Solving the Angular Momentum Operator for j=1

These eigenvectors are also orthonormal, meaning they are perpendicular to each other and have a magnitude of 1.In summary, to find the matrix representation of the angular momentum operator \vec{J_{y}}, we use the fact that it is the sum of the x and y components of angular momentum and the spin angular momentum. We can then diagonalize this matrix to find its eigenvalues and eigenvectors, which can be used to solve for the orthonormal eig
  • #1
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Homework Statement



Consider the angular momentum operator [itex]\vec{J_{y}}[/itex] in the subspace for which j=1. Write down the matrix for this operator in the usual basis (where [itex]J^{2}[/itex] and [itex]J_{z}[/itex] are diagonal). Diagonalize the matrix and find the eigenvalues and orthonormal eigenvectors.

Homework Equations



[itex]\vec{J} = \vec{L} + \vec{S}[/itex] (total angular momentum)

The Attempt at a Solution



I know J is the sum of angular momentum, L, and spin angular momentum, S, but how to we get it in matrix form? Spin would just be [itex]\hbar / 2[/itex] times the y Pauli matrix... but how do we express L in matrix form? Also, I really don't understand how to obtain eigenvalues and eigenvectors... Could someone go through the problem for me? Textbook is Griffiths. Thanks in advance.
 
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  • #2


Hello there,

First, let's start by writing down the matrix representation of the angular momentum operator in the usual basis:

\vec{J_{y}} = \begin{pmatrix} 0 & -i\hbar & 0 \\ i\hbar & 0 & -i\hbar \\ 0 & i\hbar & 0 \end{pmatrix}

To obtain this matrix, we use the fact that the raising and lowering operators for the y component of angular momentum are given by J_{+} = J_{x} + iJ_{y} and J_{-} = J_{x} - iJ_{y}, where J_{x} and J_{y} are the usual x and y components of angular momentum. Using these operators, we can write the matrix representation of J_{y} as:

J_{y} = \frac{\hbar}{2}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}

Next, we can diagonalize this matrix by finding its eigenvalues and eigenvectors. The eigenvalues of this matrix are given by the diagonal elements, which are all zero. This is because the y component of angular momentum, J_{y}, commutes with the square of the total angular momentum operator, J^{2}. Therefore, the eigenvalues of J_{y} are the same as those of J^{2}.

To find the eigenvectors, we can use the fact that the eigenvalue equation for J_{y} is given by J_{y}\psi = \lambda\psi, where \lambda is an eigenvalue and \psi is an eigenvector. Substituting the matrix representation of J_{y} into this equation, we get:

\frac{\hbar}{2}\begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \lambda \begin{pmatrix} a \\ b \\ c \end{pmatrix}

Solving this equation, we get the following three eigenvectors:

\psi_{1} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix},
 

1. What is the angular momentum operator for j=1?

The angular momentum operator for j=1 is a mathematical representation of the physical quantity of angular momentum, which is a measure of the rotation of a system. It is represented by the symbol J and is defined as J = L + S, where L is the orbital angular momentum and S is the spin angular momentum.

2. How is the angular momentum operator for j=1 solved?

The angular momentum operator for j=1 can be solved using the standard mathematical methods of quantum mechanics. This involves using the commutation relationships between the operators for L and S, as well as the eigenvalue equations for each operator. The final solution will involve a set of eigenvalues and eigenvectors that represent the possible states of the system.

3. What are the physical implications of solving the angular momentum operator for j=1?

Solving the angular momentum operator for j=1 allows us to better understand the behavior of systems with angular momentum. It can help us predict the possible states and outcomes of such systems and is a crucial tool in the study of quantum mechanics.

4. Are there any known applications of solving the angular momentum operator for j=1?

Yes, there are many applications of solving the angular momentum operator for j=1. It is used in fields such as atomic and molecular physics, nuclear physics, and solid-state physics. It is also used in technology, such as in the development of magnetic resonance imaging (MRI) machines.

5. How does the angular momentum operator for j=1 relate to other quantum mechanical operators?

The angular momentum operator for j=1 is closely related to other quantum mechanical operators, such as the position and momentum operators. It is also related to the energy operator and the total angular momentum operator. These relationships are described by the commutation relationships between the operators, which are fundamental to the principles of quantum mechanics.

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