Goodbye Determinism, Hello Heisenberg Uncertainty Principle

When the nucleus was discovered by Rutherford, it became clear the classical world was not reality, because according to classical electromagnetism, the electron should collapse to the proton. This problem was solved by Niels Bohr who showed that electrons orbit in distinct orbitals, and can only gain or lose energy in chunks, not continuously. Bohr's atom was based on Max Planck's simple equation.

Planck introduced Planck’s constant, h, which is the proportionality factor between energy and the frequency of radiation. Bohr noticed that Planck’s constant had units of angular momentum, so he guessed that this was the minimum angular momentum that an electron could have to remain in a stable orbit. He showed that the atom goes from a high energy state to a lower one, proportional to this constant.

Here's how to derive the uncertainty principle: In the double slit experiment, when monochromatic light goes through two slits we get an interference pattern just like if it was a wave. A single slit produces a similar diffraction pattern where we see a central very high peak, and very faint peaks elsewhere.

The width of the slit represents the uncertainty in position, because the electron could be anywhere along the slit. The distance to the interference pattern represents the momentum. The uncertainty in the momentum is represented by the distance from the center of the pattern to the first interference pattern.

Louis de Broglie showed that the wavelength, lambda for a particle with mass is equal to h/p, where h is Planck’s constant, and p is the momentum. Using trigonometry, we can get a series of equations showing how the uncertainty in position and uncertainty in momentum are equal to h. This is essentially what the uncertainty principle is.

This is slightly different than how the equation is typically written with is greater than or equal to h/4*pi, because of some imprecisions in this approach to the derivation. A more precise approach requires higher level math and use of Fourier transforms. Here's a link to one such derivation if you are interested:

The important thing to remember is that the uncertainty in the momentum and the position are inversely related. As one gets bigger, the other gets smaller. This is not a limitation of our ability to measure the position and momentum. It is a limitation of reality!

How does this relate to the Bohr model of the atom? The Bohr radius, where Niels Bohr calculated the electron would be at its lowest energy state in the hydrogen atom is 5.29x 10^-11 m. The velocity can be calculated to be 2.18 x 10^6 m/s for the same electron. The mass of the electron is 9.11 x 10^-31 kg.

Now we can calculate the momentum of this electron because momentum is just mass times velocity. This would be equal to about 2 x 10^-24 kg*m/s. But there is always some uncertainty. So for example if there was a 10% uncertainty in velocity, then delta p would be one tenth of this or 2 x 10^-25 kg*m/s.

Calculating from the uncertainty equation, we get: Delta X is 26 x 10^-11 meters. Compared to the Bohr radius, this is 5X larger. A 10% uncertainty in momentum results in a huge uncertainty in the position. It tells us that the Bohr radius is fundamentally wrong. The truth is that the electron doesn’t just sit at any fixed radius. There is a constant balance between the uncertainty in its momentum and position, such that it forms a cloud of probability around the proton. This cloud extends far below and far beyond the Bohr radius. And using the Schrodinger's equation, we can calculate the probability of finding it at certain locations if we measure it.

So why do we keep teaching students about Bohr’s inaccurate model? It is useful for approximate calculations, and explains a number of features of atoms on the periodic table. It is useful for chemistry.
#uncertaintyprinciple
#heisenberg
If we used the same equation and put in the mass of a tennis ball and a 1% uncertainty in momentum, we get that the uncertainty in the position, equals 1.55 x 10^-33 m. This is so small that we would never notice it. Similarly, we don’t notice it in anything else either that we can see with our eyes.

The big lesson to be learned here is that the central concept in quantum mechanics is only noticeable at very tiny scales.

[ArvinAsh]

I can see from the comments that I did not do a good enough job explaining why uncertainty exists. So I will try to explain it in words here.
WHY UNCERTAINTY: First, you have to accept that quantum "particles" such as electrons are not like little balls. They are waves. There is no distinct position or momentum. These have to be measured. The wave system is characterized by a function called Psi. The square of the absolute value of psi gives you a probability. Psi incorporates everything about the system - energy, position, momentum, quantum states, etc. When you solve for position, the value is never 1. It is always between zero and one, because a probability of one would make the momentum value for the particle infinite. This is one way to interpret the uncertainty principle. It basically means that particles do not behave classically - you can never know EXACTLY where a particle is. This is true even for macro particles, but the wave function varies so little, billionths of billionths of millimeters, that you never notice it.
ARGUMENT FOR DETERMINISM: Some argue that the many worlds interpretation of quantum mechanics makes things deterministic. This is really not true. While the probabilities of all the worlds add up to one, the world that you find yourself in is random. This is the same case with the Copenhagen interpretation. The Schrodinger equation can "predict" the probability for various outcomes, but the outcome that will occur for you when you make a measurement is random. NOTE HOWEVER, that this statement applies to quantum events, and does NOT necessarily extrapolate to determinism in terms of human decision-making or free will.
WHY NOT LOGICAL: My comment about it not being logical refers to behavior at quantum scales that does not fit with our macro scale logic - for example, things like entanglement - two particles far apart linked instantaneously, measurement changing the outcome of events, or particles not having a precisely defined position.

[itwasntidio]

"In the coming future people will quote me with things I never said" - Sir Issac Newton

[LQhristian]

The quantum weather today: Localized electron cloudiness with a probability of precipitation (wave function collapse) when measured :-)

[naterlandsw]

My dude, I can’t say it enough: We’re all super glad you’re getting this traction and payoff for the quality work you’ve been putting up since day *bleeping* one! Keep ‘em comin’, sir!

[chiranjibsaha]

You are a great teacher Arvin, nothing to say about you keep on giving us knowledge about physics . ❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️

[alancook]

Your handling of the maths - super impressive. And the science - fascinating. Keep on telling us these inconvenient truths!

[Dheeru_D_Luffy]

Hey Arvin Sir.
Just saw this video and i am glad you still jump with excitement while explaining awesome facts which we otherwise just feel regular stuffs. Your body language and your variations in loudness and pitch is what helps me to explain physics for intermediate course students.
Waiting for the next time when you say.... 'All those things are coming up right now!'
❤️

[dctdude]

I assure you that this is the best video I have ever seen. Relating every concepts and clearing the doubts one above the other..
Literally awesome..💥💥💥

[apocaRUFF]

This is the first time I've ever wanted to learn trigonometry.

[craiggordon]

Legend has it Heisenberg was a bad lover. As soon as he found the right position, he couldn't find the momentum

[seanspartan]

Thank you for stressing that even though quantum effects are imperceptible to the human eye at macro scales, they still exist and we can calculate them. I think for a lot of people there is a disconnect between quantum mechanics and our reality, as if QM is for particles and Classical mechanics is for everyday life. But in reality QM is for everything and Classical mechanics just happens to be a good approximation at large scales.

[basuraMan]

"Reality is not even logical..."

yep sounds about right, sums up my 2020 so far

[adityathestar]

@Arvin: What about Neutron stars? Does Heisenberg uncertainty principle break down since electrons fall into nucleus?

[kamilpavelka]

Your videos are simply the best. Period.

[stuglenn]

The end of Determinism, well you can never be certain about these things.

[bitmau]

I appreciate the math correlation, especially explaining Delta x and p. Thanks for value added breakdown!

[quantumrain]

Charles Duell: "everything that can be invented has been invented"

The future: im about to end this man's whole career

[hasanrahman]

I m very grateful to you for your videos.

[davidhilsee]

Man, that title really just rolls off the tounge

[channagirijagadish]

Thanks Arvin for another brilliant exposition!