This blog post will introduce you to the new and exciting world of quantum physics.

For a long time, physics has been confined to classical physics. And for decades, it’s been one of the most important subjects in colleges and universities around the world. But times are changing, and there is a new kid on the block: quantum physics. What is the remainder when (x4 + 36) is divided by (x2 – 8)?

Quantum mechanics is incredibly difficult to understand because it is built on such a different set of principles than those that govern classical mechanics like Newtonian gravity or Coulomb’s law.

1. Quantum Mechanics: The Basics

The main principle of quantum mechanics is that everything is uncertain. It’s best to think of uncertainty as being in the range from 0 to 1. A probability, if you will. 0 means the event is impossible, and 1 means that it most certainly will happen.

Let’s do an example: How much energy does a proton have? Well, no one really knows – it could be anything between 0 and 1. The more we try to measure it accurately, the more uncertain we become able to measure it. This uncertainty was brought home by physicists when they measured the decay of a neutron into a proton, an electron, and an antineutrino. They measured several of these decays, and the results were completely different. You would think that different scientists would get the same answer, but in quantum mechanics you cannot predict the outcome of a measurement. The range between 0 and 1 is what’s called the Heisenberg Uncertainty Principle, named after Werner Heisenberg.

2. Let’s Get Quantum

So now let’s look at a more concrete example: consider an electron orbiting a proton in an atom. You’re probably familiar with classical mechanics, but in quantum mechanics things work a little differently. The electron has a certain amount of energy given by:

E=hf (1)

The “h” is Planck’s constant and the “f” is the frequency of the electron. Planck’s constant is a very small number, 6.626×10^-27 erg s. And so it’s pretty big when you multiply it by something really really small like the energy an electron has when it absorbs or emits a photon of frequency f . But again, don’t worry about understanding the math too well – we’re just trying to get used to how quantum mechanics works.

We assume that the electron has an energy of E=hf, and that the electron’s frequency is f . Then, as it orbits the proton:

E = h^2f (2)

The equivalent equation for classical mechanics is:

M = Main(θ) (3)

Notice how the derivative has been changed from a velocity to a point. What this means is that things are quantized. This quantum behavior can only be observed at very low temperatures, so when you’re dealing with something like an atom, it is only by accident. Or maybe by design. Who knows?

3. Quantum States

Now, if you were to look at the world around you, it’s a very noisy place. And this noise is due to the wave nature of things. The wave nature of everything is really really easy to see with sound waves – like this

Here’s a little trick – listen to something like an opera sung in Italian and then listen to something in English and I bet you can tell which language it was sung in. This difference happens because of acoustic waves that propagate from your ear to your brain.

Invariably, classical physics deals with things that are more or less discrete . A fountain pen is not just a blurry line but an ink dot on a piece of paper. The problem is that the fountain pen is a continuous line, and what you see on your paper is made up of a bunch of ink dots. If we had perfect eyes and perfect microscopes, things would look blurry for us too.

If we were to use classical physics to describe the trajectory of an electron around the nucleus in an atom or molecule, we would get a very fuzzy picture of where the electron was. And it’s this fuzziness that allows quantum mechanics to be so effective in describing reality. 

4. The Uncertainty Principle

The uncertainty principle states that the more precise we try to measure a particle’s position, the less precise we can be about its momentum. Or put another way: The more we know about one of two quantities, the less we can know about the other quantity. The math is as follows:

h = E/c^2

E=pc/ps (4)

(h is Planck’s constant) p is momentum, and c is the speed of light. As you can see from this expression, if you want to measure something with zero error, h must be close to zero. If the electron has an energy of hf , then you have a lower bound for its momentum p.

Conclusion:

You can see that the uncertainty principle is really confusing. The low mass of the electron means it’s not that hard for things to get out of whack, and the uncertainty principle is often used to explain weird phenomena.

5. Schrodinger Equation

The Schrodinger equation describes an electron in a hydrogen atom:

E=hf + psi (5)

where psi is the kinetic energy (or a quantity equal to the momentum times velocity). If we have a number – let’s say n – electrons in an atom, then this equation tells us how many are in certain energy states when their kinetic energy is measured.

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