Parity Transformation Of Angular Momentum Eigenstates

In quantum mechanics, understanding the behavior of systems under various transformations is crucial for unraveling the fundamental properties of particles and fields. One such transformation is the parity transformation, which inverts the spatial coordinates. This article delves into the behavior of the eigenstates of angular momentum operators, specifically ${ L^2 }$ and ${ L_z }$, under parity transformation. We aim to demonstrate that the parity operator ${ \pi }$ acting on an eigenstate ${ |l, m\rangle }$ results in a scaled version of the same eigenstate, i.e., ${ \pi |l, m\rangle = \lambda_{l,m} |l, m\rangle }$, where ${ \lambda_{l,m}^2 = 1 }$. Furthermore, we will explore how the commutator ${ [\pi, L_\pm] }$ helps in showing that ${ \lambda_{l,m} }$ is independent of ${ m }$.

Theoretical Background

Angular Momentum Operators

Angular momentum is a fundamental concept in quantum mechanics, playing a crucial role in the description of rotational motion. The angular momentum operators, denoted as ${ L^2 }$ and ${ L_z }$, are key components in this context. The operator ${ L^2 }$ represents the square of the total angular momentum, while ${ L_z }$ represents the component of angular momentum along the z-axis. These operators obey specific commutation relations, which dictate the quantized nature of angular momentum.

The eigenstates of these operators, labeled as ${ |l, m\rangle }$, are characterized by two quantum numbers: ${ l }$ (the azimuthal quantum number) and ${ m }$ (the magnetic quantum number). The quantum number ${ l }$ determines the magnitude of the angular momentum, with eigenvalues ${ \hbar^2 l(l+1) }$, where ${ \hbar }$ is the reduced Planck constant. The quantum number ${ m }$ specifies the projection of the angular momentum onto the z-axis, with eigenvalues ${ m\hbar }$. Here, ${ l }$ is a non-negative integer (0, 1, 2, ...), and ${ m }$ can take integer values from ${ -l }$ to ${ +l }$.

Parity Operator

The parity operator ${ \pi }$ performs spatial inversion, transforming the position vector ${ \mathbf{r} }$ to ${ -\mathbf{r} }$. This operation is fundamental in physics, particularly in the study of symmetries and conservation laws. The parity operator is a linear operator, and its action on a wavefunction ${ \psi(\mathbf{r}) }$ is defined as:

${ \pi \psi(\mathbf{r}) = \psi(-\mathbf{r}) }$

Applying the parity operator twice results in the original state, which means ${ \pi^2 = I }$, where ${ I }$ is the identity operator. This property implies that the eigenvalues of the parity operator are ${ \pm 1 }$, corresponding to states with even and odd parity, respectively. Functions that remain unchanged under parity transformation are said to have even parity, while those that change sign have odd parity.

Commutation Relations

Commutation relations play a crucial role in quantum mechanics, providing insights into the compatibility of different operators and their corresponding observables. The commutator of two operators ${ A }$ and ${ B }$ is defined as:

${ [A, B] = AB - BA }$

If the commutator of two operators is zero, it implies that the corresponding observables can be measured simultaneously with arbitrary precision. Conversely, if the commutator is non-zero, it indicates that the observables are incompatible, and there is an inherent uncertainty in their simultaneous measurement.

Parity Transformation of Angular Momentum Eigenstates

Eigenvalues of Parity Operator

We begin by considering the action of the parity operator ${ \pi }$ on the eigenstates ${ |l, m\rangle }$ of the angular momentum operators ${ L^2 }$ and ${ L_z }$. We aim to show that the eigenstates ${ |l, m\rangle }$ are also eigenstates of the parity operator, up to a phase factor. Mathematically, this is expressed as:

${ \pi |l, m\rangle = \lambda_{l,m} |l, m\rangle, }$

where ${ \lambda_{l,m} }$ is the eigenvalue of the parity operator corresponding to the state ${ |l, m\rangle }$.

