Quantum Angular Momentum Coupler Calculator

Quantum Angular Momentum Coupler Calculator finds total j and m_j values for coupled angular momenta, aiding quantum mechanics and atomic physics studies.

First Angular Momentum (j₁):

Second Angular Momentum (j₂):

Formulas Used in Quantum Angular Momentum Coupler Calculator

The calculator uses the following formulas for coupling two angular momenta:

Total Angular Momentum Quantum Number:

\\[ j = |j_1 – j_2|, |j_1 – j_2| + 1, \ldots, j_1 + j_2 \\]

Magnetic Quantum Number:

\\[ m_j = -j, -j + 1, \ldots, j – 1, j \\]

Number of States:

\\[ N_j = 2j + 1 \\] \\[ N_{\text{total}} = (2j_1 + 1)(2j_2 + 1) \\]

Where:

\\( j_1, j_2 \\): Angular momentum quantum numbers
\\( j \\): Total angular momentum quantum number
\\( m_j \\): Magnetic quantum number for total angular momentum
\\( N_j \\): Number of states for a given \\( j \\)
\\( N_{\text{total}} \\): Total number of states

Example Calculations

Example 1: Electron-Electron Spin Coupling (j₁ = 1/2, j₂ = 1/2)

Input: j₁ = 0.5, j₂ = 0.5

\\[ j = |0.5 – 0.5|, \ldots, 0.5 + 0.5 = \{0, 1\} \\] \\[ m_j(j=0) = \{0\}, \quad m_j(j=1) = \{-1, 0, 1\} \\] \\[ N_0 = 2 \cdot 0 + 1 = 1, \quad N_1 = 2 \cdot 1 + 1 = 3 \\] \\[ N_{\text{total}} = (2 \cdot 0.5 + 1)(2 \cdot 0.5 + 1) = 4 \\]

Result: j = {0, 1}, m_j(j=0) = {0}, m_j(j=1) = {-1, 0, 1}, Number of States: {1, 3}, Total States: 4

Example 2: Spin-Orbit Coupling (j₁ = 1/2, j₂ = 1)

Input: j₁ = 0.5, j₂ = 1

\\[ j = |0.5 – 1|, \ldots, 0.5 + 1 = \{0.5, 1.5\} \\] \\[ m_j(j=0.5) = \{-0.5, 0.5\}, \quad m_j(j=1.5) = \{-1.5, -0.5, 0.5, 1.5\} \\] \\[ N_{0.5} = 2 \cdot 0.5 + 1 = 2, \quad N_{1.5} = 2 \cdot 1.5 + 1 = 4 \\] \\[ N_{\text{total}} = (2 \cdot 0.5 + 1)(2 \cdot 1 + 1) = 6 \\]

Result: j = {0.5, 1.5}, m_j(j=0.5) = {-0.5, 0.5}, m_j(j=1.5) = {-1.5, -0.5, 0.5, 1.5}, Number of States: {2, 4}, Total States: 6

Example 3: Higher Spin Coupling (j₁ = 1, j₂ = 1.5)

Input: j₁ = 1, j₂ = 1.5

\\[ j = |1 – 1.5|, \ldots, 1 + 1.5 = \{0.5, 1.5, 2.5\} \\] \\[ m_j(j=0.5) = \{-0.5, 0.5\}, \quad m_j(j=1.5) = \{-1.5, -0.5, 0.5, 1.5\}, \quad m_j(j=2.5) = \{-2.5, -1.5, -0.5, 0.5, 1.5, 2.5\} \\] \\[ N_{0.5} = 2 \cdot 0.5 + 1 = 2, \quad N_{1.5} = 2 \cdot 1.5 + 1 = 4, \quad N_{2.5} = 2 \cdot 2.5 + 1 = 6 \\] \\[ N_{\text{total}} = (2 \cdot 1 + 1)(2 \cdot 1.5 + 1) = 9 \\]

Result: j = {0.5, 1.5, 2.5}, m_j(j=0.5) = {-0.5, 0.5}, m_j(j=1.5) = {-1.5, -0.5, 0.5, 1.5}, m_j(j=2.5) = {-2.5, -1.5, -0.5, 0.5, 1.5, 2.5}, Number of States: {2, 4, 6}, Total States: 9

Formulas Used in Quantum Angular Momentum Coupler Calculator

Example Calculations

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