- by Arpan Dey
Abstract
The main objective of the present paper is to investigate the de Broglie relation [1] and determine its consequences on the time-independent Schrödinger equation [2], and whether the results are valid in terms of accommodating relativistic corrections in the Schrödinger equation. The de Broglie relation has been modified by considering the relativistic equation for energy. The goal is not to reach a particular result. Rather, some known equations are manipulated to produce a general result. The Schrödinger equation describes the wave-function [3] of a system, which is a quantum-mechanical property. The time-independent form of the Schrödinger equation is derived from de Broglie’s relation [4]. In the end, the consequences of the results obtained in this paper, on the time-independent Schrödinger equation, are determined. This paper attempts to obtain some general results involving the Planck constant, momentum, velocity, wave-function, etc. of a particle, by taking into account quantum mechanics, and verify whether the approach is valid for accommodating relativistic modifications in Schrödinger’s equation.
Introduction to the de Broglie equation
The de Broglie relation, for the first time, introduced the idea of wave-particle duality in physics. In the early twentieth century, Max Planck proved energy cannot radiate randomly and continuously, but in discrete packets called quanta (singular: quantum). The energy of each quantum is given by the equation:
(1)
Where 
(Planck constant) [5]; 
is the frequency of the radiation. This formula can also be applied to light. Light is a stream of particles called photons, each with energy, 
. The particle nature of light could explain many phenomena, such as the photoelectric effect, which the wave theory could not explain. However, the wave theory of light was also successful in explaining certain phenomena like interference, diffraction etc. Thus, in modern physics, light is treated as having both a particle nature as well as a wave nature.
De Broglie equated the above equation with the rest mass energy:
(2)
In the case of light (photons), this gives 
, where 
is the linear momentum of the particle, which is the mass multiplied by the velocity. [In the derivation, the case of light was considered. Thus, for the case of light, 
, where 
is the speed of light in a vacuum (
).] 
represents the wavelength of light. The velocity of a wave 
is given by:
(3)
where 
represents the frequency of the wave; 
represents the wavelength. [For the case of light, 
.]
Thus, the de Broglie relation is:
(4)
This formula, in general, holds not just for photons but for electrons as well. However, electrons can never reach or exceed light speed in a vacuum. Due to refraction [6], light slows down in material media, and electrons can reach near-light velocities. In general, any particle must exhibit wave-particle duality. The wave nature of particles will be felt when the probe into the particle is over regions comparable to the wavelength associated with the particle. For daily-life macroscopic objects, the wavelength is negligibly small to give rise to any perceptible wavelike phenomena.
Figure 1: Matter Waves. The de Broglie hypothesis introduced the concept of wave-particle duality in physics. In the modern view, a “wave-packet” guides the motion of the point particle in space.
According to relativistic mechanics [7], the mass-energy-momentum relation is given by:
(5)
Where 
is energy, 
is mass (more specifically, the rest mass), 
is relativistic momentum, 
is the speed of light in vacuum. Here, 
, where the Lorentz factor 
is given by:
(6)
Here, 
is the velocity of the object. 
is the relativistic mass.
Figure 2: Relativistic Mass-Energy-Momentum Relation. [Here, 
represents the relativistic mass, and 
is the rest mass. Thus, 
.]
For objects at rest, 
, and the energy 
is given by 
. For objects that have no rest mass (photons), 
; and the energy is given by 
. For light waves, 
. Thus, 
. Equating this with 
, we get the de Broglie equation: 
.
Introduction to the time-independent Schrödinger equation
After de Broglie proposed the wave nature of matter, many physicists including Heisenberg and Schrödinger, explored the consequences. The idea quickly emerged that, because of its wave character, a particle’s trajectory and destination cannot be precisely predicted for each particle individually. However, each particle goes to a definite place. After compiling enough data, one gets a distribution related to the particle’s wavelength and diffraction pattern. There is a certain probability of finding the particle at a given location, and the overall pattern is called a probability distribution.
Werner Heisenberg discovered the uncertainty principle and developed matrix mechanics. Schrödinger developed an equivalent version of quantum mechanics – wave mechanics.
Schrödinger developed an equation that gave the solution to the wave-function of a particle. The wave-function (denoted by the Greek letter 
) is an abstract mathematical function. The square of the wave-function, i.e., the product of 
and its complex conjugate 
gives the probability of finding the particle at a given region of space.
The wave-function can be expressed as:
(7)
where 
and 
are real functions; 
is the imaginary number.
The complex conjugate of 
is: 
The square of the wave-function: 
. [Since, 
, 
.]
Thus, 
is always a positive and real quantity.
