### Particle physics and the Schrodinger equation

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To get something that we can normalize we must use a superposition of such generic solutions. By adding plane waves with different oscillation frequencies we can build a wave packet that is localized in space and therefore can be normalized. Take note that using a Fourier transform a wave packet can be tought of as being built of plane waves:.

One of the most important examples is the particle in a box, since it demonstrates nicely how the quantization of energy levels happens in quantum mechanics. A particle confined in a box, here 1-dimensional, with infinitely high potential walls is one of the standard examples of quantum mechanics. Inside the box the potential is zero, outside it's infinite. Next, we use that the wave-function must be a continuous function 1. Hence the name quantum mechanics. These solutions have to be normalised again because of the probabilistic interpretation 3.

These waves describe where it is likely that we find a given particle. This is crucial in modern electronics. The equation for a single photon is precisely the same as the equation for an entire electromagnetic field.

### Visualizing Particle-In-A-Box

However, there is an important difference in the type of solution for the equations that is allowed. Classical Maxwell fields are necessarily real whereas photon states are complex. There is also a so-called 'positive frequency condition that the photon state must satisfy. This fact is responsible for the Maxwell-field-wavelike behaviour and polarization of single photons that we caught glimpses of earlier. This is known as Matrix mechanics.

The wave function can be visualized more easily than matrices and differential equations easier to handle than problems in matrix mechanics. Some physicists argue that this leads to a lot of confusion and is the source of many problems. A nice discussion of this point of view can be found in "Space is blue and birds fly through it" by Carlo Rovelli. This was proposed by Bryce deWitt in Rev. In Peter Debye's class in introductory physics Felix found what he desired and felt later that he learned more from that class than from all his other courses together.

Coming across Sommerfeld's famous book….. In an event occurred that had a great influence on his career. He described this in an article for Physics Today in December For a single particle moving around in three dimensions the equation can be written as. The solution to this equation is the wave function. The solution to the full equation is then.

These equations apply to one particle moving in three dimensions, but they have counterparts describing a system with any number of particles. And rather than formulating the wave function as a function of position and time, you can also formulate it as a function of momentum and time.

But what does this solution actually mean? It doesn't give you a precise location for your particle at a given time , so it doesn't give you the trajectory of a particle over time. Rather it's a function which, at a given time gives you a value for all possible locations. What does this value mean? In the physicist Max Born came up with a probabilistic interpretation. This probabilistic picture links in with a rather shocking consequence of de Broglie's formula for the wavelength and momentum of a particle, discovered by Werner Heisenberg in Heisenberg found that there is a fundamental limit to the precision to which you can measure the position and the momentum of a moving particle.

The more precise you want to be about the one, the less you can say about the other. And this is not down to the quality of your measuring instrument, it is a fundamental uncertainty of nature.

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This result is now known as Heisenberg's uncertainty principle and it's one of the results that's often quoted to illustrate the weirdness of quantum mechanics. It means that in quantum mechanics we simply cannot talk about the location or the trajectory of a particle. In other words, all you can expect from the mathematical representation of a quantum state, from the wave function, is that it gives you a probability.

Whether or not the wave function has any physical interpretation was and still is a touchy question. For them the wave function was only a tool for computing probabilities. Why should we believe this rather fantastical set-up? The famous physicist Richard Feynman considered this a futile question: "Where did we get that [equation] from? It's not possible to derive it from anything you know. Yet, the equation has held its own in every experiment so far. According to Ernest Rutherford's atomic model, the frequency of radiation emitted by atoms such as hydrogen should vary continuously.

Experiments showed, however, that it doesn't: the hydrogen atom only emits radiation at certain frequencies, there is a jump when the frequency changes. This discovery flew in the face of conventional wisdom, which endorsed a maxim set out by the 17th century philosopher and mathematician Gottfried Leibniz : "nature does not make jumps".

## Schrödinger equation

In Niels Bohr came up with a new atomic model in which electrons are restricted to certain energy levels. We will also explore another weird consequence of the equation called quantum tunneling. Marianne Freiberger is Editor of Plus. She interviewed Bouatta in Cambridge in May I am semi-retired, so I have the time now to learn those subjects I should have taken in high school, and college. This is just for my own curiosity, and to exercise my brain. Thanks for this article. I found it very interesting.

Was just browsing the web looking for a quick blurb about schrodinger and found this. Very nice setup and nice flow. Reminded me of my college days. Thank you very much for this very nice illustration, developed step by step chronically. Reading this nice article, one can see how quantum mechanics evolves and understand it better. Thank you very much. A very concise article giving a big picture description of the basic tenets of QM. Would have been more impactful if the article was written straight instead of writing quotes from the interview.

Although I am a PhD student in physics and have been studying quantum physics for several years, this article gives me a better view and summary of quantum concepts.

## 3: The Schrödinger Equation and a Particle in a Box - Chemistry LibreTexts

With special thanks. Thanks Marianne. I was looking for a straight forward explanation of Schrodinger's equation, and here is your thoughtful article. The best article on schrodinger's equation ever read. Helped me a lot.