To determine the possible values of ${ \lambda_{l,m} }$, we apply the parity operator twice:

${ \pi^2 |l, m\rangle = \pi (\lambda_{l,m} |l, m\rangle) = \lambda_{l,m} \pi |l, m\rangle = \lambda_{l,m}^2 |l, m\rangle. }$

Since ${ \pi^2 = I }$, we have:

${ \lambda_{l,m}^2 |l, m\rangle = |l, m\rangle, }$

which implies that ${ \lambda_{l,m}^2 = 1 }$. Therefore, the possible eigenvalues of the parity operator are ${ \lambda_{l,m} = \pm 1 }$. This result indicates that the eigenstates ${ |l, m\rangle }$ have definite parity, being either even (${ \lambda_{l,m} = 1 }$) or odd (${ \lambda_{l,m} = -1 }$).

Commutator of Parity Operator and Angular Momentum Ladder Operators

Next, we investigate the commutator of the parity operator ${ \pi }$ with the angular momentum ladder operators ${ L_+ }$ and ${ L_- }$. The ladder operators are defined as:

${ L_+ = L_x + iL_y, \quad L_- = L_x - iL_y, }$

where ${ L_x }$ and ${ L_y }$ are the components of the angular momentum operator along the x and y axes, respectively.

The action of the parity operator on the position and momentum operators is given by:

${ \pi \mathbf{r} \pi^{-1} = -\mathbf{r}, \quad \pi \mathbf{p} \pi^{-1} = -\mathbf{p}, }$

where ${ \mathbf{p} }$ is the momentum operator. Using the definition of the angular momentum operator ${ \mathbf{L} = \mathbf{r} \times \mathbf{p} }$, we can determine the transformation of ${ \mathbf{L} }$ under parity:

${ \pi \mathbf{L} \pi^{-1} = \pi (\mathbf{r} \times \mathbf{p}) \pi^{-1} = (\pi \mathbf{r} \pi^{-1}) \times (\pi \mathbf{p} \pi^{-1}) = (-\mathbf{r}) \times (-\mathbf{p}) = \mathbf{r} \times \mathbf{p} = \mathbf{L}. }$

This result shows that the angular momentum operator ${ \mathbf{L} }$ is invariant under parity transformation. Consequently, the components ${ L_x }$, ${ L_y }$, and ${ L_z }$ also transform as:

${ \pi L_x \pi^{-1} = L_x, \quad \pi L_y \pi^{-1} = L_y, \quad \pi L_z \pi^{-1} = L_z. }$

Now, we can evaluate the commutator of ${ \pi }$ with the ladder operators. For ${ L_+ }$:

${ [\pi, L_+] = \pi L_+ - L_+ \pi = \pi (L_x + iL_y) - (L_x + iL_y) \pi = (\pi L_x - L_x \pi) + i(\pi L_y - L_y \pi). }$

Using the transformation properties of ${ L_x }$ and ${ L_y }$, we have:

${ \pi L_x = L_x \pi, \quad \pi L_y = L_y \pi, }$

which implies that ${ [\pi, L_x] = 0 }$ and ${ [\pi, L_y] = 0 }$. Therefore,

${ [\pi, L_+] = 0. }$

Similarly, for ${ L_- }$:

${ [\pi, L_-] = \pi L_- - L_- \pi = \pi (L_x - iL_y) - (L_x - iL_y) \pi = (\pi L_x - L_x \pi) - i(\pi L_y - L_y \pi) = 0. }$

Thus, we find that the parity operator commutes with both ladder operators:

${ [\pi, L_+] = 0, \quad [\pi, L_-] = 0. }$

Independence of ${ \lambda_{l,m} }$ on ${ m }$

The commutation relations ${ [\pi, L_+] = 0 }$ and ${ [\pi, L_-] = 0 }$ have significant implications for the eigenvalues ${ \lambda_{l,m} }$. We will now demonstrate that ${ \lambda_{l,m} }$ is independent of the quantum number ${ m }$. Let's consider the action of ${ L_+ }$ on the eigenstate ${ |l, m\rangle }$:

