A note on what makes solutions discretized?

When one stumbles upon the words ‘Discretized solution’, one is inclined to think of Quantum Mechanics. In quantum mechanics, the following are fundamentally discrete:

  • Electric charge
  • Weak hypercharge
  • Colour charge
  • Baryon number
  • Lepton number
  • Spin

BUT not energy. One only finds discrete spectra in bound states or where there are boundary conditions.

Discrete spectra and Boundary conditions

Consider a string that is clamped at x = 0 and x= L undergoing traverse vibrations. And you would like to know the motion of the string.

image

Maybe you know a priori that the solutions are sinusoids but you have no information on its wave number.

So you start trying out every single possibility of the wave number.

image

The important thing to understand here is that If there weren’t any boundary conditions that was imposed on the string then all possible sinusoidal wave would be a solution to the problem.

But the existence of a boundary condition ruins it.

image

This is the case with energy as well.

If
you have an electron in a hydrogen atom, there are only specific energy
levels it can be observed to occupy when its energy is measured.

But
if the electron is unbound because its energy exceeds the ionization
energy of the atom, then it’s in a scattering state and its energy and
angular momentum have continuous spectra.

image

       The solution of the Schrodinger equation for the hydrogen atom


Sources and more:

Solution to the wave equation by method of separation of variables

Brain Bi’s answer to ‘What quantities are always quantized?

Mathematical Methods for Physicists( Chapter – 8), George B. Arfken, Hans J. Weber, Frank E. Harris

Energy is a continuous analytic function

… mom’s spaghetti…

… mom’s spaghetti…

Ask Ethan: Why don’t comets orbit the same way…

Ask Ethan: Why don’t comets orbit the same way planets do?

“Why [do] comets orbit the Sun in a parabolic path, unlike planets which orbit in an elliptical one? Where do comets get the energy to travel such a long distance, from the Oort cloud to the Sun & back? Also, how could interstellar comets/asteroids come out of their parent star [system] and visit other ones?”

When we see comets in our Solar System, they can be either periodic, passing near the Sun and then extending very far away, to return many years later, or they could be a one-shot deal. But comets are driven by the same gravitational laws that drive the planets, which simply make fast-moving, nearly-circular ellipses around the Sun. So what makes these orbits so different, particularly if they’re obeying the same laws? Believe it or not, most of the would-be comets out there are moving in exactly the same nearly-circular paths, only they’re far more tenuously held by the Sun. Gravitational interactions might make small changes in their orbits, but if you’re already moving very slowly, a small change can have a very big effect!

Why don’t comets orbit the same way as the planets? Find out on this edition of Ask Ethan!

Emmy NoetherAmalie Emmy Noether was a German mathematician known…

Emmy Noether

Amalie Emmy Noether was a German mathematician known for her landmark contributions to abstract algebra and theoretical physics. And noether’s theorem is one of the most beautiful equation in all of theoretical physics.

The theorem explains the connection between symmetry and conservation laws.

It is remarkably surprising that there are a lot of people who are not aware of Noether’s contribution to physics.

This video by ‘Looking Glass Universe’ does a good job (but does not cover the math) in explaining the essence of the theorem.

Have fun!

spaceshipenterprise: Having a NCC Enterprise …

spaceshipenterprise:

Having a NCC Enterprise bottle opener and watching Star Trek Discovery makes the easiest things the best!✨☄🌽🎬🛰

In the Pacific Island nation of Vanuatu, women…

In the Pacific Island nation of Vanuatu, women have a tradition of water music, accompanying their singing with a percussive use of water. This video explores the physics behind this music. Performers use three basic motions – a slap, a plunge, and a plow – that each have distinctive acoustics thanks to the interaction of hand, water, and air. High pitches come from the initial impact on the water, whereas lower pitches come mostly from the collapse of the air cavity in the hand’s wake. By altering the rhythms and patterns of these three building blocks, the musicians create a rich harmony to accompany their singing. (Video credit: R. Hurd et al.)

The Year’s Best Meteor Shower Is Here,…

The Year’s Best Meteor Shower Is Here, And The Geminids Are Better Than Ever

“This year, at the peak of the Geminids, the Moon will be a waning crescent, not even rising until well after midnight. Even when it does, it will be thin enough and far enough away from the origin of the Geminids that you’ll still have a spectacular show. If you have dark, cloudless skies, you should be able to see up to two or three meteors per minute once the sky reaches full darkness this year. While the cold snaps affecting much of the country might make it a little unpleasant to be outside for too long, it also provides the best viewing conditions for the night sky. This year, you won’t want to pass up the opportunity.”

