Quantum Addition Of Angular Momenta J1=1 And J2=1 Eigenstate Derivation

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Introduction

In the captivating domain of quantum mechanics, the concept of angular momentum stands as a cornerstone, playing a pivotal role in deciphering the behavior of atomic and subatomic entities. This intrinsic property, a measure of an object's rotational inertia and angular velocity, dictates the spatial orientation and interactions of particles within the quantum realm. When confronted with systems comprising multiple sources of angular momentum, such as electrons orbiting an atomic nucleus or nucleons within an atomic nucleus, the quantum mechanical framework necessitates a precise methodology for combining these individual angular momenta into a total angular momentum. This intricate process, known as the addition of angular momenta, unveils a tapestry of quantum states, each characterized by its unique angular momentum properties.

Delving into the intricacies of angular momentum addition, we embark on a journey to unravel the quantum mechanical underpinnings that govern the behavior of composite systems. Our focus lies on a specific scenario the addition of two angular momenta, denoted as j1 = 1 and j2 = 1. This seemingly simple case serves as a gateway to understanding the more complex scenarios encountered in atomic and nuclear physics. By meticulously dissecting this example, we aim to illuminate the fundamental principles and techniques employed in the quantum mechanical treatment of angular momentum addition. We will learn how to construct the resultant eigenstates, labeled by their total angular momentum j and its projection m, and express them as linear combinations of the product states formed from the individual angular momentum eigenstates. This process involves employing sophisticated mathematical tools such as the ladder operator method and recursion relations, which will be meticulously explored and applied to this specific case.

The central objective of this exploration is to derive explicit expressions for all nine eigenstates, denoted as |j, m>, that arise from the addition of angular momenta j1 = 1 and j2 = 1. These eigenstates correspond to the possible values of the total angular momentum quantum number j, which can be 2, 1, or 0, and the corresponding values of the magnetic quantum number m, which ranges from -j to +j in integer steps. To achieve this, we will leverage the powerful techniques of ladder operators and recursion relations, providing a comprehensive and pedagogical approach to constructing the eigenstates and expressing them in terms of the basis states |j1, m1; j2, m2>. This endeavor will not only solidify our understanding of angular momentum addition but also provide valuable insights into the structure and behavior of quantum systems.

Theoretical Foundation

Before we embark on the practical task of constructing the eigenstates, let us lay the theoretical groundwork by revisiting the fundamental principles of angular momentum in quantum mechanics. In the quantum realm, angular momentum is not a classical vector but rather an operator whose components satisfy specific commutation relations. These commutation relations dictate the quantized nature of angular momentum, meaning that its magnitude and direction can only take on discrete values. The square of the total angular momentum operator, denoted as J^2, and its projection along a chosen axis, typically the z-axis denoted as J_z, are the key operators that define the angular momentum state of a quantum system. The eigenvalues of these operators are quantized and characterized by the quantum numbers j and m, respectively.

The quantum number j, often referred to as the total angular momentum quantum number, determines the magnitude of the angular momentum. It can take on non-negative integer or half-integer values, such as 0, 1/2, 1, 3/2, and so on. For a given value of j, the square of the total angular momentum is given by ฤง^2j(j+1), where ฤง is the reduced Planck constant. The magnetic quantum number m, on the other hand, specifies the projection of the angular momentum along the chosen axis. For a given value of j, m can take on 2j+1 values, ranging from -j to +j in integer steps. These values correspond to the different possible orientations of the angular momentum vector in space. The eigenvalue of the J_z operator is given by ฤงm.

When dealing with multiple sources of angular momentum, such as two particles with individual angular momenta j1 and j2, the total angular momentum of the system is obtained by vectorially adding the individual angular momenta. However, in the quantum realm, this addition is not as straightforward as in classical mechanics. The individual angular momenta j1 and j2 can combine to form a total angular momentum j that can take on a range of values, from |j1 - j2| to j1 + j2 in integer steps. This is known as the triangle inequality, which dictates the possible values of the total angular momentum. For each value of j, there are 2j+1 possible values of the magnetic quantum number m, ranging from -j to +j. The process of combining individual angular momenta to form the total angular momentum is known as the addition of angular momenta, and it is a fundamental concept in quantum mechanics.

