Acta Polytechnica
doi:10.14311/AP.2014.54.0127
Acta Polytechnica 54(2):127–129, 2014 © Czech Technical University in Prague, 2014
available online at http://ojs.cvut.cz/ojs/index.php/ap
DIRAC AND HAMILTON
P. G. L. Leacha, b
a Department of Mathematics and Statistics, University of Cyprus, Lefkosia, Republic of Cyprus
b School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, Republic of
South Africa
correspondence: leachp@ukzn.ac.za
Abstract. Dirac devised his theory of Quantum Mechanics and recognised that his operators
resembled the canonical coordinates of Hamiltonian Mechanics. This gave the latter a new lease of life.
We look at what happens to Dirac’s Quantum Mechanics if one starts from Hamiltonian Mechanics.
Keywords: Dirac, quantum mechanics, Hamiltonian mechanics.
1. Introduction
When Dirac was developing his theory of Quantum
Mechanics, see [1], he observed that the operators
he had introduced reminded him of something from
Mechanics. He looked to his personal collection in his
study and could find nothing suitably advanced, such
as Whitaker’s Treatise [2] which would contain such
information. It was a Sunday afternoon and naturally
in those more civilised days the College Library would
not be open. Dirac had to wait until Monday morning
to find that in fact the very properties he had in his
operators, apart from the occasional i, were precisely
those of the Poisson Brackets of Hamiltonian Mechan-
ics, which had been established and developed some
120 years before by Poisson and incorporated into the
new mechanics some 30 years later.
It is a little difficult to draw a line in the history of
the development of Mechanics and to claim what is
the terminus a quo, but to keep this narrative compact
we start with the Mechanics of Newton [3], for it is
on his Laws of Motion that the subsequent evolution
of Mechanics has been based. One of the problems
in practice is the solution of equations and it soon
became apparent that the number of problems which
could be solved at any particular epoch was rather
limited. The theoreticians were always trying to think
of a new way to solve the unsolvable.
The Calculus of Variations provided one way to look
for solutions from a different direction. Developing
upon the work of Euler, Lagrange in his Mécanique
analytique of 1788 introduced his equations of motion
based upon the variation of a functional called the
Action Integral. As far as Classical Mechanics was
concerned, Lagrange’s Equations of Motion were of the
second order. In principle these equations reduced to
the corresponding Newtonian equations, but Lagrange
had introduced the idea of generalised coordinates and
generalised forces which always have the possibility
of giving equations which look simpler than the naive
Newtonian equations.
Then along came Hamilton, who decided to reduce
Mechanics to a system of first-order equations. In
a sense he was returning to the original formulation
of Newton II. As the Hamiltonian, as it came to be
called, was derived by introducing a momentum as
a derivative of the Lagrangian and then obtained
by using a Legendre transformation, it inherited the
generalised coordinates and generalised forces of La-
grangian Mechanics except that now there were gener-
alised coordinates and generalised momenta. One of
the attractions of Hamilton’s approach was the intro-
duction of a theory of transformations whereby one
could transform from one Hamiltonian to another by
means of transformations which obeyed certain rules.
Hamiltonian Mechanics was and is a marvellous
theoretical construct which does have some practical
uses. However, as a general tool in elementary me-
chanics it is not of much practical use and could well
have faded into near-oblivion had it not been for the
observation of Dirac. Now Hamiltonian Mechanics
provided a lodestone to the new Quantum Mechanics.
2. A Problem
We have remarked that the basis of these theoretical
constructs can be seen in the Equations of Motion of
Newton, particularly in the perturbation theory of or-
bital mechanics. The construction of an Hamiltonian
from a Lagrangian is a quite definite procedure. How-
ever, the construction of a Lagrangian which leads to
the proper Newtonian Equation of Motion, which in
itself begs the question of properness, is by no means
unique.
We take a very elementary example, namely the
simple harmonic oscillator in one space dimension.
The Newtonian Equation of Motion is
q̈ + q = 0 (1)
in which we have scaled the time to remove a distract-
ing Ω2.
Equation (1) has a plethora of Lagrangians. For a
modest sampling see [4]. Here we consider just three,
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P. G. L. Leach Acta Polytechnica
namely
L1 =
1
2
(
q̇2 − q
)
, (2)
L2 =
1
2 sin2 t (q̇ sin t − q cos t)
and (3)
L3 =
1
2 cos2 t (q̇ cos t + q sin t)
. (4)
These Lagrangians share a common property in that
they all possess five point Noether Symmetries, which
is the maximal number for a one-degree-of-freedom
system. In each case the algebra of the symmetries is
sl(2, R) ⊕s 2A1, which is a subalgebra of the sl(3, R)
algebra of (1).
The standard formalism leads to three Hamiltonians.
These are
H1 =
1
2
(
p2 + q2
)
(5)
H2 = pq cot t +
i
√
p
sin3 /2t
and (6)
H3 = −pq tan t +
i
√
2p
cos3/2 t
(7)
with the canonical momentum in each case being
L1 p = q̇
L2 p = −
cosec t
2 (q̇ sin t − q cos t)2
and
L3 p = −
sec t
2 (q sin t + q̇ cos t)2
.
The ‘quantisation procedure’ for H1 (5) is well
known and leads to the equation
2iut = −uqq − q2u (8)
about which any competent undergraduate can write
at length.
