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Space Quantisation of an Atom

Hello friends welcome once again on our blog. In our last post we’ve discussed Larmor Precession . In this post we are going to discuss Space Quantisation of an atom,  which is another important topic in Atomic Physics. 

In Bohr-Sommerfeld atomic model, the space around nucleus is quantised, i.e. the distribution
of electrons in these orbits is in accordance with quantum number.

However, when an electron rotates in an orbit it produces a magnetic field and possesses a certain magnetic moment. 
Therefore, when an atom is placed in external magnetic field B ,the electron orbit precesses about the field direction as axis(Larmor Precession).

Fig 1. Vector L traces a cone
around vector B. 

The electron orbital angular momentum vector traces a cone around such that angle ϴ between and remains constant as shown in figure. 

If is along z-axis, the component of parallel to the field is :

According to the quantisation principle :

This angle ϴ between and the z-axes is determined by the quantum number ‘l’ & ‘ml’.

Since for a given l, there are (2l+1) possible values of ml(0,±1,±2,…..±l) 
The angle ϴ can assume (2l+1) discrete values. i.e. 

In other words, the angular momentum vector can have (2n+1) discrete orientations, with respect to the magnetic field. 

The quantisation of the orientation of atom in space is known as space quantisation. 

The space quantisation of the orbital angular momentum vector corresponding to l=2 is shown in fig. below:

Fig 2.Space quantisation of
vector L corresponding to L=2.

For l=2 we have,

  ml = 2,1,0,-1,-2
so that Lz = 2h/2π ,h/2π, 0, -h/2π ,-2h/2π.
Alternatively, the orientation ϴ owith respect to field are given by :

Note:- can never be aligned exactly parallel or antiparallel to B, since |ml| is always smaller than √l(l+1).

This is Space Quantisation. You can ask any question related to this post in comment section. 

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