Foundations of Quantum Mechanics*
The whole of quantum mechanics can be expressed in terms of a small set of postulates. This
chapter introduces the postulates and
illustrates how they are used.
Postulates of Quantum Mechanics
Now we turn to an application of the preceding material, and move into the foundations of
quantum mechanics.
Quantum mechanics is based on a series of postulates which lead to a
very good description of the microphysical realm. Quantum mechanics is a very powerful
theory which has led to an accurate description of
the microphysical mechanisms. It is founded on a set of postulates from which the main
processes pertaining to its application domain are derived. A challenging issue in physics is
therefore to exhibit the underlying principles from which these postulates might emerge.
The set of statements we find in the literature as ‘postulates’ or ‘principles’ can be split into
three subsets: the main postulates which cannot be derived from more fundamental ones, the
secondary postulates which are often presented as ‘postulates’ but can actually be derived
from the main ones, and then statements often called ‘principles’ which are well known to be
as mere consequences of the postulates.
Main postulates
(1)
State or Wavefunction
.
Each physical system is described by a state function which
determines all can be known about the system. The coordinate realization of this state
function, the wavefunction
(uppercase psi) plays a central role in quantum mechanics.
Two wavefunctions represent the same state if they differ only by a phase factor. The
wavefunction has to be finite and single valued throughout position space, and furthermore, it
must also be a continuous and continuously differentiable function. We shall see that the
wavefunction of a system will be specified by a set of labels called quantum numbers, and
may then be written ca,b, . . . , where a, b, . . . are the quantum numbers. The values of these
quantum numbers specify the wavefunction and thus allow the values of various physical
observables to be calculated.
(2)
Equation for the wave function (Schrödinger equation)
.
The time evolution of the
wavefunction
of
a
nonrelativistic physical
system
is
given
by
the
timedependent
Schrödinger equation
where the Hamiltonian
is a linear Hermitian operator, whose
expression is constructed from the correspondence principle. The time independent form of
the equation can be written as:
We shall have a great deal to say about the Schrödinger equation and its solutions in the rest
of the text.
(3)
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 Spring '08
 staff
 Physical chemistry, pH

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