Angular Momentum Addition J1=1 And J2=1 Forming J=2, 1, And 0 States
Introduction
In the realm of quantum mechanics, the addition of angular momenta is a fundamental concept, particularly when dealing with systems involving multiple sources of angular momentum, such as multiple particles or a single particle possessing both orbital and spin angular momentum. This article delves into the process of combining two angular momenta, specifically j1 = 1 and j2 = 1, to form states with total angular momentum j = 2, 1, and 0. We will explore how to express all nine |j, m⟩ eigenkets in terms of the |j1, m1; j2, m2⟩ basis using methods such as the ladder operator and recursion relations. Understanding this process is crucial for comprehending the behavior of complex quantum systems and forms the backbone for various applications in atomic and nuclear physics.
Theoretical Background
Before diving into the specifics, let's establish the theoretical framework. Angular momentum in quantum mechanics is quantized, meaning it can only take on discrete values. An angular momentum j has (2j + 1) possible projections along a chosen axis, typically the z-axis, denoted by m, where m ranges from -j to +j in integer steps. When combining two angular momenta j1 and j2, the possible values of the total angular momentum j are given by |j1 - j2| ≤ j ≤ (j1 + j2), also in integer steps. For our case, with j1 = 1 and j2 = 1, this yields j = 2, 1, and 0.
The states are expressed in two common bases: the uncoupled basis |j1, m1; j2, m2⟩ and the coupled basis |j, m⟩. The uncoupled basis represents the individual angular momenta and their projections, while the coupled basis represents the total angular momentum and its projection. The transformation between these bases is facilitated by Clebsch-Gordan coefficients, which encode the probability amplitudes for different combinations of m1 and m2 to form a specific m for a given j. Our goal is to find these relationships and express the |j, m⟩ states in terms of |j1, m1; j2, m2⟩.
Methodology: Ladder Operators and Recursion Relations
There are two primary methods for determining the relationships between the coupled and uncoupled bases: the ladder operator method and the recursion relation method. Both methods leverage the properties of angular momentum operators, particularly the raising and lowering operators, J+ and J-, respectively. These operators, also known as ladder operators, change the m value of a state by +1 or -1, while leaving j unchanged.
Ladder Operator Method
The ladder operator method begins by identifying the highest weight state, which is the state with the maximum possible m value (m = j). This state is typically straightforward to express in the uncoupled basis. For example, the highest weight state for j = 2 is |2, 2⟩, which can only be formed by combining the maximum projections of j1 and j2, i.e., |1, 1; 1, 1⟩. Applying the lowering operator J- to this state generates the state |2, 1⟩. By repeatedly applying J-, we can obtain all states with j = 2. The lowering operator is defined as J- = J1- + J2-, where J1- and J2- act on the j1 and j2 subspaces, respectively. The action of the lowering operator on a state |j, m⟩ is given by:
J- |j, m⟩ = ħ√((j(j + 1) - m(m - 1)) |j, m - 1⟩
Where ħ is the reduced Planck constant. This equation allows us to calculate the coefficients in the expansion of |j, m - 1⟩ in terms of the uncoupled basis states.
To find the states for other j values, such as j = 1 and j = 0, we need to identify states that are orthogonal to the states already found. For instance, the state |1, 1⟩ must be orthogonal to |2, 1⟩. This orthogonality condition, along with normalization, allows us to determine the coefficients for |1, 1⟩ in the uncoupled basis. We can then apply the lowering operator to obtain |1, 0⟩ and |1, -1⟩. Finally, the state |0, 0⟩ is found by ensuring it is orthogonal to all the j = 2 and j = 1 states.
Recursion Relation Method
The recursion relation method provides a systematic way to calculate the Clebsch-Gordan coefficients. These coefficients, denoted as ⟨j1, m1; j2, m2|j, m⟩, represent the overlap between the uncoupled and coupled basis states. The recursion relations are derived from the action of the angular momentum operators on the states and provide a set of equations that can be solved iteratively to find the coefficients. The general recursion relations are somewhat complex, but they simplify for specific cases like the one we are considering.
