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.

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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.

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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.

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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.

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       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

How to fake a credit card number ? – 101

If you are thinking – ‘Dude, its a bunch of numbers, can’t i just make my own credit card number?’. Well, then this post is for you.

You see in order to ensure that people don’t game the system, credit card companies have a simple set of rules. ( a checksum )

“Mod-10″ Algorithm

The Luhn algorithm or the mod-10 algorithm is what you need to beat. Its used to validate a lot of things from credit cards to social security numbers.

Say you have a credit card whose number is 4012-8888-8888-1881. In order to
check whether the credit card is a valid one, then we have to do a
really simple mathematical operation. 

image

                                              Source:  CodeProject

Double every other number and add them. Call this sum – x

Add the rest of the numbers. Call this sum -y

If (x+y) is a multiple of 10, then its a valid card, otherwise its not.

image

Valid credit card numbers

The last number in a credit card is known as a ‘checksum’ and it plays a vital role.

According to this, only 4012-8888-8888-1881 is a valid card number. not 4012-8888-8888-1882, 4012-8888-8888-1883,4012-8888-8888-1884, 4012-8888-8888-1885 …… 4012-8888-8888-1889.

( because they their sum is not a multiple of 10 )

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Of course, although this was good enough algorithm in the 20th century, with high processing power you can do much more than a simple mod 10 test.

Therefore, the present day check sum although is based on this, is packed with much more fascinating mathematics.

( Verhoeff algorithm, Damm algorithm, Luhn mod N algorithm are some good places to start on your quest )

Have a great day!

** Post Banner Credits : Oh you credit card by Linn Fritz

Minimal Math Concepts +

These are the work of Marlon Tenório, who runs a blog called minimalmathconcepts. What he does is take concepts and illustrate them in a exquisite fashion. Check him out! His work is remarkable.

And if you really like his work, you can also buy his goodies on-line here.

Kudos MMC+ on such a good work and all the very best to you mate 🙂

** When FYP surfaced on Tumblr, we were just another blog. And big giants like space-exploration and mind-blowing-science, were so generous to uplift us. And we are just following the Tumblr tradition 🙂 Cheers!