After a short break due to a busy period in my offline life, here I am, back with a new post on quantum mechanics.
In the previous episodes, I have described how a wave (or actually light) could also possess the behavior of a particle (see here). In a second article, I swapped those concepts and I discussed the fact that a wave could also be attached to any particle (see here).
This is the concept of wave-particle duality. With each wave comes a particle and vice-versa: which each particle comes a wave. I have given more details about that in my last quantum mechanics post, here.
It is now time to move one step further and get our hands dirty: a wave obeys to a wave equation. But what is the equation governing the behavior of the matter waves?
What? Equations? Aouch…
However, please do not be afraid. I am in a partial half-good mood and I will try to make this post only half-dirty. Some equations will be there, as they cannot be avoided entirely (at least I do not know how to do so).
I will however try to have way more text than equations so that the reading of the post will be kind of easy going. Please let me know whether I succeeded. And do not hesitate to complain and ask for clarifications if needed. ^^
THE CONSTRUCTION OF THE SCHRODINGER EQUATION
Once physicists were convinced that matter waves were existing, the next natural step was to find an equation which those matter waves were solutions of.
It is Erwin Schrödinger (the guy of the picture below) who has built this equation, about 25 years after that the notion of a quantum of energy was introduced.
The way in which Schrödinger designed his equation was not a proof, but a heuristic approach.
He started from the information he had in the beginning of the 20th century and from these pieces of information, he imagined an equation that could work and describe the behavior of the matter waves.
By virtue of its success, this equation now carries Schrödinger’s name.
The amazing thing is that Schrödinger managed to construct an equation way more general than the basic elements which he started from.
The two ingredients used by Schrödinger were the following. First, he knew that the relations proposed by de Broglie have to be reproduced. This means that (beware, first equation ahead!)
p = ℏ k and that E = ℏ ω
In this equation, the vector p denotes the momentum of the particle and E its energy. I recall that the momentum is a quantity that mixes mass and motion and is defined as the product of the particle mass by its velocity. Moreover, the constant ℏ is the reduced Planck constant (ℏ= h/2π), k is the wave vector (or in other words the direction in which the wave propagates), and ω the wave pulsation (or its frequency, more or less).
The second ingredient used by Schrödinger was the fact that he knew that classical mechanics is working at the macroscopic level. Therefore, all the relations induced by classical mechanics are valid and must be recovered in some limit.
MERGING CLASSICAL MECHANICS WITH MATTER WAVES
The reasoning of Schrödinger started by considering a particle of mass m moving freely, not interacting with anything. Its energy is thus simply its kinetic energy, and here is the second equation of this post,
E = p2 / (2 m)
Classical mechanics at work! Nothing less, nothing more. The energy equals the norm of the momentum squared, over twice the particle mass.
But the squared momentum and the energy are also the two quantities given by the relations of de Broglie shown above.
Putting everything together, one derives the fact that the pulsation of the matter wave is proportional to the wave-vector squared,
ω = ℏ k2 / (2 m)
And this is an amazing relation.
Waves are usually introduced as light waves. In this case, the pulsation is proportional to the speed of light and the norm of the wave vector. Here, we have a totally different relation. The pulsation ω is proportional to the norm of the wave-vector squared k2…
TOWARDS THE FORM OF SCHRODINGER EQUATION
So what have we learned so far? Let us summarize.
We have a wave that propagates in a direction (cf. its wave-vector k) that is aligned with the particle momentum p (cf. the relations of de Broglie ), or in other words with its velocity (as the momentum is proportional to the velocity).
All this to say that the wave propagates parallel to the motion of the particle.
A wave is a function that depends on the position and on the time, and the simplest possible wave is a plane wave.
I could give the equation… But actually, I will not (enough equations here… for now).
What is important to know is that a plane wave has two important properties.
When a plane wave is derived once with respect to the time, one gets the pulsation of the wave. When a plane wave is derived twice with respect to the position, one gets the norm of the wave-vector squared.
But we have just explained in the previous section that in the case of matter waves, the pulsation is proportional to the norm of the wave-vector squared.
Yes! We have our wave equation: the first derivative of the wave with respect to time is roughy equal to the second derivative of the wave with respect to the position,
GENERALISATION - THE CORRESPONDENCE RULES
The equation above (I know, we start to have a lot of equations around here) can be derived immediately from the classical definition of the energy by using so-called correspondence rules. These rules consist of replacing physical quantities with operators (or derivatives).
We start from the classical definition of the energy already given at the beginning of this post,
E = p2 / (2 m)
We then add a wave 𝜑 on both sides, replace the energy by a time derivative and the momentum by a space derivative, and we get the last relation.
[image credits: homemade]
These correspondence rules allow for an easy generalization to the case where we have interactions, as shown on the left.
The quantity on the right-hand side of the equation H is now the Hamiltonian. It is a mathematical object that contains the kinetic energy and the potential energy (relevant when interactions are in order) and whose exact form is derived using the correspondence rules.
Of course, the reality is slightly more complex to what I have sketched here. But not that much actually :)
A SHORT DISCUSSION AND REFERENCES
With this post, I wanted to explain how the main equation of quantum mechanics was born. It has been originally found by Schrödinger by using classical physics (as a working limit of quantum mechanics) and de Broglie findings for matter waves.
It is a bit hard to me to introduce Schrödinger equation without relying too much on equations. I tried to do my best to make this post simple enough. Please, let me know whether I have succeeded :)
For more information (and references), one can check either this book, or the various links provided in the post.
Table of contents of this series of posts
II. The origins of quantum mechanics
1. Fundamental physics at the beginning of the 20th century
2. The mysteries of the atomic spectra
3. The mysteries of the black bodies
4. Quantization of the electromagnetic radiation and the birth of the photons
5. The Compton effect
6. The planetary model of the atom
7. The Bohr model of the atom
8. Matter waves
9. More on the wave-particle duality
10. The Schrödinger equation (this post).
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