Quantization Of Angular Momentum

All about the quantization in the magnitude, direction  of orbital, spin, total angular momentum in quantum mechanical systems.  

Quantization Of Angular Momentum | Magnitude Quantization - Space quantization  

Introduction:

    Angular momentum plays a very important role in physics. It is a conserved quantity. Conserved quantities are those quantities that remain constant as the system evolves, unless some kind of an external force acts upon that system.

    More explicitly we can say that - if there is a certain body that has a rotational motion with respect to an origin or an axis then there is a physical quantity known as angular momentum associated with it's rotation. It will remain constant until and unless some kind of an external torque acts upon that system.

    Because of this very reason angular momentum plays a very important role not only in classical mechanics ( i.e. In classical systems where you might study how planets revolve around the sun in the solar system ) but also it plays a very important role in quantum mechanical systems ( like an electron revolving around the nucleus).

    In the case of nuclear physics it is also quite important to study the spin and the angular momentum of the nucleus and the properties associated with that (like nuclear magnetic moment etc.).

    To study the angular momentum of the nucleons and the electrons, first of all, we have to look into the orbital rotational momentum then the spin of these particles. In both these two cases, we have the quantization of both the magnitude and direction of these particular quantities. In this part we are going to study about it in detail.

   

Angular Momentum of a Classical system:

    Before discussing quantum mechanical systems, let’s first look at a classical system. For example let us consider the solar system. Now, in the case of the Solar System, the Earth is revolving around the Sun. Besides the revolving, the earth rotates about its own axis.
    
    Thus, the earth has two types of angular momenta - one is associated with the revolution of earth around the sun that is known as orbital angular momentum, and the other is associated with the rotation about earth’s own axis which is known as the spin angular momentum.

    1. Orbital angular Momentum of a Classical system:

We may write the earth’s orbital angular momentum(L) as -

Where, = Position vector of earth from the axis of rotation.

and, represents the linear momentum of Earth’s orbital motion.

Thus,   is perpendicular to both  and  i.e. it is perpendicular to the plane of the orbit.

     Earth has a certain orbital velocity tangential to this particular orbit and that corresponds to the linear momentum  . Angular or rotational momentum has a direction perpendicular to the plane in which the Earth moves around the Sun.

    Angular momentum is a vector quantity. Then it has both magnitude and a particular direction. The magnitude tells us about the value of the angular momentum.

    In the case of the solar system, the system itself does not put any restriction on the magnitude or the direction of the angular momentum. This means, if the earth suddenly starts spinning at a higher velocity then the magnitude of the angular momentum of Earth will increase.

    Thus, angular momentum of Earth can take any value, depending upon the distance from the axis of rotation and the orbital velocity of Earth here. Therefore, the magnitude of angular momentum of earth can take a continuous range of values. A given planet can revolve in any elliptical plane and it can have any direction of angular momentum pointing in any direction in the three-dimensional space. This also means that angular momentum of Earth is not quantized.
    
    Thus, the orbital angular momentum of classical systems are not quantized. That means they do not take discrete values. They can take any value depending upon the position vector of the rotating object with respect to the axis of rotation.

    2. Spin angular Momentum of a Classical system:

The earth as a body itself is spinning on its own axis passing through its center. Then we have a spin motion of earth here. In that case also, we will have  the magnitude of the spin angular momenta and the direction.

Spin angular momentum is also not quantized for all classical systems. 

 

Quantization of Angular Momentum in Quantum-mechanical Systems: 

Whenever we look into the quantum mechanical systems, the angular momentum can not take any magnitude or direction. It can take some discrete values only.

Let us take a simple example of an electron revolving around the nucleus. In this case we can see that the electron is not allowed to take any magnitude or direction of angular momentum. It can take some particular values only i.e. it is quantized here. We are going to discuss this phenomenon in detail in the next section.

    The solution of the Schrodinger’s equation for the case of an electron in an atom results in certain quantum numbers and these quantum numbers put restrictions on the magnitude and the direction of the angular momentum of the electron.

Orbital angular Momentum of a Quantum-Mechanical System:

Let’s, first look at the orbital angular momentum of electron in hydrogen atom.  

