Coupling refers to the interaction between two systems such that their angular momenta combine. There are two types of coupling in physics - L-S coupling and J-J coupling. L-S coupling specifies how the orbital and spin angular momenta of electrons in an atom combine. The orbital angular momenta of electrons are first combined into a resultant L, and their spin angular momenta into a resultant S. L and S then interact via spin-orbit coupling to form the total angular momentum J. Quantum numbers like L, S, J are used to describe atomic states under L-S coupling. Selection rules determine the allowed transitions between these states.

1. By: Muhammad Naeem

2. Coupling
• Coupling is the act of combining two things.
• Couplings are mechanical elements that ‘couples’ two drive
elements which enables motion to be transferred from one
element to another.
• In order to transmit torque between two shafts that either
tend to lie in the same line or slightly misaligned, a
coupling is used.

3. Coupling in Physics
➢In physics, two systems are coupled if they are interacting
with each other.
• There are two types of coupling
i. L-S coupling
ii. J-J coupling

4. L-S coupling or Russell-Saunders coupling
In atomic spectroscopy, Russell–Saunders coupling,
also known as LS coupling, specifies a coupling scheme of
electronic spin- and orbital-angular momenta. The coupling
scheme is named after H. N. Russell and F. A. Saunders (1925).

5. Total angular Momentum of an atom
• Total angular momentum associated with the atom is
measured along its projection along observation axis (z-axis).
• If more than one source of angular
momentum (spin or orbital motion
contributes to A.M), we will only
observe the total angular momentum
from this source included.

6. Total angular Momentum of an atom
• Contributions come from various sources that “couple” or
combine together to give the resulting total.
• Here we’ll discuss the angular momentum for many electron
system.

7. Quantum Numbers
➢ The principalquantum number (n) describes the size of the orbital.
➢ The angular quantum number (l) describes the shape of the orbital.
➢ The magnetic quantum number (ml) describe the orientation in
space of a particular orbital.
➢ The Spin quantum number (ms) describe the spin.

8. Total Angular Momentum
In this scheme, the usual pattern for all atoms is that the orbital angular
momenta L of the various electrons are coupled together in to a single
resultantL (combine vectorially).
𝐿=𝐿1+𝐿2+𝐿3+…………
The spin angular momenta si are also coupled together into another single
resultantS.
S=𝑆1+𝑆2+𝑆3+……….

9. Total Angular Momentum
So,
L=σ 𝐿𝑖 And S=σ 𝑆𝑖
➢ The momenta L and S then interact via the spin-orbit effect to form the
total angular momentum J. This scheme is called LS coupling.
•The momenta L and S combine to form the total angularmomentumJ as
𝐽=𝐿+𝑆

10. The angular momentum magnitudes L, S, J and their z components Lz, Sz
and Jz are all quantized with their respective quantum numbers L, S, J, ML,
Ms, and MJ. Hence
𝐿 = 𝐿(𝐿 + 1) ℏ
𝐿𝑍 = 𝑀𝐿ℏ
𝑆 = 𝑆(𝑆 + 1) ℏ
𝑆𝑍 = 𝑀𝑆ℏ
𝐽 = 𝐽(𝐽 + 1) ℏ
𝐽𝑍 = 𝑀𝐽ℏ

11. ➢ Both L and ML are always integers or 0, while the other quantum
numbers are half integral if an odd number of electrons is involved and
integral or 0 if even number of electrons is involved.
➢ When two orbital angular momenta having quantum numbers l1 and l2
combine, the allowed values of L are
𝐿=( 𝑙1+ 𝑙2 ), ( 𝑙1+ 𝑙2−1 ),…..,|𝑙1−𝑙2|
➢ Similarly the allowed values are calculated for S and J.

12. ➢ For a given value of L, the allowed values of J are
𝐽=𝐿 + 𝑆, 𝐿 + 𝑆−1, ….,|𝐿−𝑆|
➢ For L>S, there are 2S+1 values of J.
➢ For L<S, the number of possible values of J are 2L+1.
➢ The value of 2S+1 is called the multiplicity of the state.
➢ When S=0, we have 2S+1=1, such states are referred to as singletstates
➢ When S=1/2 and 2S+1=2, these are referred to as doubletstate.
➢ When S=1 and 2S+1=3, these are referred to as triplet state.

13. Term symbols
➢ To describe states conveniently, one requires a
notation. The symbol nLj used for a single electron is
changed as follows.
• State symbol or Term symbol = n2S+1Lj
Where,
• The superscript 2S+1 represents the multiplicity of the
state.
• Subscript J is the total angular momentum quantum
number, and L stand for S, P, D, …….

14. ❑ How to read L-S coupling notation ?
• If S=1/2, L=1, J=3/2, 1/2. The corresponding states will be
2P3/2 and 2P1/2, read as doublet P three halves and doublet P
half respectively.
• Multiplicities associated with different number of electrons
in the outer shell are given by the rule which states that the
terms of atoms or ions with even number of valence
electrons have odd multiplicities and vice versa.

15. L-S coupling’ Selection Rules
ΔS = 0
ΔL = 0,±1
ΔJ = 0,±1
However, the J=o to J=0 transition is forbidden
Δ mJ = 0,±1
However, the mJ = 0 to mJ = 0 is forbidden if ΔJ = 0

16. Reference
• ModernPhysics by G. Aruldhas(university of kerala) & P.
Rajagopal (MahatmaGandhi University).
• Principles of Modern Physics by Ajay K Saxena.
• ModernPhysics by S. L. Kakani & Shubhra Kakani.
• Physics of atoms and moleculesby B. H. Bransden and C. J.
Joachain.
• Wikipedia