The quantum states that describe the individual angular momenta, denoted as |j1, m1> and |j2, m2>, form a complete orthonormal basis for the individual angular momentum spaces. Similarly, the quantum states that describe the total angular momentum, denoted as |j, m>, form a complete orthonormal basis for the total angular momentum space. The key challenge in the addition of angular momenta is to express the total angular momentum eigenstates |j, m> as linear combinations of the product states formed from the individual angular momentum eigenstates |j1, m1; j2, m2>. The coefficients in this linear combination are known as the Clebsch-Gordan coefficients, and they encode the quantum mechanical rules for combining angular momenta. These coefficients play a crucial role in calculating transition probabilities and selection rules in atomic and nuclear physics.

Methods for Eigenstate Construction

To explicitly construct the eigenstates |j, m> in terms of the product states |j1, m1; j2, m2>, we can employ two powerful techniques: the ladder operator method and recursion relations. Both methods leverage the commutation relations of angular momentum operators and the properties of eigenstates to systematically build up the eigenstates. The ladder operator method is particularly useful for finding the highest weight state, which is the state with the maximum value of m for a given j. Once the highest weight state is known, the other eigenstates with the same j can be obtained by repeatedly applying the lowering operator. Recursion relations, on the other hand, provide a more general approach for relating different Clebsch-Gordan coefficients, allowing us to calculate them systematically.

The ladder operator method hinges on the introduction of raising and lowering operators, denoted as J+ and J-, respectively. These operators are defined as linear combinations of the angular momentum operators J_x and J_y, specifically J+ = J_x + iJ_y and J- = J_x - iJ_y, where i is the imaginary unit. The crucial property of these operators is that they raise or lower the magnetic quantum number m by one unit, while leaving the total angular momentum quantum number j unchanged. That is, when the raising operator J+ acts on an eigenstate |j, m>, it produces an eigenstate with the same j but with m increased by one, i.e., J+|j, m> โˆ |j, m+1>. Similarly, when the lowering operator J- acts on an eigenstate |j, m>, it produces an eigenstate with the same j but with m decreased by one, i.e., J-|j, m> โˆ |j, m-1>. These operators act as ladders, stepping us up or down the ladder of m values for a given j.

The highest weight state, denoted as |j, j>, is the eigenstate with the maximum value of m, which is equal to j. This state is annihilated by the raising operator, meaning that J+|j, j> = 0. This property provides a crucial starting point for constructing the eigenstates. We can start by finding the highest weight state and then repeatedly apply the lowering operator to generate the other eigenstates with the same j. The normalization constants that appear when applying the raising and lowering operators can be determined by ensuring that the eigenstates are normalized to unity.

Recursion relations provide an alternative approach for calculating the Clebsch-Gordan coefficients. These relations connect different Clebsch-Gordan coefficients, allowing us to calculate them systematically. The recursion relations are derived from the commutation relations of the angular momentum operators and the properties of the eigenstates. By applying the raising and lowering operators to both sides of the equation that expresses the total angular momentum eigenstates in terms of the product states, we can obtain a set of equations that relate the Clebsch-Gordan coefficients. These equations can then be solved iteratively to determine the coefficients.

Both the ladder operator method and recursion relations are powerful tools for constructing the eigenstates in the addition of angular momenta. The ladder operator method is particularly useful for finding the highest weight state, while recursion relations provide a more general approach for calculating the Clebsch-Gordan coefficients. In the following sections, we will apply these methods to the specific case of adding angular momenta j1 = 1 and j2 = 1.

Application to j1=1 and j2=1

Now, let us apply the theoretical framework and the methods discussed to the specific case of adding angular momenta j1 = 1 and j2 = 1. This scenario is of particular interest as it arises in various physical contexts, such as the coupling of the orbital angular momentum and spin angular momentum of an electron in an atom, or the coupling of the spins of two nucleons in a nucleus. The individual angular momenta, j1 = 1 and j2 = 1, each correspond to three possible values of the magnetic quantum number, m1 and m2, which are -1, 0, and +1. Therefore, there are a total of 3 x 3 = 9 possible product states |j1, m1; j2, m2>, which form a basis for the combined angular momentum space.

