The task is, not so much to see what no one has yet seen,
but to think what nobody has yet thought, about that which everybody sees. Arthur
Schopenhauer
(1788-1860)
The celebrated Schrödinger equation
is merely what the most ordinary wave equation
becomes when the celerity of a wave
(i.e., the product u = ln of its wavelength
by its frequency) is somehow equated to the ratio (E/p) of the energy to the
momentum of an "associated" nonrelativistic particle.
This surprising relation is essentially what the (relativistic)
de Broglie principlepostulates.
In an introductory course, it might be more pedagological and more economical to invoke the
de Broglie principle in order to
deriveSchrödinger's equation...
However, it's enlightening to present how Erwin Schrödinger himself introduced
the subject:
Following Hamilton, he showed how the relation u = E/p
can be obtained, by equating the classical principles previously stated
by Fermat for waves (least time)
and Maupertuis for particles (least "action").
This is an idea which made the revolutionary concepts of wave mechanics acceptable
to physicists of a bygone era, including Erwin Schrödinger himself.
Also, the more recent "sum over histories" formulation of quantum
mechanics by Richard Feynman is arguably
based on the same variational principles.
(2002-11-02) Hamilton's Analogy: Paths to the Schrödinger Equation
Equating the principles of Fermat and Maupertuis yields the celerity u.
Schrödinger took seriously an analogy
attributed to William Rowan Hamilton (1805-1865)
which bridges the gap between well-known features of two aspects of physical reality,
classical mechanics and wave theory. Hamilton's analogy states that, whenever waves conspire to
create the illusion of traveling along a definite path (like "light rays" in geometrical optics),
they are analogous to a classical particle:
The Fermat principle for waves may then be equated with
the Maupertuis principle for particles.
Equating also the velocity of a particle with
the group speed of a wave, Schrödinger drew the
mathematical consequences of combining it all with Planck's quantum hypothesis
(E = hn).
These ideas were presented (March 5, 1928)
at the Royal Institution of London, to start a course of
"Four Lectures on Wave Mechanics"
which Schrödinger dedicated to his late teacher,
Fritz Hasenöhrl.
Maupertuis' Principle of Least "Action" (1744, 1750)
Adding up the masses of all bodies multiplied by their respective speeds
and the distances they travel yields the quantity called action,
which is always the least possible in any natural motion. Pierre-Louis Moreau de Maupertuis.
"Sur les lois du mouvement " (1746).
When a point of mass m moves at a speed v in a force field described
by a potential energy V (which depends on the position):
The kinetic energy is T = ½ mv2
The total energy E = T+V remains constant.
The actual trajectory from a point A to a point B will minimize
the quantity that Maupertuis
(1698-1759) dubbed action, namely the integral
ò 2T dt.
(Maupertuis' Principle is thus also called the Least-Action Principle.)
Now, introducing the curvilinear abscissa (s) along the trajectory, we have:
2T = mv2 = m (ds/dt)2
= 2(E-V)
Multiply the last two quantities by m and take the square roots of those:
m (ds/dt) = Ö
2m (E-V)
Multiply that by (ds/dt) and use the previous equation. This gives:
2T = (ds/dt) Ö
2m (E-V)
so the action
is:
ò Ö
2m (E-V)
ds
The time variable (t) has thus disappeared from the integral to be minimized,
which is now a purely static function of the spatial path from A to B.
Fermat's Principle: Least Time (c. 1655)
When some quantity j propagates in
3 dimensions at some celerity u
(also called phase speed), it verifies the well-known wave equation:
1
¶ 2 j
=
¶ 2 j
+
¶ 2 j
+
¶ 2 j
u 2
¶ t 2
¶ x 2
¶ y 2
¶ z 2
=
Dj
[D is the Laplacian operator]
The speed u may depend on the properties of the medium in which the "thing" propagates,
and it may thus vary from place to place.
When light goes through some nonhomogeneous medium with a varying
refractive index (n>1), it propagates at a speed
u = c/n and will travel along
a path (a "ray", in the approximation of geometrical optics) which is always such
that the time (òdt)
it takes to go from point A to point B is minimal
[among "nearby" paths].
This is Fermat's Principle, first stated by
Pierre
de Fermat (1601-1665) for light in the context of geometrical optics,
where it implies both the
law of reflection
and Snell's law
for refraction.
This principle applies quite generally to any type of wave,
in those circumstances where some path of propagation can be defined.
If we introduce a curvilinear abscissa s for a wave that follows some path in
the same way light propagate along rays [in a smooth enough medium],
we have u = ds/dt. This allows us to express the time it takes to go
from A to B as an integral of ds/u.
The conclusion is that a wave will [roughly speaking]
take a "path" from A to B along which the following integral is minimal:
ò 1/u ds
Hamilton's Analogy :
The above shows that, when a wave appears to propagate along a path,
this path satisfies a condition of the same mathematical form as that
obeyed by the trajectory of a particle.
In both cases, a static integral along the path has to be minimized.
If the same type of "mechanics" is relevant,
it seems the quantities to integrate should be proportional.
The coefficient of proportionality cannot depend on the position, but it may very well
depend on the total energy E (which is constant
in the whole discussion).