In principle we may follow the same procedure for
H2 and H3. However, there are two problems. The
first is the presence of
√
p, which doubtless makes the
process a little complicated. It is true that there exist
procedures for dealing with nonstandard forms, but
this is not the place to deal with such things as we
are considering a very elementary problem. The other
difficulty is that, even neglecting the fractional power,
the resulting Schrödinger equation would be linear in
the spatial derivative.
The question is what to do? There are various
possibilities:
(1.) A mistake was made in the calculation of the sec-
ond and third Hamiltonians. This is unlikely as the
algorithm is particularly simple and Mathematica
is much better at arithmetic than I am.
(2.) The process was initiated under false pretences.
One recalls that the simple harmonic oscillator
has three linearly independent quadratic integrals.
They are
I1 =
1
2
(
q̇2 + q2
)
,
I2 =
1
2
e2it
{(
q̇2 − q2
)
− 2iqq̇
}
and
I3 =
1
2
e−2it
{(
q̇2 − q2
)
+ 2iqq̇
}
.
The first integral, I1, corresponds to the Hamilto-
nian and equally leads to the Schrödinger Equation
given above.
I2 and I3 can also be used to construct Schrödinger
Equations, but the question is that of meaning [5].
One feature of the Schrödinger Equation for H1
is that it is a parabolic equation which has the
same specific algebra as L1. One could take the
Noether Symmetries of L2, respectively L3, and
construct the corresponding Schrödinger Equation
with the additional requirement that it be linear.
The meaning of the outcome could be of interest [6].
One could conclude that the similarity of the op-
erators introduced by Dirac and their identification
with those of Hamiltonian Mechanics is an accident
and one should read Dirac’s book carefully.
3. Conclusion
This simple example already shows that there is the
potential for ambiguity in the interpretation of the
properties of a classical Hamiltonians, in this case that
of the Simple Harmonic Oscillator. A critical aspect
is the interpretation of how one should progress from
Classical Mechanics to Quantum Mechanics. In the ap-
proach adopted by Dirac the Hamiltonian correspond-
ing to the ‘standard’ Lagrangian, ie a Lagrangian of
the form L = T −V in the case of simple systems was
used to construct an operator which fitted into the
expectation for the corresponding quantum mechani-
cal system. If one views the Newtonian equation of
motion, (1), as the fundamental source of the problem,
that equation has eight Lie point symmetries. With-
out going into anything fanciful one can construct a
large number of Lagrangians using these symmetries,
two at a time, with the vector field of the equation of
motion to determine Jacobi Last Multipliers and the
relationship,
∂2L
∂ẋ2
= M,
where M is a Last Multiplier, to calculate a whole
pile of Lagrangians for the single Newtonian Equation,
(1). One can then apply Noether’s Theorem to each
of these Lagrangians in turn to determine the Noether
Symmetries. These vary in number up to a maximum
of five. There are three such Lagrangians [7], the L1,
L2 and L3 listed above. The first, L1, is the one which
is usually used to construct the Hamiltonian for the
Simple Harmonic Oscillator and hence a Schrödinger
Equation. Why is it that this Lagrangian should be
chosen when there are two other Lagrangians equally
well endowed with Noether Symmetries?
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vol. 54 no. 2/2014 Dirac and Hamilton
One can reasonably claim that this choice leads to
physically acceptable results. However, this does not
really gel well with the idea of symmetry and oper-
ators. Consequently one must argue that the choice
of Lagrangian, hence Hamiltonian and corresponding
operators for Quantum Mechanics must be predicated
on other considerations. In Dirac’s book he writes of
the energy being the source of the operator to be used
for quantisation. It is an accident that in elementary
mechanics the energy is the Hamiltonian in the usual
meaning of the word. That the operators required for
Dirac’s Quantum Mechanics had essentially the same
properties as the canonically conjugate variables of
Classical Hamiltonian Mechanics is perhaps a cause
for jumping on a bandwagon without checking to see
if the horses had been harnessed.
Acknowledgements
This paper was prepared while PGLL was enjoying the
hospitality of Professor Christodoulos Sophocleous and
the Department of Mathematics and Statistics, University
of Cyprus. The continuing support of the University of
KwaZulu-Natal and the National Research Foundation of
South Africa is gratefully acknowledged. Any opinions
expressed in this talk should not be construed as being
those of the latter two institutions.
References
[1] P. Dirac. The Principles of Quantum Mechancis. First
edition. Cambridge, at the Clarendon Press, 1932.
[2] E. Whittaker. A Treatise on the Analytical Dynamics
of Particles and Rigid Bodies. Fourth Edition, First
American printing. Dover, New York, 1944.
[3] I. Newton. Principia. Fourth Edition, Second printing.
Editor: Cajori F, University of California Press,
Berkeley, Voll I & II (translation by Motte A), 1962.
[4] N. M. . L. PGL. Lagrangians galore. Journal of
Mathematical Physics 48(123510):1–16, 2007.
[5] N. M. . L. PGL. Classical integrals as quantum
mechanical differential operators: a comparison with the
symmetries of the schrödinger equation. In preparation.
[6] N. M. . L. PGL. Lagrangians of the free particle and
the simple harmonic oscillator of maximal noether point
symmetry and their corresponding evolution equations
of schrödinger type. In preparation.
[7] P. Winternitz. Subalgebras of Lie algebras. example of
sl(3, R). Centre de Recherches Mathématiques CRM
Proceedings and Lecture Notes 34:215–227, 2004.
129
Acta Polytechnica 54(2):127–129, 2014
1 Introduction
2 A Problem
3 Conclusion
Acknowledgements
References