The recursion relations typically involve the Clebsch-Gordan coefficients for neighboring m values. By starting with a known coefficient, such as the coefficient for the highest weight state, we can use the recursion relations to calculate the other coefficients. This method is particularly useful when dealing with more complex angular momentum additions where the ladder operator method becomes cumbersome.
Expressing the Eigenkets
Now, let's apply these methods to express the nine |j, m⟩ eigenkets for j = 2, 1, 0 and m ranging from -j to +j in terms of the |j1, m1; j2, m2⟩ basis. We will use a combination of the ladder operator method and the orthogonality conditions.
j = 2 States
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|2, 2⟩: As mentioned earlier, the highest weight state is straightforward:
|2, 2⟩ = |1, 1; 1, 1⟩
-
|2, 1⟩: Applying the lowering operator J- to |2, 2⟩:
J- |2, 2⟩ = ħ√(2(2 + 1) - 2(2 - 1)) |2, 1⟩ = 2ħ |2, 1⟩
(J1- + J2-) |1, 1; 1, 1⟩ = J1- |1, 1; 1, 1⟩ + J2- |1, 1; 1, 1⟩ = ħ√(1(1 + 1) - 1(1 - 1)) |1, 0; 1, 1⟩ + ħ√(1(1 + 1) - 1(1 - 1)) |1, 1; 1, 0⟩ = ħ√2 |1, 0; 1, 1⟩ + ħ√2 |1, 1; 1, 0⟩
Equating the two expressions and normalizing:
|2, 1⟩ = (1/√2) |1, 0; 1, 1⟩ + (1/√2) |1, 1; 1, 0⟩
-
|2, 0⟩: Applying J- to |2, 1⟩:
J- |2, 1⟩ = ħ√(2(2 + 1) - 1(1 - 1)) |2, 0⟩ = ħ√6 |2, 0⟩
(J1- + J2-) [(1/√2) |1, 0; 1, 1⟩ + (1/√2) |1, 1; 1, 0⟩] = (1/√2) J1- |1, 0; 1, 1⟩ + (1/√2) J2- |1, 0; 1, 1⟩ + (1/√2) J1- |1, 1; 1, 0⟩ + (1/√2) J2- |1, 1; 1, 0⟩ = (1/√2) ħ√(1(1 + 1) - 0(0 - 1)) |1, -1; 1, 1⟩ + (1/√2) ħ√(1(1 + 1) - 1(1 - 1)) |1, 0; 1, 0⟩ + (1/√2) ħ√(1(1 + 1) - 1(1 - 1)) |1, 0; 1, 0⟩ + (1/√2) ħ√(1(1 + 1) - 0(0 - 1)) |1, 1; 1, -1⟩ = ħ |1, -1; 1, 1⟩ + ħ√2 |1, 0; 1, 0⟩ + ħ |1, 1; 1, -1⟩
Equating the two expressions and normalizing:
|2, 0⟩ = (1/√6) |1, -1; 1, 1⟩ + (√2/√3) |1, 0; 1, 0⟩ + (1/√6) |1, 1; 1, -1⟩
-
|2, -1⟩: Applying J- to |2, 0⟩:
J- |2, 0⟩ = ħ√(2(2 + 1) - 0(0 - 1)) |2, -1⟩ = ħ√6 |2, -1⟩
After applying J- and normalizing, we get:
|2, -1⟩ = (1/√2) |1, -1; 1, 0⟩ + (1/√2) |1, 0; 1, -1⟩
-
|2, -2⟩: Applying J- to |2, -1⟩:
J- |2, -1⟩ = ħ√(2(2 + 1) - (-1)(-1 - 1)) |2, -2⟩ = 2ħ |2, -2⟩
After applying J- and normalizing, we get:
|2, -2⟩ = |1, -1; 1, -1⟩
j = 1 States
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|1, 1⟩: The state |1, 1⟩ must be orthogonal to |2, 1⟩. Therefore:
|1, 1⟩ = α |1, 0; 1, 1⟩ + β |1, 1; 1, 0⟩
Orthogonality with |2, 1⟩ implies:
⟨2, 1|1, 1⟩ = 0 = (1/√2)α + (1/√2)β
Thus, α = -β. Normalizing the state, we get:
