Intensity of Hydrogen spectral line

[Angelos K]

I would like to calculate the intensities of Hydrogen spectral lines for the case of Hydrogene at temperature T contained inside a volume V.

I reckon that what I need is

1)The mean population of the upper energy level
2)The probability of transition between the levels

I believe that the mean population of a quantum state will be given by Bose statistics, so that of an energy level will be the same multiplied by the degeneracy of the energy level.

For the transition probabilities I have no clue. When we deal with atoms subjected to a field those are usually calculated by perturbation theory, for which one needs a perturbative Hamiltonian. I have no clue how to arrive at the latter here. Nor do I think that the influence of the whole surrounding would be small enough to be a perturbation.

Help is really appreciated!

 

[eljose79]

Thank you for your inquiry regarding calculating the intensities of Hydrogen spectral lines for a specific case. You are correct in your understanding that the mean population of the upper energy level and the transition probabilities are key factors in determining the intensities of spectral lines. Allow me to provide some guidance on how to approach this problem.

Firstly, in order to calculate the mean population of the upper energy level, you will need to know the temperature and volume of your system, as well as the energy levels of Hydrogen. These can be found in tables or calculated using the Bohr model or quantum mechanics. Once you have this information, you can use the Boltzmann distribution to determine the relative populations of each energy level. The degeneracy of each energy level will also need to be taken into account in this calculation.

Next, for the transition probabilities, you are correct in noting that perturbation theory is typically used in cases where there is an external field. However, in this case, we can use the Einstein coefficients to calculate the transition probabilities between energy levels. These coefficients take into account the spontaneous emission, stimulated emission, and absorption processes. They can be expressed in terms of the energy levels and the wavelength of the spectral line.

It is also important to note that the surrounding environment can affect the spectral lines, as you mentioned. This can be taken into account by considering the pressure, density, and temperature of the surrounding gas. These factors can alter the transition probabilities and therefore impact the intensities of the spectral lines.

I hope this information helps guide you in your calculations. If you have further questions, please do not hesitate to ask for clarification. Best of luck with your research!

 

[denian]

I can provide some guidance on how to calculate the intensities of Hydrogen spectral lines in this scenario.

Firstly, you are correct in identifying the mean population of the upper energy level as a key factor in calculating the intensity of a spectral line. This can be determined using the Boltzmann distribution, which gives the relative probabilities of a particle being in a particular energy state at a given temperature. The degeneracy of the energy level also plays a role in this calculation, as you mentioned.

Next, the transition probability between energy levels can be calculated using the Einstein coefficients. These coefficients describe the probability of an atom transitioning from one energy level to another, either by absorption or emission of a photon. The Einstein coefficients depend on the properties of the atom and the surrounding environment, such as the temperature and density of the gas.

However, in order to accurately calculate the intensities of Hydrogen spectral lines, we must also consider the effects of collisions and broadening mechanisms. Collisions between atoms can alter the energy levels and affect the transition probabilities, while broadening mechanisms such as Doppler and pressure broadening can affect the shape and width of the spectral lines.

In this case, it may be helpful to use a computer program or simulation to model the system and calculate the intensities of the spectral lines. This would take into account all the relevant factors and provide a more accurate result.

In summary, to accurately calculate the intensities of Hydrogen spectral lines in this scenario, you will need to consider the mean population of the upper energy level, the transition probabilities between energy levels, and the effects of collisions and broadening mechanisms. I hope this helps guide your calculations.