14. Crafting the Quantum: Chaps 6-7, Conclusion.
I. Prinzipienfuchser and Virtuosen.
1. Einstein on Principles.
• 1925: Einstein letter to Paul Ehrenfest.
"There exist Prinzipienfuchser and Virtuosi.
We belong, all three of us [Ehrenfest, Einstein,
Bohr] to the first category and have (at least
certainly we two) little virtuosic ability."
• "Prinzipienfuchser" = those obsessed by principles
(in an artistic, as opposed to technical, way).
• "Virtuosi" = those possessed of great technical skill
(that, perhaps, lacks genuine insight).
Sommerfeld
Born
• 1919 article in Nature: Constructive theories vs. theories of principle.
"Constructive theories attempt to build up
a picture of the more complex phenomena
out of the materials of a relatively simple
scheme from which they start out."
• Example: kinetic theory of gases.
"[Theories of priniciple] ...employ the analytic, not the
synthetic method. The elements which form their basis
and starting point are not hypothetically constructed but
empirically discovered ones, general characteristics of
natural processes, principles that give rise to mathematically formulated criteria which separate processes or the
theoretical representations of them have to satisfy."
• Examples: thermodynamics, special relativity, general relativity.
• Eddington (1918) on general relativity as a theory of principle:
Sir Arthur Eddington
"The nearest parallel to it is found in the applications of the
second law of thermodynamics, in which remarkable
conclusions are deduced from a single principle without any
inquiry into the mechanism of the phenomena; similarly, if
the principle of equivalence is accepted, it is possible to
sride over the difficulties due to ignorance of the nature of
gravitation and arrive directly at physical results."
• Product of 19th century Cambridge pedagogical system: problem solving
based on mechanical modeling.
• But: Uniquely sympathetic to Einstein's principled approach (Warwick 2003).
• Later influence: Paul Dirac's (1930) Principles of Quantum Mechanics:
Paul Dirac
(Eddington student)
"The growth of the use of transformation theory as applied
to first relativity and later to the quantum theory, is the
essence of the new method in theoretical physics. Further
progress lies in the direction of making our equations
invariant under wider and still wider applications."
2. Ehrenfest's (1913) Adiabatic Hypothesis.
In an infinitely slow, reversible transformation,
allowed quantum motions are converted into
other allowed quantum motions.
Paul Ehrenfest
• What this means:
 If: You know the "allowed motions" (i.e., "adiabatic invariants") of a simple
quantum system.
 And: If you transform the simple system into a more complex system by an
"adiabatic" transformation (i.e., an infinitely slow, reversible transformation).
 Then: The "allowed motions" of the simple system remain the "allowed
motions" of the more complex system!
• A way of constructing quantum descriptions of complex systems from the
quantum descriptions of simple systems.
Example:
• Let the simple quantum system be a Hertzian
resonator oscillating in an electric field.
((
))
• Then the following quantity is an "adiabatic invariant" (remains constant
under an adiabatic transformation):
P
T
= ∫ dt ⋅T
0
ω
T is the time average of the kinetic energy over period P.
ω is the frequency of the motion.
• T is half the total energy, so Planck's (original) Quantum Hypothesis entails:
T
1
= nh
ω
2
• Now: Suppose we adiabatically decrease the electric field.
• Result: The original harmonic oscillation of the resonator
is adiabatically transformed into a uniform rotation.
Initial allowed oscillatory
motions (ellipses)
Final allowed
rotational motions
• Any adiabatic invariant associated with
the initial oscillatory motions remains
the same for the final rotational motions:
⎛T ⎞⎟
⎜⎜ ⎟
⎜⎝ ω ⎟⎟⎠
initial
⎛T ⎞⎟
= ⎜⎜ ⎟⎟
⎜⎝ ω ⎟⎠
final
1
= nh
2

The final rotatational frequency is ω = q/4π.

The final kinetic energy is T = pq/2.
• So: The final momentum is p = ±nh/4π.
Moral: You can calculate properties for a complex quantum system from
properties of a more simple quantum system if the two are connected by
an adiabatic transformation and possess adiabatic invariants.
3. Bohr on Principles.
• Recall: Problematic aspects of Bohr's 1913 model of the atom.
 Specification of stationary states doesn't
obey classical mechanics!