Every year, there are two meteor showers reliably worth checking out if the conditions are favorable: August’s Perseids and December’s Geminids. This year, with the Moon in a waning crescent phase and with clear skies anticipated across most of the country, the Geminids just might shape up to be spectacular. Created by the debris of asteroid 3200 Phaethon, the Geminids are a relatively young shower at under 200 years, and they continue to get more spectacular over time, peaking at over 150 meteors per hour the last few years. There’s an incredible scientific story behind where these showers come from, and an incredible show to be had if you can find clear, dark skies. The year’s best meteor shower is here, with the peak coming in just a few days.

Come find out how to get the most out of your Geminid experience, Don’t pass up the opportunity to see one of nature’s most spectacular shows!

Recently, NASA Goddard released a visualizatio…

Recently, NASA Goddard released a visualization of aerosols in the Atlantic region. The simulation uses real data from satellite imagery taken between August and October 2017 to seed a simulation of atmospheric physics. The color scales in the visualization show concentrations of three major aerosol particles: smoke (gray), sea salt (blue), and dust (brown). One of the interesting outcomes of the simulation is a visualization of the fall Atlantic hurricane season. The high winds from hurricanes help pick up sea salt from the ocean surface and throw it high in the atmosphere, making the hurricanes visible here. Fires in the western United States provide most of the smoke aerosols, whereas dust comes mostly from the Sahara. Tiny aerosol particles serve as a major nucleation source for water droplets, affecting both cloud formation and rainfall. With simulations like these, scientists hope to better understand how aerosols move in the atmosphere and how they affect our weather. (Image credit: NASA Goddard Research Center, source; submitted by Paul vdB)

Repealing Fuel Economy Standards To Cost Ame…

Repealing Fuel Economy Standards To Cost American Households Nearly $1000 Per Year

“Across the board — cars, crossovers, SUVs, minivans, and pickups — the fuel efficiency of vehicles have continued to climb steadily, even just during the past decade. An economic analysis details the impact that the impact that this can have on the typical American household. For middle-income families, defined as households making under $100,000 in 1980 and under $120,000 in 2014, transportation costs make up a total of approximately 20% of total expenditures, a figure that rises to 30% for low-income families. Yet the increased fuel economy alone has saved families a tremendous amount: an estimated $17,000 over the 1980-2014 time period.”

Earlier this year, EPA administrator Scott Pruitt and the Trump Administration announced their plans to roll back the Obama-era regulations on fuel economy standards. Since 1975, the first year that fuel standards were enacted, the savings to Americans has been tremendous: a total of 1.5 trillion gallons of gas have been saved, resulting in a direct cost savings of approximately $4 trillion to US consumers. Meanwhile, the costs have been borne by the automotive industry, not passed along to customers, resulting in a huge boon to the American economy across-the-board. It’s the low-and-middle income American families that benefit the most, and a rare case where the science and math support a universal conclusion, regardless of political affiliation.

So why are we talking about repealing the regulations? Don’t let this issue fly under the radar; get informed today!

Many insects are known to quest underwater, bu…

Many insects are known to quest underwater, but few are as adept at it as the alkali fly. This species has taken common attributes among flies – being covered in tiny hairs and a waxy layer – and really upped the ante. Their extra hairiness and extra waxiness make them extremely difficult to get wet, even in the excessively salty and alkaline waters of California’s Mono Lake, which are enough to defeat all but algae, brine shrimp, bacteria, and alkali flies. 

Staying dry is a challenge, but only one of many this insect tackles. The combination of hair and wax over the insect makes it superhydrophobic, coating it in a silvery layer of air as it crawls below the surface. All that air is buoyant, so to walk underwater, the fly has to exert forces up to 18 times its body weight just to keep from popping back up to the surface. 

The shimmering bubble also helps the fly breathe. Insect respiratory systems use openings all over the exoskeleton to exchange oxygen with the ambient atmosphere via diffusion. While diffusion of oxygen does still happen underwater, it’s a much slower process there. The air sheath around the fly creates a large surface area for oxygen to diffuse, which helps counter the lower diffusion rate. Inside the sheath, the fly breathes as it normally does. (Image and research credit: F. van Breugel and M. Dickinson; via Gizmodo; submitted by @1307phaezr)