Let's get back to the energy levels and the simple pictorial model of the hydrogen atom suggested by Bohr. As you probably remember he explained the positions of the spectral lines and how they arise. This explanation was based on the presumption that electrons can rotate only on some orbits. The other orbits are inaccessible for electrons. An electron moving on some orbit has its energy in the state characteristic for that orbit. An electron circulating in a hydrogen atom on the orbit closest to the nucleus has its energy equal to -13,6 eV. The 'minus' symbol means here that one has to give the energy of 13,6 eV (electron volt) to the electron to free it from the atom (to ionise the atom). And on the second orbit the energy of an electron is equal to -3,4 eV. An electron can absorb a quantum of energy equal to the difference between some higher energy level (orbit) and the present one. At the moment of the absorption of an energy quantum the electron jumps from his present orbit to a higher one. For example if an electron was situated on the first orbit then if jumping to the second one, it has to absorb an energy quantum equal to:
If an electron situated on the first orbit is jumping to the third one, which has the energy level of -1,51 eV, then it has to absorb an energy quantum equal to:
This rule holds also for all the other orbits. Of course, an electron situated on the second orbit also can jump to a different orbit; for example to the third one if absorbing an energy quantum of 1,89 eV. If an electron rotating on the first orbit is given an energy quantum of 13,6 eV or more, then it leaves the atom; the atom will then have positive charge surplus, and so will become positive ion.
When all of the electrons in an atom are placed on their lowest orbit we say the atom is on its basic energy level. All the other energy states are called the excited states. An atom can't stay long in an excited state. Usually after a very short moment an electron gets back to a lower
energy state, and finally achieves the lowest of its allowed orbits. Such "jump down", as you know, is connected with the emission of a quantum of energy equal to the difference between the energy of the higher and the lower states.
Watching the emissive spectrum of an element one sees the lines representing the quantum of light (photons) of the energies equal to the energies of various transitions. Niels Bohr was the one who explained the emissive spectrum this way. He also assigned some number n, called then the main (or sometimes the total) quantum number, to the successive orbits.
Soon it was discovered that the spectral lines are not homogenous but consist of several lines placed very close to each other. Arnold Sommerfeld explained the effect. He assumed that except for the spherical orbits there can also be some elliptic ones. Electrons can move only on some, allowed ellipses. He assigned a second number l that was called the secondary (or the azimuthal) quantum number. The number defines the shape - the "oblateness" of an orbit. For n=1 an orbit can be only the spherical one (1=0). For n=2 an orbit can be spherical (1=0) or elliptic (1=1). For n=3 an orbit can be spherical (1=0) or can take one of two elliptic shapes (1=1 or 1=2). And so on. Generally for any n there are n kinds of shapes of orbits. Electrons moving on orbits of the same n number but of different shapes differ a bit by energy. That explains the composite nature of the spectral lines. It is accepted that the n number is a symbol for the shell, and the l number is a symbol for the subshell. It is also customary that the successive subshells are symbolised by letters.
Another improvement in the Bohr model was the discovery that orbits don't have to lay in the same plane. They can be oriented in space in some defined directions. Their orientation is defined by the magnetic quantum number ml and shows appears by splitting spectral lines in outer magnetic field (when placing a sample of the examined gas between opposite magnetic poles). It is called the Zeeman effect. Sommerfeld proved that there was only some limited number of possible situations of the orbits in space. The number of the possible positions is equal to 2*l+1. In a magnetic field each position has a bit different energy. The m number can take values from l to –l.
To let you better understand the quantum numbers n, l, ml, their possible values for n=1, n=2, n=3, and n=4 are given in the chart below:
Beside the described facts one more was discovered - that the spectral line consists of two lines placed much closer to each other than those defining shapes of orbits. The Bohr-Sommerfeld model couldn't explain the fact. The two Danish physicists - Uhlenbeck and Goudshmit, found the explanation. They noticed that an electron not only rotates around the nucleus but also on its own axis (Just like Earth rotating both around the sun and on its own axis.). The electron can spin left or right. It induces the circular current flowing in the direction of the spin. The current causes the magnetic field, which is in accordance with , or on the contrary to the field created by the electron moving on its orbit. So the fields sum or subtract. The spinning electron causing the contrary fields has its energy a bit lower than the one causing the consistent fields. So the spectral line splits. One line represens the electron spinning left, and the other - right. For symbolising the spin direction the spin quantum number ms is used. For the electron it can be equal to +1/2 or -1/2.
So there are four quantum numbers in total: the main, the azimuthal, the magnetic, and the spin quantum number. Those four numbers describe each and every electron in an atom. So if two electrons of the three identical numbers differ in the fourth one then their energies are also a bit different from each other.
In 1925 Wolfgang Pauli (1900-1958) formulated a principle called the Pauli exclusion principle. According to it there can't be electrons of the same state in an atom. That means there can't be electrons of the same all four quantum numbers (n,l,ml,mS) in one atom.
