Quantum Mechanics of PointParticles
In developing quantum mechanics of pointlike particles one is faced
with a curious, almost paradoxical situation: One seeks a more general
theory which takes proper account of Planck’s quantum of action
h and which encompasses classical mechanics, in the limit h →0,
but for which initially one has no more than the formal framework
of canonical mechanics. This is to say, slightly exaggerating, that one
tries to guess a theory for the hydrogen atom and for scattering of
electrons by extrapolation from the laws of celestial mechanics. That
this adventure eventually is successful rests on both phenomenological
and on theoretical grounds.
On the phenomenological side we know that there are many experimental
findings which cannot be interpreted classically and which
in some cases strongly contradict the predictions of classical physics.
At the same time this phenomenology provides hints at fundamental
properties of radiation and of matter which are mostly irrelevant
in macroscopic physics: Besides its classically well-known wave nature
light also possesses particle properties; in turn massive particles
such as the electron have both mechanical and optical properties.
This discovery leads to one of the basic postulates of quantum theory,
de Broglie’s relation between the wave length of a monochromatic
wave and the momentum of a massive or massless particle in uniform
rectilinear motion.
Another basic phenomenological element in the quest for
a “greater”, more comprehensive theory is the recognition that measurements
of canonically conjugate variables are always correlated.
This is the content of Heisenberg’s uncertainty relation which, qualitatively
speaking, says that such observables can never be fixed
simultaneously and with arbitrary accuracy. More quantitatively, it
states in which way the uncertainties as determined by very many
identical experiments are correlated by Planck’s quantum of action. It
also gives a first hint at the fact that observables of quantum mechanics
must be described by noncommuting quantities.
A further, ingenious hypothesis starts from the wave properties
of matter and the statistical nature of quantum mechanical processes:
Max Born’s postulate of interpreting the wave function as
an amplitude (in general complex) whose absolute square represents
a probability in the sense of statistical mechanics.
Regarding the theoretical aspects one may ask why classical
Hamiltonian mechanics is the right stepping-stone for the discovery
of the farther reaching, more comprehensive quantum mechanics. To
this question I wish to offer two answers:
(i) Our admittedly somewhat mysterious experience is that Hamilton’s
variational principle, if suitably generalized, suffices as a formal
framework for every theory of fundamental physical interactions.
(ii) Hamiltonian systems yield a correct description of basic,
elementary processes because they contain the principle of energy
conservation as well as other conservation laws which follow from
symmetries of the theory.
Macroscopic systems, in turn, which are not Hamiltonian, often provide
no more than an effective description of a dynamics that one
wishes to understand in its essential features but not in every microscopic
detail. In this sense the equations of motion of the Kepler
problem are elementary, the equation describing a body falling freely
in the atmosphere along the vertical z is not because a frictional term
of the form −κ ˙ z describes dissipation of energy to the ambient air,
without making use of the dynamics of the air molecules. The first
of these examples is Hamiltonian, the second is not.
In the light of these remarks one should not be surprised in
developing quantum theory that not only the introduction of new,
unfamiliar notions will be required but also that new questions will
come up regarding the interpretation of measurements. The answers
to these questions may suspend the separation of the measuring
device from the object of investigation, and may lead to apparent
paradoxes whose solution sometimes will be subtle. We will turn
to these new aspects in many instances and we will clarify them
to a large extent. For the moment I ask the reader for his/her patience
and advise him or her not to be discouraged. If one sets out
to develop or to discover a new, encompassing theory which goes
beyond the familiar framework of classical, nonrelativistic physics,
one should be prepared for some qualitatively new properties and interpretations
of this theory. These features add greatly to both the
fascination and the intellectual challenge of quantum theory
|
with a curious, almost paradoxical situation: One seeks a more general
theory which takes proper account of Planck’s quantum of action
h and which encompasses classical mechanics, in the limit h →0,
but for which initially one has no more than the formal framework
of canonical mechanics. This is to say, slightly exaggerating, that one
tries to guess a theory for the hydrogen atom and for scattering of
electrons by extrapolation from the laws of celestial mechanics. That
this adventure eventually is successful rests on both phenomenological
and on theoretical grounds.
On the phenomenological side we know that there are many experimental
findings which cannot be interpreted classically and which
in some cases strongly contradict the predictions of classical physics.
At the same time this phenomenology provides hints at fundamental
properties of radiation and of matter which are mostly irrelevant
in macroscopic physics: Besides its classically well-known wave nature
light also possesses particle properties; in turn massive particles
such as the electron have both mechanical and optical properties.
This discovery leads to one of the basic postulates of quantum theory,
de Broglie’s relation between the wave length of a monochromatic
wave and the momentum of a massive or massless particle in uniform
rectilinear motion.
Another basic phenomenological element in the quest for
a “greater”, more comprehensive theory is the recognition that measurements
of canonically conjugate variables are always correlated.
This is the content of Heisenberg’s uncertainty relation which, qualitatively
speaking, says that such observables can never be fixed
simultaneously and with arbitrary accuracy. More quantitatively, it
states in which way the uncertainties as determined by very many
identical experiments are correlated by Planck’s quantum of action. It
also gives a first hint at the fact that observables of quantum mechanics
must be described by noncommuting quantities.
A further, ingenious hypothesis starts from the wave properties
of matter and the statistical nature of quantum mechanical processes:
Max Born’s postulate of interpreting the wave function as
an amplitude (in general complex) whose absolute square represents
a probability in the sense of statistical mechanics.
Regarding the theoretical aspects one may ask why classical
Hamiltonian mechanics is the right stepping-stone for the discovery
of the farther reaching, more comprehensive quantum mechanics. To
this question I wish to offer two answers:
(i) Our admittedly somewhat mysterious experience is that Hamilton’s
variational principle, if suitably generalized, suffices as a formal
framework for every theory of fundamental physical interactions.
(ii) Hamiltonian systems yield a correct description of basic,
elementary processes because they contain the principle of energy
conservation as well as other conservation laws which follow from
symmetries of the theory.
Macroscopic systems, in turn, which are not Hamiltonian, often provide
no more than an effective description of a dynamics that one
wishes to understand in its essential features but not in every microscopic
detail. In this sense the equations of motion of the Kepler
problem are elementary, the equation describing a body falling freely
in the atmosphere along the vertical z is not because a frictional term
of the form −κ ˙ z describes dissipation of energy to the ambient air,
without making use of the dynamics of the air molecules. The first
of these examples is Hamiltonian, the second is not.
In the light of these remarks one should not be surprised in
developing quantum theory that not only the introduction of new,
unfamiliar notions will be required but also that new questions will
come up regarding the interpretation of measurements. The answers
to these questions may suspend the separation of the measuring
device from the object of investigation, and may lead to apparent
paradoxes whose solution sometimes will be subtle. We will turn
to these new aspects in many instances and we will clarify them
to a large extent. For the moment I ask the reader for his/her patience
and advise him or her not to be discouraged. If one sets out
to develop or to discover a new, encompassing theory which goes
beyond the familiar framework of classical, nonrelativistic physics,
one should be prepared for some qualitatively new properties and interpretations
of this theory. These features add greatly to both the
fascination and the intellectual challenge of quantum theory
|
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