Atomic Spectra of Hydrogen Atom

If an electric discharge is passed through hydrogen gas in atomic state taken in a discharge tube under low pressure and the emitted radiations is analyzed with the help of spectrograph the spectrum does not have radiations of all the wavelengths but radiations of only certain wavelengths. The spectrum of hydrogen is found to consist of a series of sharp lines in the UV visible and IR regions.
When gaseous hydrogen in a glass tube is excited by a 5000-volt electrical discharge, four lines are observed in the visible part of the emission spectrum: red at 656.3 nm, blue-green at 486.1 nm, blue violet at 434.1 nm and violet at 410.2 nm (F.g.3):





Figure 3. Visible spectrum of atomic hydrogen.
Other series of lines have been observed in the ultraviolet and infrared regions. Rydberg (1890) found that all the lines of the atomic hydrogen spectrum could be fitted to a single formula:

…(20)

where R, known as the Rydberg constant, has the value 109,677 cm-1 for hydrogen. The reciprocal of wavelength, in units of cm-1, is in general use by spectroscopists. This unit is also designated wave numbers, since it represents the number of wavelengths per cm. The Balmer series of spectral lines in the visible region, shown in Fig. 3, correspond to the values n1 = 2; n2 = 3; 4; 5 and 6. The lines with n1 = 1 in the ultraviolet make up the Lyman series. The spectra are summarized in Fig. 4. Figure 5. shows energy levels of atomic hydrogen.












Figure 4. The atomic spectra of hydrogen.










Figure 5. Energy levels of atomic hydrogen.
2. Schr?dinger’s equation
The wave equation of a particle (e.g., an electron) moving in one direction (e.g., x direction) is given below:

…….(21)
where m is the mass of an electron, V is the potential energy of the system as a function of coordinates, and ? is the wave function, E is the total energy of a system.

One-dimensional square-well potential
The simplest example of Schr?dinger’s equation is the one for an electron in a one-dimensional square-well potential. Suppose the potential energy V of an electron confined in a box (the length is a) is 0 in the box (0 < x < a) and ? outside the box. The Schr?dinger equation within the box is:

………. (22)
…….. (23)

The following equation is obtained by solving the above equations.
?(x) = (?2/a)sin(n?x/a) ……..(24)
Note that n arouse automatically. The wave function ? itself does not have any physical meaning. The square of the absolute value of ?, ?2, is, however, a mathematical indication of the possibility to find the electron in question at a given position, and hence very important because it is related to the electron density. If the possibility of existence of the electron existing at a certain position is integrated all over the active space, the result should be unity: i.e., ??2dx = 1.
The energy (Eigenvalue) is
E = n2h2/8ma2 n = 1, 2, 3 ... (25)
It is clear that the energy value of the particle is discontinuous.

Hydrogen-like atom
….(26)

…..(27)

The three-dimensional Schr?dinger’s equation will become as below:
…..(28)
or
…..(29)

The potential energy of a hydrogen-like atom is given below where Z is the electric charge.
……(30)

……(31)

bohrs:

and energy in hartrees:


Conversion to atomic units is equivalent to setting
h = e = m = 1
in all formulas containing these constants. Rewriting the Schr?dinger equation in atomic units:

…..(32)

Since the potential energy is spherically symmetrical (a function of r alone), it is obviously advantageous to treat this problem in spherical polar coordinates. Expressing the Laplacian operator in these coordinates

..(33)