The wave-function, in general, is given by:
(8)
in one-dimension. Here, 
represents the amplitude of the wave; 
is Euler’s number, 2.71828; 
; 
(angular frequency).
In Schrödinger’s equation, some terms contain and its derivatives but no terms independent of the wave-function or that involve higher powers of the wave-function or its derivatives.
The time-independent Schrödinger equation is:
(9)
Where 
is the reduced Planck’s constant; 
represents the mass of the particle; 
is the Laplacian operator (which describes the wave-function, or any other function in three-dimensions); 
is the potential energy of the particle. The 
on the right-hand side of equation (9) represents the total energy. The equation states that the Hamiltonian operator operated on the wave-function gives energy as a result, 
. Here, 
is the Hamiltonian operator, 
. Here, 
is an eigenvalue.
Schrödinger’s equation has been universally recognized as one of the greatest achievements of 20th-century science, containing much of physics and in principle, all of chemistry. It is a mathematical tool of power equivalent to the Einstein field equations for gravity, if not more, for dealing with problems of the quantum mechanical model of the atom.
Using 
in the de Broglie equation
If the equation 
is used to find a relation in terms of 
, 
and 
, the result obtained is different.
Using 
,
[Here, 
is the relativistic mass, and 
is the rest mass. We know that 
.]
Multiplying both sides by 
,
Using 
, and 
,
Squaring both sides,
Using 
,
Using 
,
Using 
,
The result of 
is 
multiplied by a constant, 
. Thus, 
(11)
This expression may be simplified as follows:
(12)
Thus, the result is:
(13)
It should be noted that the modified equation is still dimensionally correct, as the original equation. This is because the dimension of both 
and 
is the same (both being metres per second, velocity). Thus, the constant 
, being 
, is dimensionless.
This expression collapses to 
only at 
. This makes sense, since, in the original derivation of the de Broglie relation, the assumption was that 
.
At 
, however, this expression goes undefined. That is acceptable since at 
, the momentum 
also becomes zero.
Modifications on the Schrödinger equation based on the above result
In the derivation of the time-independent Schrödinger equation, the wave-function is taken to be:
(14)
Taking the second-order derivative of the wave-function, 
,
(15)
[
indicates that the wave-function is one-dimensional (x-dimension). For convenience, we may also express this as just 
.]
Let the total energy be 
and potential energy be 
. The relativistic kinetic energy is given by:
(16)
where 
; 
is rest mass and 
is the speed of light in vacuum.
This is because, in relativity, the total energy is given by 
, which must be the sum of the kinetic and potential energies. The potential energy is the same as the rest-mass energy 
, (when the body is at rest, no kinetic energy). Thus, the kinetic energy would be 
.
(17)
Manipulating equation (17),
Where 
; 
is the rest mass, 
is the speed of light in vacuum, 
is the linear momentum and 
is the Lorentz factor 
.
[The kinetic energy is expressed as the product of some term with 
since 
is the kinetic energy in classical mechanics. (The Schrödinger equation does not take into account relativistic modifications, and is classical in nature.) This will make it more convenient to determine the consequences of using the modified de Broglie equation in the Schrödinger equation.]
Using 
,
(18)
This is obvious, since 
(both represent the kinetic energy).
Using 
,
(19)
Putting the value 
in equation (15),
Using 
,
Using 
,
Using 
,
Thus,
(20)
In three dimensions, this can be expressed as:
(21)
[For convenience, 
can be expressed as just 
.]
(22)
Here, our Hamiltonian is:
(23)
[The original Hamiltonian 
of the time-independent Schrödinger equation has, thus, been modified by multiplying the first term with 
(24). It should be noted that this extra term is dimensionless. (This is because the dimension of 
is just the square of metres per second, which when multiplied with the 
in the numerator, becomes (metres/second) raised to the fourth power. This cancels out the 
in the denominator. The 
in the numerator is dimensionless.) Thus, the modified equation remains dimensionally correct, like the original equation.]
In terms of 
, this can also be written as:
(25)
At 
, the first term of the left-hand side of the equation becomes zero. [This is because 
, at 
; and in this case, the term 
in the numerator becomes zero.]
At 
, the first term of the Left Hand Side goes undefined. [This is because 
is undefined at 
.]
Using relativistic kinetic energy in the Schrödinger equation
If the standard de Broglie relation was used in the derivation of the Schrödinger equation; but the kinetic energy was assumed to be 
, the result would have been:
(26)
In this case, the Hamiltonian is:
(27)
The original Hamiltonian is modified by multiplying the first term with: 
(28)
This expression is undefined both at 
and at 
.
