Problems on Photons: UNIZOR.COM - Physics4Teens - Waves - Photons and Matter
Channel: Zor Shekhtman on UNIZOR Education
Date: 2022-12-11

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com

Problems on Photons and Matter

Problem 1

A hydrogen atom has one electron that is located in a certain shell.

It can be in a "normal" state on the ground level of energy (energy level #1 or shell #1).
When it's excited, it can absorb certain amount of energy and jump to level #2 or #3 etc. to shells for higher energy electrons.

The increase in energy level number corresponds to absorption of more energy to jump to this level.
If electron emits some energy in a form of radiation, it jumps from higher energy shell to a lower energy.

The energy level number N for a hydrogen atom can contain an electron that carry an amount of energy equaled to
EN = Eground /N²
where Eground = 13.6 eV
(electron-volts)

Calculate the energy differences between shells, restricting the calculations only to the first four shells with energy levels 1, 2, 3 and 4.


Solution

To jump from a shell Li with energy level i to a shell Lj with energy level j electron needs to absorb (positive) or emit (negative) a photon with energy
13.6·(1/i² − 1/j²) eV

According to this formula, the table with calculated energy differences is

i\j L1 L2 L3 L4
L1 0 10.20 12.09 12.75
L2 −10.20 0 1.89 2.55
L3 −12.09 −1.89 0 0.66
L4 −12.75 −2.55 −0.66 0

As you see, the energy difference between two consecutive levels rapidly diminishes with an increase in energy level.
It equals to 10.20 eV between levels 1 and 2, 1.89 eV between levels 2 and 3 and 0.66 eV between levels 3 and 4.


Problem 2

For all cases when the radiation is emitted by electrons of a hydrogen atom (consider only the first four shells, like in Problem 1) determine the frequency and wave length of electromagnetic oscillations and the color of light emitted.

Here are the data you might need:
(a) charge of an electron:
e = −1.60217663·10−19 C
(b) Planck's Constant:
h=6.62607015·10−34 J·s
(c) speed of light:
c=2.99792458·108 m/s
(d) colors for wave lengths in nanometers, according to one (out of many different) sources on the Web:
less-400 Ultraviolet
380-440 Violet
440-485 Blue
485-510 Cyan
510-565 Green
565-590 Yellow
590-625 Orange
625-740 Red
740-more Infrared


Solution

Based on energy levels derived in Problem 1, we can obtain the frequencies of emitted light based on the formula
E = h·f
from which follows
f = E/h
Considering the energy in Problem 1 was calculated in electron-volts, to convert it to Joules we have to multiply the given energy by a charge of an electron e getting
f = e·E/h
For a frequency f the wavelength λ is calculated based on the speed of light
λ = c/f

Using the results of Problem 1 above, we obtain the frequencies of emitted light, depending on the difference in energy level between shells of the hydrogen atom:
i\j L1 L2 L3
L2 2466·1012  
L3 2923·1012 457·1012 
L4 3083·1012 617·1012 160·1012


Converting this to wave length (in nanometers), we obtain
i\j L1 L2 L3
L2 122  
L3 103 656 
L4 97 486 1878


Electrons transitioning to the energy level #1 from any other level emit ultraviolet light (wave length is too short, invisible).
Electrons transitioning to the energy level #2 from level #3 emit red light of 656 nanometers wave length.
Electrons transitioning to the energy level #2 from level #4 emit cyan light of 486 nanometers wave length.
Electrons transitioning to the energy level #3 from level #4 emit infrared light (wave length is too long, invisible).