|1, 1⟩ = (1/√2) |1, 1; 1, 0⟩ - (1/√2) |1, 0; 1, 1⟩
-
|1, 0⟩: Applying J- to |1, 1⟩:
J- |1, 1⟩ = ħ√(1(1 + 1) - 1(1 - 1)) |1, 0⟩ = ħ√2 |1, 0⟩
After applying J- and normalizing, we get:
|1, 0⟩ = (1/√2) |1, -1; 1, 1⟩ - (1/√2) |1, 1; 1, -1⟩
-
|1, -1⟩: Applying J- to |1, 0⟩:
J- |1, 0⟩ = ħ√(1(1 + 1) - 0(0 - 1)) |1, -1⟩ = ħ√2 |1, -1⟩
After applying J- and normalizing, we get:
|1, -1⟩ = (1/√2) |1, 0; 1, -1⟩ - (1/√2) |1, -1; 1, 0⟩
j = 0 State
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|0, 0⟩: The state |0, 0⟩ must be orthogonal to |2, 0⟩ and |1, 0⟩. Therefore:
|0, 0⟩ = α |1, -1; 1, 1⟩ + β |1, 0; 1, 0⟩ + γ |1, 1; 1, -1⟩
Orthogonality with |2, 0⟩ and |1, 0⟩ gives two equations, and normalization provides the third. Solving these equations, we get:
|0, 0⟩ = (1/√3) |1, -1; 1, 1⟩ - (1/√3) |1, 0; 1, 0⟩ + (1/√3) |1, 1; 1, -1⟩
Summary of Eigenket Expressions
Here's a summary of all nine |j, m⟩ eigenkets expressed in terms of the |j1, m1; j2, m2⟩ basis:
- |2, 2⟩ = |1, 1; 1, 1⟩
- |2, 1⟩ = (1/√2) |1, 0; 1, 1⟩ + (1/√2) |1, 1; 1, 0⟩
- |2, 0⟩ = (1/√6) |1, -1; 1, 1⟩ + (√2/√3) |1, 0; 1, 0⟩ + (1/√6) |1, 1; 1, -1⟩
- |2, -1⟩ = (1/√2) |1, -1; 1, 0⟩ + (1/√2) |1, 0; 1, -1⟩
- |2, -2⟩ = |1, -1; 1, -1⟩
- |1, 1⟩ = (1/√2) |1, 1; 1, 0⟩ - (1/√2) |1, 0; 1, 1⟩
- |1, 0⟩ = (1/√2) |1, -1; 1, 1⟩ - (1/√2) |1, 1; 1, -1⟩
- |1, -1⟩ = (1/√2) |1, 0; 1, -1⟩ - (1/√2) |1, -1; 1, 0⟩
- |0, 0⟩ = (1/√3) |1, -1; 1, 1⟩ - (1/√3) |1, 0; 1, 0⟩ + (1/√3) |1, 1; 1, -1⟩
Conclusion
In this article, we have successfully expressed all nine |j, m⟩ eigenkets resulting from the addition of angular momenta j1 = 1 and j2 = 1 in terms of the |j1, m1; j2, m2⟩ basis. We employed both the ladder operator method and the concept of orthogonality to systematically derive these expressions. This process highlights the fundamental principles of angular momentum addition in quantum mechanics, which is essential for understanding the behavior of multi-particle systems and various phenomena in atomic, nuclear, and particle physics. The explicit expressions for these states provide a foundation for further calculations and analyses, such as determining transition probabilities and selection rules in quantum systems.
By understanding how to combine angular momenta, physicists can better predict and interpret the properties of complex quantum systems. The techniques described here are widely applicable and form a cornerstone of quantum mechanical calculations in many areas of physics.
