Problems on Photons: UNIZOR.COM - Physics4Teens - Waves - Photons and Ma...

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
E_N = E_ground /N²
where E_ground = 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 L_i with energy level i to a shell L_j 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:

< 400	Ultraviolet
380-440	Violet
440-485	Blue
485-510	Cyan
510-565	Green
565-590	Yellow
590-625	Orange
625-740	Red
> 740	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).