetic energy
T ¼ðmv2
xÞ=2, potential energy V(x), total energy E¼(TþV), etc. of this system
depend only on x and vx. ‘‘The state of the system is known initially’’ means that the
numerical values of x(0) and vx(0) are given. The key concept ofNewtonianmechanics
is that the action on the particle can be specified in terms of a ‘‘force’’, Fx, acting on the
particle, and this force is proportional to the acceleration, ax ¼d2
x /dt
2
, where the
proportionality constant is the mass, m, of the particle, or
Fx ¼ max ¼ md2
x
dt2
: (1:1)
This means that once the force acting on a particle of known mass is specified, the
second derivative of its position with respect to time, or the acceleration, is known
from (1.1).With the acceleration known, one will know the numerical value of vx(t)at
all times by simple integration. By further integrating vx(t), one will then also know the
numerical value of x(t), and hence what happens to the particle for all times. Thus, if
the initial conditions on x and vx are given and the action, or the force, on the particle
is specified, one can always predict the state of the particle for all times, and the
initially posed problem is solved.
The crucial point is that, because the state of the particle is specified by x and its first
time-derivative vx to begin with, in order to know how x and vx change with time, one
only has to know the second derivative of x with respect to time, or specify the force.
This is a basic concept in calculus which was, in fact, invented by Newton to deal with
the problems in mechanics.
A more complicated dynamic system is composed of many constituent parts, and
its motion is not necessarily limited to any one-dimensional space. Nevertheless, no
matter how complicated the system and the actions on the system are, the dynamics of
the system can, in principle, be understood or predicted on the basis of these same
principles. In the macroscopic world, the validity of these principles can be tested
experimentally by direct measurements. Indeed, they have been verified in countless
cases. The principles ofNewtonianmechanics, therefore, describe the ‘‘laws ofNature’’
in the macroscopic world