The possible values of the total angular momentum quantum number j are determined by the triangle inequality, which states that |j1 - j2| โ‰ค j โ‰ค j1 + j2. In this case, we have |1 - 1| โ‰ค j โ‰ค 1 + 1, which gives us the possible values of j as 0, 1, and 2. For each value of j, there are 2j+1 possible values of the magnetic quantum number m, ranging from -j to +j. Therefore, for j = 2, there are 5 possible values of m: -2, -1, 0, 1, and 2. For j = 1, there are 3 possible values of m: -1, 0, and 1. And for j = 0, there is only 1 possible value of m: 0. In total, we have 5 + 3 + 1 = 9 states, which matches the number of product states, as expected.

Our goal is to express the eigenstates |j, m> for j = 2, 1, and 0 in terms of the product states |j1, m1; j2, m2>. We will employ the ladder operator method and recursion relations to achieve this. Let us start by finding the highest weight state for each value of j. For j = 2, the highest weight state is |2, 2>. This state must have m = m1 + m2 = 2, which can only be achieved if m1 = 1 and m2 = 1. Therefore, the highest weight state |2, 2> is simply the product state |1, 1; 1, 1>. This gives us our first eigenstate:

|2, 2> = |1, 1; 1, 1>

Now, we can apply the lowering operator J- to this state to generate the other eigenstates with j = 2. The lowering operator acts on the total angular momentum state as J-|j, m> = ฤงโˆš(j(j+1) - m(m-1))|j, m-1>. Applying this to |2, 2>, we get:

J-|2, 2> = ฤงโˆš(2(2+1) - 2(2-1))|2, 1> = ฤงโˆš4|2, 1>

On the other hand, the lowering operator acts on the product states as J- = J1- + J2-, where J1- and J2- are the lowering operators for the individual angular momenta. Applying this to |1, 1; 1, 1>, we get:

J-|1, 1; 1, 1> = J1-|1, 1; 1, 1> + |1, 1; J2-|1, 1>

Using the action of the lowering operator on the individual angular momentum states, J-|j, m> = ฤงโˆš(j(j+1) - m(m-1))|j, m-1>, we get:

J1-|1, 1; 1, 1> = ฤงโˆš2|1, 0; 1, 1> J2-|1, 1; 1, 1> = ฤงโˆš2|1, 1; 1, 0>

Therefore, J-|1, 1; 1, 1> = ฤงโˆš2(|1, 0; 1, 1> + |1, 1; 1, 0>). Equating the two expressions for J-|2, 2>, we get:

ฤงโˆš4|2, 1> = ฤงโˆš2(|1, 0; 1, 1> + |1, 1; 1, 0>)

Dividing by ฤงโˆš4, we obtain the eigenstate |2, 1>:

|2, 1> = (1/โˆš2)(|1, 0; 1, 1> + |1, 1; 1, 0>)

We can continue this process, applying the lowering operator repeatedly to generate the other eigenstates with j = 2. Similarly, we can find the highest weight states for j = 1 and j = 0 and apply the lowering operator to generate the other eigenstates. Alternatively, we can use recursion relations to calculate the Clebsch-Gordan coefficients directly. The final result will be a complete set of nine eigenstates |j, m> expressed in terms of the product states |j1, m1; j2, m2>. These eigenstates provide a complete description of the possible quantum states that arise from the addition of angular momenta j1 = 1 and j2 = 1.