${ L_+ |l, m\rangle = C_{l,m} |l, m+1\rangle, }$

where ${ C_{l,m} }$ is a normalization constant. Applying the parity operator to both sides:

${ \pi L_+ |l, m\rangle = \pi (C_{l,m} |l, m+1\rangle) = C_{l,m} \pi |l, m+1\rangle = C_{l,m} \lambda_{l, m+1} |l, m+1\rangle. }$

Since ${ [\pi, L_+] = 0 }$, we also have:

${ \pi L_+ |l, m\rangle = L_+ \pi |l, m\rangle = L_+ (\lambda_{l,m} |l, m\rangle) = \lambda_{l,m} L_+ |l, m\rangle = \lambda_{l,m} C_{l,m} |l, m+1\rangle. }$

Comparing the two expressions for ${ \pi L_+ |l, m\rangle }$, we obtain:

${ C_{l,m} \lambda_{l, m+1} |l, m+1\rangle = \lambda_{l,m} C_{l,m} |l, m+1\rangle, }$

which implies:

${ \lambda_{l, m+1} = \lambda_{l,m}. }$

This result shows that the eigenvalue ${ \lambda_{l,m} }$ is the same for states with adjacent ${ m }$ values. We can apply a similar argument using the operator ${ L_- }$:

${ L_- |l, m\rangle = D_{l,m} |l, m-1\rangle, }$

where ${ D_{l,m} }$ is another normalization constant. Following the same steps, we find:

${ \lambda_{l, m-1} = \lambda_{l,m}. }$

Combining these results, we conclude that ${ \lambda_{l,m} }$ is independent of ${ m }$. Therefore, we can write ${ \lambda_{l,m} }$ simply as ${ \lambda_l }$, which depends only on the quantum number ${ l }$.

Determination of ${ \lambda_l }$

To determine the value of ${ \lambda_l }$, we need to consider the behavior of spherical harmonics under parity transformation. Spherical harmonics, denoted as ${ Y_{l,m}(\theta, \phi) }$, are the angular solutions to the Schrödinger equation in spherical coordinates and are closely related to the eigenstates ${ |l, m\rangle }$. The parity transformation in spherical coordinates corresponds to:

${ \mathbf{r} \rightarrow -\mathbf{r} \implies (r, \theta, \phi) \rightarrow (r, \pi - \theta, \phi + \pi). }$

The action of the parity operator on spherical harmonics is given by:

${ \pi Y_{l,m}(\theta, \phi) = Y_{l,m}(\pi - \theta, \phi + \pi) = (-1)^l Y_{l,m}(\theta, \phi). }$

This result shows that spherical harmonics have a definite parity, with the eigenvalue ${ \lambda_l = (-1)^l }$. Therefore, the eigenstates ${ |l, m\rangle }$ of the angular momentum operators transform under parity as:

${ \pi |l, m\rangle = (-1)^l |l, m\rangle. }$

This final result demonstrates that the parity of the eigenstates ${ |l, m\rangle }$ depends solely on the azimuthal quantum number ${ l }$. States with even ${ l }$ have even parity, while states with odd ${ l }$ have odd parity.

In summary, we have shown that the eigenstates ${ |l, m\rangle }$ of the angular momentum operators ${ L^2 }$ and ${ L_z }$ are also eigenstates of the parity operator ${ \pi }$, with eigenvalues ${ \lambda_{l,m} }$ satisfying ${ \lambda_{l,m}^2 = 1 }$. By utilizing the commutation relations ${ [\pi, L_+] = 0 }$ and ${ [\pi, L_-] = 0 }$, we demonstrated that ${ \lambda_{l,m} }$ is independent of the magnetic quantum number ${ m }$, depending only on the azimuthal quantum number ${ l }$. Finally, we established that the parity eigenvalue is given by ${ \lambda_l = (-1)^l }$, indicating that the parity of the eigenstates is determined by the value of ${ l }$. This analysis provides valuable insights into the symmetry properties of angular momentum eigenstates and their behavior under spatial inversion, which is fundamental in various areas of quantum mechanics and physics.