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 Erwin Schroedinger
Erwin Schrödinger

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© 2000-2016   Gérard P. Michon, Ph.D.

The Schrödinger Equation

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)

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Schrödinger's Equation in 1-D  by  Michael Fowler  |  Physics Applets.
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Electron in a Finite Square Well Potential  (calculator).
The Hydrogen Atom
HyperPhysics  by  Carl R. (Rod) Nave :   Hydrogen Atom   |   Spherical Well

Video:  Particles and Waves  (MU50)  by  David L. Goodstein   1 | 2 | 3 | 4
Atoms to Quarks  (MU51)  by  David L. Goodstein   1 | 2 | 3 | 4 | 5


Justifying Schrödinger's Equation

 Schroedinger's 126th Birthday

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 principle  postulates.  In an introductory course, it might be more pedagological and more economical to invoke the de Broglie principle in order to derive  Schrödinger'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)

 Pierre-Louis Moreau de 
 Maupertuis (1698-1759)

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), its 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 turns out to be such as to 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 their square roots to obtain an expression for  m (ds/dt) ,  which you may plug back into the whole thing to get an interesting value for 2T:

2T  =  (ds/dt) Ö 2m (E-V)so the action is:ò Ö2m (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.  Pierre de Fermat 

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:

 Pierre-Simon Laplace 
   2 j     =     2 j   +   2 j   +   2 j  
Vinculum Vinculum Vinculum Vinculum Vinculum
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:

 (n / u)
vinculum vinculum

We enter the quantum realm by postulating  Planck's formula :  E = hn.  This proportionality of energy and frequency turns the previous equation into:

 (E / u)
vinculum vinculum

On the other hand, since  ½ mv2 = E-V,  the following relation also holds:

  Ö  2m (E-V)
vinculum vinculum

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:

u   =   E  /  Ö  2m (E-V)

Mathematically, this equation and Planck's relation  (E = hn)  turn the general wave equation into the stationary  equation of Schrödinger, discussed below.

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)


Schrödinger's  Stationary  Equation

Dy   +   (8 p2 m / h2 ) (E - V)  y     =     0

(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.

 Finite one-dimensional well

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
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:

( 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.

 Come back later, we're
 still working on this one...

(2005-07-10)   Harmonic Oscillator
Quantization of energy in a parabolic well  (Hooke's law).

 Come back later, we're
 still working on this one...

Hermite Polynomials   |   Charles Hermite (1822-1901; X1842)

(2005-07-10)   Angular Momentum
The angular momentum of a rotator is quantized.

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 still working on this one...

(2005-07-10)   Coulomb Potential
Classification of the orbitals corresponding to a Coulomb potential.

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 still working on this one...

Legendre Polynomials   |   Laguerre Polynomials

(2015-11-22)   The Wallis Formula for p   (John Wallis, 1655).
A quantum derivation by  Tamar Friedmann  and  C. R. Hagen  (2015).

 Come back later, we're
 still working on this one...

Tamar Friedmann  &  C. R. Hagen,   AIP Journal of Mathematical Physics56, 112101 (2015).
New derivation of pi links quantum physics and pure math  (AIP, 2015-11-10).

(2016-01-16)   How tough is Schrödinger's equation, really?
Any homogeneous second-order linear differential equation reduces to it.

In one dimension, an second-order linear differential equation can be tranformed into a Schrödinger equation or a Ricatti equation, and vice-versa.

 Come back later, we're
 still working on this one...

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