Expressing all (nine) {j, m} Eigenkets

Continuing the process outlined above, we can systematically derive all nine eigenstates |j, m> for the addition of angular momenta j1 = 1 and j2 = 1. We have already found the eigenstates |2, 2> and |2, 1>. Now, let's apply the lowering operator again to |2, 1> to find |2, 0>:

J-|2, 1> = ฤงโˆš(2(2+1) - 1(1-1))|2, 0> = ฤงโˆš6|2, 0>

Applying the lowering operator to the expression for |2, 1>, we get:

J-|(1/โˆš2)(|1, 0; 1, 1> + |1, 1; 1, 0>)| = (1/โˆš2)(J-|1, 0; 1, 1> + J-|1, 1; 1, 0>)

= (1/โˆš2)(J1-|1, 0; 1, 1> + |1, 0; J2-|1, 1> + J1-|1, 1; 1, 0> + |1, 1; J2-|1, 0>)

= (1/โˆš2)(ฤงโˆš2|1, -1; 1, 1> + ฤงโˆš2|1, 0; 1, 0> + ฤงโˆš2|1, 0; 1, 0> + ฤงโˆš2|1, 1; 1, -1>)

= ฤง(|1, -1; 1, 1> + 2|1, 0; 1, 0> + |1, 1; 1, -1>)/โˆš2

Equating the two expressions for J-|2, 1>, we get:

ฤงโˆš6|2, 0> = ฤง(|1, -1; 1, 1> + 2|1, 0; 1, 0> + |1, 1; 1, -1>)/โˆš2

Dividing by ฤงโˆš6, we obtain the eigenstate |2, 0>:

|2, 0> = (1/โˆš6)(|1, -1; 1, 1> + 2|1, 0; 1, 0> + |1, 1; 1, -1>)

We can continue this process to find |2, -1> and |2, -2>:

|2, -1> = (1/โˆš2)(|1, -1; 1, 0> + |1, 0; 1, -1>) |2, -2> = |1, -1; 1, -1>

Now, let's move on to the j = 1 states. The highest weight state for j = 1 is |1, 1>. This state must be orthogonal to |2, 1>, which we have already found. We can write a general linear combination of product states with m = 1:

|1, 1> = a|1, 0; 1, 1> + b|1, 1; 1, 0>

To be orthogonal to |2, 1> = (1/โˆš2)(|1, 0; 1, 1> + |1, 1; 1, 0>), we must have:

<2, 1|1, 1> = (1/โˆš2)(a + b) = 0

This implies that a = -b. We also require the state to be normalized, so a^2 + b^2 = 1. This gives us a = 1/โˆš2 and b = -1/โˆš2 (or vice versa). Choosing a = 1/โˆš2 and b = -1/โˆš2, we get:

|1, 1> = (1/โˆš2)(|1, 0; 1, 1> - |1, 1; 1, 0>)

Applying the lowering operator to this state, we find |1, 0>:

|1, 0> = (1/โˆš2)(|1, -1; 1, 1> - |1, 1; 1, -1>)

And applying the lowering operator again, we find |1, -1>:

|1, -1> = (1/โˆš2)(|1, 0; 1, -1> - |1, -1; 1, 0>)

Finally, we consider the j = 0 state. The only state with j = 0 is |0, 0>. This state must be orthogonal to both |2, 0> and |1, 0>. We can write a general linear combination of product states with m = 0:

|0, 0> = c|1, -1; 1, 1> + d|1, 0; 1, 0> + e|1, 1; 1, -1>

To be orthogonal to |2, 0> = (1/โˆš6)(|1, -1; 1, 1> + 2|1, 0; 1, 0> + |1, 1; 1, -1>), we must have:

(1/โˆš6)(c + 2d + e) = 0

To be orthogonal to |1, 0> = (1/โˆš2)(|1, -1; 1, 1> - |1, 1; 1, -1>), we must have:

(1/โˆš2)(c - e) = 0

This implies that c = e. Substituting this into the first equation, we get:

(1/โˆš6)(2c + 2d) = 0

This implies that d = -c. We also require the state to be normalized, so c^2 + d^2 + e^2 = 1. This gives us c^2 + c^2 + c^2 = 1, or 3c^2 = 1. This gives us c = 1/โˆš3, d = -1/โˆš3, and e = 1/โˆš3. Therefore,

|0, 0> = (1/โˆš3)(|1, -1; 1, 1> - |1, 0; 1, 0> + |1, 1; 1, -1>)

We have now expressed all nine eigenstates |j, m> in terms of the product states |j1, m1; j2, m2>. These eigenstates provide a complete description of the possible quantum states that arise from the addition of angular momenta j1 = 1 and j2 = 1.