 Motion within a stationary state obeys
classical mechanics, but not classical
electrodynamics!
 Transitions between stationary states
don't obey classical mechanics or
classical electrodynamics!
Principle of Mechanical Transformability (1918):
Ordinary mechanical laws prevail in infinitely slow
(adiabatic) transformations of stationary states.
• Based on Ehrenfest's Adiabatic Hypothesis.
"[Ehrenfest's] principle allows us to overcome a fundamental difficulty.
In fact we have assumed that the direct transition between two such
[stationary] states cannot be described by ordinary mechanics, while on
the other hand we possess no means of defining an energy difference between two states if there exists no possibility for a continuous mechanical
connection between them. It is clear, however, that such a connection is
afforded by Ehrenfest's principle which allows us to transform mechanically the stationary states of a given system into those of another..."
• The Principle of Mechanical Transformability provided "...a justification for
the application of specific arguments from mechanics to what were, by
definition, non-mechanical systems." (Seth, pg. 196.)
Correspondence Principle (1920):
Individual quantum transitions between
stationary states correspond to components
of the classical frequency spectrum.
• "...a means of applying arguments from classical electrodynamics to systems
defined by their failure to accord with Maxwell's equations" (Seth pg. 196).
• What does it mean? What is the nature of the "correspondence"?
Classical Picture:
• The path x(t) of an electron undergoing periodic motion
with orbital frequency ω is given by a Fourier series:
∞
x(t) = ∑ c τ cos(τωt)
τ =1
= c1 cos(ωt) + c2 cos(2ωt) + c3 cos(3ωt) + 
each term is called a "harmonic"
• The radiation emitted by an accelerating electron is given by
all the frequences in the harmonics of its motion: ω, 2ω, 3ω, ...
Bohr's Picture:
• Radiation is not emitted by an electron in a
stationary state, but by an electron
transitioning between stationary states:
• For a transition between states with energies
En' and En'', the emitted radiation has a single
frequency given by νn',n'' = (En' − En'')/h.
n'
n''
• Three versions of the Correspondence Principle:
Correspondence Principle (Frequency Version):
νn',n'' = τω, for large values of n, where n' − n'' = τ.
For large n, the frequency of emitted radiation for a transition between the n' and
n'' states is given by the frequency of the τth harmonic of the classical motion.
Correspondence Principle (Intensity Version):
Pn',n'' ∝ |cτ|2, for large values of n, where n' − n'' = τ, and Pn',n''
is the probability of a transition between the n' and n'' states.
For large n, the probability of a transition between the n' and n'' states is
proportional to the intensity |cτ|2 of the τth harmonic of the classical motion.
Correspondence Principle (Selection Rule Version):
A transition from n' to n'' is allowed if and only if there exists a
τth harmonic between the n' and n'' states, where n' − n'' = τ.
"Although the process of radiation cannot be described on the
basis of the ordinary theory of electrodynamics, according to
which the nature of the radiation emitted by an atom is directly
related to the harmonic components occurring in the motion of
the system, there is found, nevertheless, to exist a far-reaching
correspondence between the various types of possible transitions between the stationary states on the one hand, and the
various harmonic components of the motion on the other hand."
"Bohr has discovered in his principle of correspondence a
magic wand (which he himself calls a formal principle),
which allows us immediately to make use of the results of
the classical wave theory in the quantum theory." (1919)
"The magic of the correspondence principle has proved itself
generally through the selection rules of the quantum numbers, in the series and band spectra… Nonetheless I cannot
view it as ultimately satisfying on account of its mixing of
quantum-theoretical and classical viewpoints.." (1924)
• "...a very peculiar idea of a principle" (Seth, pg. 198).
"variable and groping...[but in a good non-rigid way]" (1921)
Ehrenfest
"I always liked [it] just because it gave that kind of
lack of rigidity, that flexibility in the picture, which
could lead to real mathematical schemes." (1963)
Elder Heisenberg
Three different views on principles (Seth, pg. 200):
• Planck: Principles have the lasting validity of a holy commandment in a
world of absolute disunity.
• Einstein: Principles are fixed truths of Being in a world of absolute unity.
• Bohr: Principles are both approximate and necessarily changeable in a world
of dynamic change.