In other words, the proportionality between the integrand of
the principle of Maupertuis and its Fermat counterpart (1/u)
implies that the following quantity is a function of the energy E alone:
f (E) = u Ö
2m (E-V)
Combined with Planck's formula, the next assumption implies
f (E) = E ...
Schrödinger's Argument :
Schrödinger assumed that the wave equivalent of the speed v of
a particle had to be the so-called group velocity,
given by the following expression:
1
d
(n / u)
v
dn
We enter the quantum realm by postulating Planck's formula :
E = hn.
This proportionality of energy and frequency turns the previous equation into:
1
d
(E / u)
v
dE
On the other hand, since ½ mv2 = E-V,
the following relation also holds:
1
d
Ö
2m (E-V)
v
dE
Recognizing the square root as the quantity we denoted f (E) / u
in the above context of Hamilton's analogy
[it's actually the momentum p, if you must know]
the equality of the right-hand sides
of the last two equations implies that the following quantity C
does not depend on E:
( f (E) - E ) / u =
C =
[ 1 - E / f (E) ]
Ö
2m (E-V)
This means f (E) = E / ( 1 - C
[ 2m (E-V) ] -1/2 which is, in
general, a function of E alone only if C vanishes
(as V depends on space coordinates).
Therefore f (E) = E, as advertised,
which can be expressed by the relation:
In 1928, Schrödinger quoted only as "worth mentioning" the fact that
the above relation boils down to u = E/p,
without identifying that as the nonrelativistic counterpart of
the formally identical relation for
the celerity u = ln
obtained from the 1923 expression
of a de Broglie wave's momentum
(p = h/l)
using E = hn.
Nowadays, it's simpler to merely invoke
de Broglies's Principle to
establish mathematically
the formal
stationary equation of Schrödinger,
given below.
English translations of the 9 papers and 4 lectures that Erwin Schrödinger published
about his own approach to Quantum Theory ("Wave Mechanics")
between 1926 and 1928 have been compiled in:
" Collected Papers on Wave Mechanics " by E. Schrödinger
(Chelsea Publishing Co., NY, 1982)
(2005-07-08) Partial Confinement in a Box by a Finite Potential
Solutions for a single dimension yield the three-dimensional solutions.
Consider a particle confined within a rectangular box by a finite potential, so that
(8 p2 m / h2 )
(V - E)
is -l-2 inside the box,
and m-2 outside of it.
For a single dimension, we'd be looking at a box
with boundaries at
x = ± L
and a bounded and continuous solution
y of the following type:
y(x)
=
[ A cos (L/l) -
B sin (L/l) ]
exp ( [L+x] / m )
for x < -L
=
A cos (x/l) +
B sin (x/l)
for |x| < L
=
[ A cos (L/l) +
B sin (L/l) ]
exp ( [L-x] / m )
for x > L
The continuity of the derivative of
y at
x = ± L
translates into the relations:
(A/l) sin (L/l) +
(B/l) cos (L/l)
=
(1/m)
[ A cos (L/l) -
B sin (L/l) ]
(-A/l) sin (L/l) +
(B/l) cos (L/l)
=
(1/m)
[ -A cos (L/l) -
B sin (L/l) ]
We may replace these by their sum and their difference, which boil down to:
B = 0 or
m cos (L/l)
=
-l sin (L/l)
A = 0 or
m sin (L/l)
=
l cos (L/l)
Since lm does not vanish,
either A or B does (not both).
A nonzero solution is thus either even (B=0) or odd (A=0)
with the matching condition derived from the above,
which is dubbed "quantization" in the following table:
Single-Dimensional Well of Width 2L and
Energy Depth V
1 / l2
+
1 / m2
=
(8p2 m / h2 ) V
Solution
Even
Odd
Symmetry
y(-x) = y(x)
y(-x) = -y(x)
Quantization
l / m = tan (L / l)
m / l = -tan (L / l)
y(x)
x < -L
cos (L/l)
exp ( [L+x] / m )
-sin (L/l)
exp ( [L+x] / m )
-L < x < L
cos ( x / l )
sin ( x / l )
L < x
cos (L/l)
exp ( [L-x] / m )
sin (L/l)
exp ( [L-x] / m )
ò
|y| 2 dx
m cos2 (L/l) +
L +
½ l sin (2L/l)
m sin2 (L/l) +
L -
½ l sin (2L/l)
m + L
Any solution is proportional to the function expressed in
either of the above columns.
The last line indicates that (because of their respective quantization conditions)
the norms of both tabulated functions have a unified expression.
This is just a coincidence, since we merely took a priori
the simplest choices among proportional
expressions... Normalized functions are thus
obtained by multiplying the above expressions by
e / Ö(m+L)
for some complex unit e
( |e| = 1 ).
The probability
P ( |x| > L ) to find the particle outside the box also
has a unified expression, valid for either parity of the wavefunction:
m3
( m2 + l2 )
(m + L)
Wavefunctions for a 3-dimensional box of dimensions
a, b, c are obtained as products of the above types
of functions of x, y or z, respectively.
(2005-07-10) Harmonic Oscillator
Quantization of energy in a parabolic well (Hooke's law).