Discussion
The results obtained by modifying the de Broglie equation are based on an assumption that would not work directly for radiations, like light. It must be noted that light is a form of electromagnetic radiation. Light does not have any rest mass. Putting 
, in the equation 
, the result obtained is 
. Equating this with 
directly produces the de Broglie equation 
. This equation relates a particle’s wave nature with its particle nature. The equation is mostly applied to photons and electrons; and in both of these cases, 
works just fine. If the energy was assumed to be 
instead of 
, the result would have been:
(29)
Thus, it is clear that equating 
with 
is, though tempting, actually not practical while dealing with any kind of radiation (electromagnetic radiation or light). However, the results obtained might be useful in certain circumstances. For instance, radiation can be approximated as a stream of particles with significant rest mass, as well as significant velocity. The results derived offer scope for further investigation.
The calculation on the time-independent Schrödinger equation is, again, of no direct practical importance. This approach is not valid when it comes to accommodating relativistic corrections in the Schrödinger equation. Though the Schrödinger equation does not take into account relativistic corrections, it produces acceptable results in most cases. The formal approach taken in uniting special relativity with quantum mechanics is different. The relation between mass, energy and momentum in Einstein’s Special Theory of Relativity can be used in quantum mechanics. The corresponding equations are given by the Klein-Gordon equation [8] and the Dirac equation [9], instead of the Schrödinger equation.
The Klein-Gordon equation can be derived from the formula: 
. In place of 
and 
, one can put the energy and momentum operators, respectively, to derive the Klein-Gordon equation. The energy operator is given by 
; the momentum operator is given by 
. Using these results in 
, we get 
. On simplification, this gives the Klein-Gordon equation:
(30)
However, the Klein-Gordon equation fails to account for the intrinsic property of spin. Thus, the Klein-Gordon equation was modified into the Dirac equation.
Conclusion
It can be concluded the de Broglie relation works well for the cases it is meant for. The results obtained in this paper are of limited practical application as of now. However, the results can be investigated further for certain cases, where that might produce more accurate results than the standard equations.
Thus, the approach taken in this paper is not correct, when it comes to applying relativistic corrections in the Schrödinger equation. For that, entirely different concepts are used, as in Dirac’s theory. However, finally, the results do not seem paradoxical; and the first result reproduces the standard equation at 
, as expected.
Acknowledgements
My school teachers, Pooja Mazumdar and Anupa Bhattacharya, played a vital role in reviewing the equations. I am also blessed to have this paper reviewed by Dr. Saumen Datta, a high-energy physicist at the Tata Institute of Fundamental Research. His advice and insights have been invaluable.
References
[1] The Editors of Encyclopaedia Britannica. “De Broglie wave”. 1998. https://www.britannica.com/science/de-Broglie-wave
[2] Marianne Freiberger. “Schrödinger’s equation – what is it?”. 2012. https://plus.maths.org/content/schrodinger-1
[3] Lisa Zyga. “Does the quantum wave function represent reality?”. 2012. https://phys.org/news/2012-04-quantum-function-reality.html
[4] Arpan Dey. “Quantum Mechanics: Derivation of Schrödinger\’s Equation by Arpan Dey”. 2020. https://youtu.be/m-KwIyaRVYw
[5] Patrick J. Kiger. “What Is Planck’s Constant, and Why Does the Universe Depend on It”. 2019. https://science.howstuffworks.com/dictionary/physics-terms/plancks-constant.htm
[6] The Editors of Encyclopaedia Britannica. “Refraction”. 1998. https://www.britannica.com/science/refraction
[7] Gary William Gibbons. “Relativistic mechanics”. 1999. https://www.britannica.com/science/relativistic- mechanics#ref611479
[8] Robert G. Littlejohn. “Introduction to Relativistic Quantum Mechanics and the Klein-Gordon Equation†”. 2019. http://bohr.physics.berkeley.edu/classes/221/1112/notes/kleing.pdf
[9] Ethan Siegel . “This Is Why Quantum Field Theory Is More Fundamental Than Quantum Mechanics”. 2019. https://www.forbes.com/sites/startswithabang/2019/04/25/this-is-why-quantum-field-theory-is-more-fundamental-than-quantum-mechanics/#3b4700d02083
Figure References
[1] “PM [D01] Matter Waves”. https://image.slidesharecdn.com/pmd01matterwaves-151027082009-lva1-app6892/95/pm-d01-matter-waves-10-638.jpg?cb=1445934312
[2] “File:Relativistic Dynamics.png”. https://upload.wikimedia.org/wikipedia/en/thumb/8/8d/Relativistic_Dynamics.png/640px-Relativistic_Dynamics.png