Summary of Eigenkets

Here is a summary of all nine eigenkets |j, m> expressed in terms of the product states |j1, m1; j2, m2> for the addition of angular momenta j1 = 1 and j2 = 1:

  • j = 2 states:
    • |2, 2> = |1, 1; 1, 1>
    • |2, 1> = (1/โˆš2)(|1, 0; 1, 1> + |1, 1; 1, 0>)
    • |2, 0> = (1/โˆš6)(|1, -1; 1, 1> + 2|1, 0; 1, 0> + |1, 1; 1, -1>)
    • |2, -1> = (1/โˆš2)(|1, -1; 1, 0> + |1, 0; 1, -1>)
    • |2, -2> = |1, -1; 1, -1>
  • j = 1 states:
    • |1, 1> = (1/โˆš2)(|1, 0; 1, 1> - |1, 1; 1, 0>)
    • |1, 0> = (1/โˆš2)(|1, -1; 1, 1> - |1, 1; 1, -1>)
    • |1, -1> = (1/โˆš2)(|1, 0; 1, -1> - |1, -1; 1, 0>)
  • j = 0 states:
    • |0, 0> = (1/โˆš3)(|1, -1; 1, 1> - |1, 0; 1, 0> + |1, 1; 1, -1>)

These expressions provide a complete description of the quantum states that arise from the addition of angular momenta j1 = 1 and j2 = 1. They are essential for understanding the behavior of quantum systems with multiple sources of angular momentum, such as atoms and nuclei. The Clebsch-Gordan coefficients, which are the coefficients in these linear combinations, encode the quantum mechanical rules for combining angular momenta. They play a crucial role in calculating transition probabilities and selection rules in atomic and nuclear physics.

Conclusion

In this comprehensive exploration, we have delved into the intricate world of angular momentum addition, focusing on the specific case of combining two angular momenta, j1 = 1 and j2 = 1. We have meticulously dissected the theoretical underpinnings of angular momentum in quantum mechanics, emphasizing the quantized nature of angular momentum and the role of commutation relations. Furthermore, we have elucidated the fundamental principles governing the addition of angular momenta, including the triangle inequality and the concept of Clebsch-Gordan coefficients. Through the application of the powerful ladder operator method and the utilization of recursion relations, we have successfully constructed all nine eigenstates |j, m>, expressing them as linear combinations of the product states |j1, m1; j2, m2>. These eigenstates, meticulously derived and summarized, offer a complete and insightful portrayal of the quantum states that emerge from the amalgamation of angular momenta j1 = 1 and j2 = 1.

This endeavor has not only illuminated the mathematical techniques involved in angular momentum addition but also underscored the profound physical implications of this process. The eigenstates derived in this analysis serve as the building blocks for understanding the behavior of a wide array of quantum systems, ranging from atoms and molecules to atomic nuclei and elementary particles. The Clebsch-Gordan coefficients, which meticulously quantify the contributions of individual angular momentum states to the total angular momentum eigenstates, play a pivotal role in predicting transition probabilities and elucidating selection rules that govern quantum transitions. The detailed understanding of angular momentum addition, as exemplified in this exploration, provides a robust foundation for unraveling the intricacies of quantum phenomena and their manifestation in the physical world.

Moreover, the insights gained from this specific example can be readily extended to more complex scenarios involving the addition of multiple angular momenta or the coupling of angular momentum with other quantum mechanical degrees of freedom. The techniques and concepts presented here serve as a versatile toolkit for tackling a diverse range of problems in quantum mechanics, empowering researchers and students alike to delve deeper into the fascinating realm of quantum phenomena. The exploration of angular momentum addition stands as a testament to the elegance and power of quantum mechanics, providing a framework for understanding the fundamental properties of matter and the forces that govern their interactions.