        1. Quantization of Magnitude:

      The electron moves around the nucleus of the atom and  it has a particular angular momenta(). The magnitude of  is basically restricted by a particular equation which comes from the solution of the Schrodinger’s equation. The equation is given by,

Here,  l =0,1,2,3, ….. (n-1)

[ Where n is the principal quantum number.]

    l is known as the Azimuthal quantum number.

Thus , L can not take a continuous range of values; it can only take certain discrete  values which is allowed by this particular equation given above. This is known as the quantization of the magnitude of the angular momentum.



For example, let's look at the S orbital. In the case of S orbital , we have azimuthal quantum number l=0. In that case the magnitude of the angular momentum would be equal to -

Similarly, for p orbital we have azimuthal quantum number(l)=1 . In that case the magnitude of the angular momentum would be equal to -

Similarly, for d orbital :

Thus, the magnitude of the angular momentum of the electron can only possess certain discrete values. Given by  L=  

 

        2. Direction quantization or space quantization for Orbital Electron:

    

 

    This quantum mechanical equation puts restriction on the direction in which a particular electron orbital plane can orient itself in three dimensional space. 
gives the z-component of the angular momentum of that particular orbit. This restriction is known as Direction quantization or space quantization .

    let's look at this particular diagram below - (The points at which the value of  is same are represented by the particular red circles in the figure.) 

    For example, let us take the p orbital. In the case of a p orbital the azimuthal quantum number(l) has a value of 1. Then magnetic quantum number() can have the values -1,0,+1 . 

    Then,  for the case of p orbital z component of orbital angular momentum() can have only three values,

      We know that the electron's motion is confined in three dimensional space. Here ,  is the  z component of orbital angular momentum.     

    If there is an external magnetic field present then we take the standard z-axis along the field direction. The electron is a charged particle moving in an orbit so it has a particular magnetic moment. So it will orient itself in such a direction that the z component of its angular momentum can take only the three values, as mentioned above.

    Now, for example let us take   =1. In this case angular momentum of the orbit can precess in such a direction that the z component of it can take only the value . 

    In terms of 3D visualization the  component can have a certain fixed value for a particular conical section. So there are three different conical sections which put a restriction on the direction of the angular momentum of the electron orbit.

Quantization of Spin Angular Momentum:

    1. Quantization of Magnitude:

    Apart from the revolving around the nucleus the electron is also spinning about its own axis as the earth revolving around the Sun is also spinning about its own axis. However this kind of analogy is not necessarily entirely correct because electron is a quantum particle, with point mass. It is not necessarily spinning on its own axis in the same way as the planet Earth is spinning on its own axis. We use these analogies of classical mechanics only for the sake of convenience. It helps us to understand or visualize what is happening. Purely from the point of view of physics that is not necessarily true. We cannot always use the analogies of classical mechanics with the quantum particles because quantum particles behave in a completely different way. 

    The rotation of the electron about its own axis, contributes towards an angular momentum of the electron and that angular momentum is also quantized with respect to its magnitude and direction.

The expression which gives the quantization of the magnitude of this kind of a spin angular momentum is written as -

    s represents a particular quantum number it has only one value  (magnitude) -

                   

    Therefore the angular momentum or the spin angular momentum of the electron will also have one value -

 

    Thus, Electrons will always have an intrinsic property known as spin angular momentum associated with its existence. The magnitude of the spin angular momentum will always be -

    In fact other elementary particles like protons and neutrons which also have spin 1/2 have a spin angular momentum of this value associated with their existence. 

    2. Quantization of direction:

  Direction of this particular spin angular momentum is also quantized. It can be obtained from the z component of spin angular momentum.

     is the spin quantum number. it has the values +1/2 or -1/2.

    Thus, space quantization of spin angular momentum leads to two different orientations. 

Spin angular momentum and orbital angular momentum are not necessarily conserved quantities separately. The conserved quantity of any kind of an electron system is the total angular momentum of a system. Total angular momenta comes from the vector addition of these two kinds of angular momenta.

Then, the total angular momenta is given by, 

    The quantization rules in the previous sections are applicable to orbital angular momentum and spin angular momentum. Similar kinds of quantization rules are also applicable to the magnitude and direction of  .

.

    j is nothing but a quantum number which can have values (l+1/2) or |(l-1/2)| . 

Hope, you have got an overall idea about the magnitude and space quantization of angular momentum.