Why Physicists Call This Equation the GOLDEN RULE of Physics.

#goldenrule #physics #fermisgoldenrule #goldenruleofphysics

Fermi's Golden Rule is mathematical rule within the theory of quantum mechanics. It is used to calculate the transition probability between any two quantum states. In simpler terms, when we are studying systems with more than one possible state, like atoms which have multiple energy levels, then the Golden Rule formula can tell us how likely an electron is to transition from one state to another at any point in time. But the situation is a little bit more complicated than that.

First, we see that a system's "allowed" energy levels can be calculated using the Schrodinger equation. This is done by inputting information about the system, such as its kinetic and potential energies, into the equation, and then solving for the "allowed" wave functions or eigenstates.

These are the energy levels in which our system will be found whenever we make a measurement on it. For example, we will find each electron in one of the shells around the nucleus. However if the system is "perturbed" slightly, such as by a negatively charged electron passing relatively far away from the atom, then the allowed energy levels of the atom will also change.

This is because the negatively charged electron will repel all the electrons in the atom, and attract all the protons in the nucleus. Thus, the potential energies we used to calculate the allowed energy levels of the atom will have changed, and hence the allowed energy levels themselves will have changed.

When this is the case, however, the electrons in the atom are still in the "old" energy levels. And they will need to transition into the new energy levels. So which exact level will each electron transition into?

Well, each electron could technically transition into any one of the new levels. However, there are two factors that affect the likelihood of a single electron to transition into a new state.

Firstly, the "coupling" between the old state of an electron, and the new state we're studying. This is given by factors such as how close in energy the old and the new state are. The closer in energy, the strong the coupling, and the more likely our electron is to transition into this new state.

Secondly, the transition probability also depends on the "density of states" around the new energy level. This means that states which are surrounded by other states very close in energy are more likely to receive the electron, because there are more states for the electron to transition into in the vicinity of that energy.

Combining these two factors, we can figure out the probability per unit time that an electron will transition into a particular state. This is exactly what Fermi's Golden Rule studies and gives us a formula for.

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Timestamps:
0:00 - Energy Transitions and the Schrodinger Equation
2:42 - Big Thanks to Shortform for Sponsoring This Video!
4:39 - Changing the Energies in Our System
6:28 - Fermi's Golden Rule for Transitions

This video was sponsored by Shortform #ad

[omicronceti]

A more in depth video about the mathematical background would be great!
Such an interesting topic.

[Snarlydowrong]

In Fermi's golden rule or any time dependent perturbation theory, you assume that the states of the reference Hamiltonian (H_original in this video) are approximately the true states of the system. However, these states are weakly coupled to each other by some perturbation (H') and Fermi's golden rule describes the transition rate between them due to this coupling. It doesn't describe the rate between the old (states of H_original) and new states (states of H_original + H') as said in the video. This is because the new and old states are linear combinations of each other and the <f|H'|i> is an element of the perturbation operator expanded in the basis of H_original. To exemplify this, here's an important application of Fermi's golden rule: if you have a molecule containing a ground and excited electronic state and this molecule is in an electromagnetic field which it interacts weakly with, you can determine the transition rate between an excitation in the molecule and the large number electromagnetic field modes interacting with the molecule. This gives the rate of emission from the molecule (or Einstein A coefficient). Here, the <i| (or initial state) in Fermi's golden rule, is the electronic excited state in the molecule and |f> (or final state) is some electromagnetic field mode. These are both states of H_original. The whole point of Fermi's golden rule is to avoid calculating the states of H_original + H', or new states as referred to in the video. In the example given it is impossible to calculate the new states as the number of electromagnetic field modes is practically infinite. One other issue, and maybe this is pedantic, but the video uses the term "strong coupling" but this has specific terminology in quantum mechanics that is different than used here and could potentially lead to confusion. In fact, strong coupling specifically refers to the case when the coupling between states of H_original is large enough such that time dependent perturbation theory fails to describe things in a meaningful way. I get that the video is saying that if the coupling is stronger the transition rate is stronger, which is 100% true, but in Fermi's golden rule, the states should by no means be strongly coupled. I would say this video gives the basic idea of Fermi's golden rule but for people trying to understand it at a mathematical level, for say a class, the issue with initial and final states could lead to significant confusion. I wanted to mention this, especially the first part because it's important not to loose truthfulness even when distilling things down and I think it could lead to confusions.

[mridulacharya]

Can U make a video to mention the level of mathematics required and specifications in order to understand quantum mechanics properly...?

[maxgallegos]

Hello from 🇨🇱 Love your videos, the only place where i can learn next level stuff in a understandable way

[akagordon]

It's worth noting that this rule leads to important factors in modeling chemical reactions, in particular the HOMO/LUMO energy states predict the likelihood of reactions, be it excitations or bonding, directly because of this principle.

[FermionPhysics]

Bro looks like Schrodinger with mustache 😂 love your work

[ahmedmukhtar]

Hello there from 🇳🇬. Could u make a vid on renormalization and regularization? The difference btwn the two and on how solid they r in Physics

[LeoTaxilFrance]

I’d like to wish you Happy New Year!
You have enthusiastic, curious, paradox, deep, original approach to the physics and mathematics. It is very entertaining to follow your reasoning and it is very useful because you stay precise and deep enough in your mathematical “justifications”.

[annaclarafenyo]

I had a prof named Mitch Golden, who took pains to explain that, although he was teaching it, and it was his name, he wasn't the actual creator of the Golden Rule. You should point out that it only works asymptotically, for finite times, the Golden rule is violated, according to the time-energy uncertainty. The final transition probability is caused by cancellations in phase between states that don't conserve energy.

[lewisbulled]

Why not just write h instead of 2*pi/h_bar?

[sheetalmadi]

Please make a detailed video on this topic.

[fabianquevedo]

I see this topic one year ago in the university and you remembered me the fascinating that it is! Btw in general the Quantum Perturbation theory is very cool topic (for curios the "Zetilli - Quantum Mechanics" book have a very good description of this theory)

[KevinToppenberg]

So, the Golden Rule is ... Treat other Hamiltonians as your Hamiltonian would want to be treated?? I'm so confused! (Just Kidding) Great video. Thanks!

[SteveGouldinSpain]

I think Karl would have approved of your YouTube channel very much indeed. Happy holidays!

[rohitrohj]

Brother, you are even better than most profs.

[amnqetu_Cipher-da-Builder]

Bro. The stash adds about 30 years to your appearance. Great videos. You inspire me

[photon]

Parth, I liked your example with the extra electron whizzing by your atom. Is it possible that the waves are probable due to chaotic and random interaction with other waves? 🌪

[jaysavitrimaai]

Spin of electron is 1/2, why it's not 3/2, 5/2, 1, 2 or any other number ? And where this factor has been calculated from??

[markharder]

It reminds me of entropy in macroscopic systems. If one thermodynamic system can exist in more states than another one with the same energy, then the first one will be found with a higher probability than the latter one. I.e. the more degenerate system, which has the greater entropy, will occur more often than the one with fewer ways of distributing the same amount of energy.

[SN]

Please please please make video on Drichlet and nuemen boundary conditions 🙏🙏🙏🙏