Question

Show that \(\mathbf{L}^{2}=\mathbf{L}_{+} \mathbf{L}_{-}+\mathbf{L}_{z}^{2}-\mathbf{L}_{z}\) and \(\mathbf{L}^{2}=\mathbf{L}_{-} \mathbf{L}_{+}+\mathbf{L}_{z}^{2}+\mathbf{L}_{z}\).

Short answer

The two equalities are obtained by substituting the definitions of the raising and lowering operators into the expression for \(\mathbf{L}^{2}\) and simplifying using the commutation relations for angular momentum operators.

Step-by-step solution

Identify the raising and lowering operators

In quantum mechanics, \(\mathbf{L}_{+}\) and \(\mathbf{L}_{-}\) are known as the raising and lowering operators, respectively. They are defined as:\(\mathbf{L}_{+} = \mathbf{L}_{x} + i\mathbf{L}_{y} \),\(\mathbf{L}_{-} = \mathbf{L}_{x} - i\mathbf{L}_{y} \).

Apply the commutation relations

Use the commutation relations for angular momentum operators:\([ \mathbf{L}_{x}, \mathbf{L}_{y} ] = i\mathbf{L}_{z}\),\([ \mathbf{L}_{y}, \mathbf{L}_{z} ] = i\mathbf{L}_{x}\),\([ \mathbf{L}_{z}, \mathbf{L}_{x} ] = i\mathbf{L}_{y}\),to simplify the expression.

Show the first equality

\(\mathbf{L}^{2}\) is defined as \(\mathbf{L}_{x}^{2}+\mathbf{L}_{y}^{2}+\mathbf{L}_{z}^{2}\). Insert the expressions for \(\mathbf{L}_{x}\) and \(\mathbf{L}_{y}\) from the definitions of \(\mathbf{L}_{+}\) and \(\mathbf{L}_{-}\) and simplify using the commutation relations. This leads to the first result \(\mathbf{L}^{2}=\mathbf{L}_{+}\mathbf{L}_{-}+\mathbf{L}_{z}^{2}-\mathbf{L}_{z}\).

Show the second equality

To prove the second result, again replace \(\mathbf{L}_{x}\) and \(\mathbf{L}_{y}\) with the definitions of \(\mathbf{L}_{-}\) and \(\mathbf{L}_{+}\) in the expression for \(\mathbf{L}^{2}\). Simplify using the commutation relations to obtain \(\mathbf{L}^{2}=\mathbf{L}_{-}\mathbf{L}_{+}+\mathbf{L}_{z}^{2}+\mathbf{L}_{z}\).

Key concepts

Angular Momentum Operators

In the realm of quantum mechanics, angular momentum is a core concept crucial for understanding rotational dynamics. Angular momentum operators denote the quantized versions of classical angular momentum components. These are usually represented by \( \mathbf{L}_x \), \( \mathbf{L}_y \), and \( \mathbf{L}_z \), corresponding to the \( x \), \( y \), and \( z \) axes, respectively. These operators are essential as they describe how systems evolve when subjected to rotational symmetry.
The combined magnitude of these operators is represented by \( \mathbf{L}^2 \), which for quantum systems is given by:

• \( \mathbf{L}^2 = \mathbf{L}_x^2 + \mathbf{L}_y^2 + \mathbf{L}_z^2 \)

Understanding this construction allows us to determine the total angular momentum of a quantum system, vital for predicting system behavior under rotational transformations. Each of the components commutes uniquely, leading us to further explore their relationships with raising and lowering operators.

Raising and Lowering Operators

Raising and lowering operators, \( \mathbf{L}_{+} \) and \( \mathbf{L}_{-} \), play a fascinating role within quantum mechanics. They are operators that change the quantum state of a system related to its angular momentum. These are specifically defined as follows:

• \( \mathbf{L}_{+} = \mathbf{L}_x + i \mathbf{L}_y \)
• \( \mathbf{L}_{-} = \mathbf{L}_x - i \mathbf{L}_y \)

The purpose of these operators is to increase or decrease the magnetic quantum number \( m \) without altering the principal quantum number \( l \), essentially "raising" or "lowering" the state's energy level. By applying \( \mathbf{L}_{+} \) to an angular momentum state, you effectively increase \( m \) by one unit, while the \( \mathbf{L}_{-} \) decreases \( m \) by one unit.
These features are integral for deriving relationships such as \( \mathbf{L}^2 = \mathbf{L}_{+} \mathbf{L}_{-} + \mathbf{L}_z^2 - \mathbf{L}_z \) and \( \mathbf{L}^2 = \mathbf{L}_{-} \mathbf{L}_{+} + \mathbf{L}_z^2 + \mathbf{L}_z \), fundamental identities in angular momentum datasets.

Commutation Relations

Commutation relations form the foundation of how quantum mechanical operators interact. For angular momentum, these relations are vital for understanding the intrinsic properties that define quantum states. The general commutation relations between angular momentum operators are:

• \( [ \mathbf{L}_x, \mathbf{L}_y ] = i \mathbf{L}_z \)
• \( [ \mathbf{L}_y, \mathbf{L}_z ] = i \mathbf{L}_x \)
• \( [ \mathbf{L}_z, \mathbf{L}_x ] = i \mathbf{L}_y \)

These equations reveal that unlike certain other operators, angular momentum operators do not commute with each other. Instead, they follow a cyclic pattern where the commutator of two components results in the third, oriented in a 90-degree transformation of angular momentum axes.
These relationships can be leveraged to simplify complex expressions within quantum mechanics, such as those found in the original exercise, effectively showcasing that through specific manipulations and the use of the raising and lowering operators, one can show \( \mathbf{L}^2 = \mathbf{L}_{+} \mathbf{L}_{-} + \mathbf{L}_z^2 - \mathbf{L}_z \) and \( \mathbf{L}^2 = \mathbf{L}_{-} \mathbf{L}_{+} + \mathbf{L}_z^2 + \mathbf{L}_z \). Understanding and applying these relationships are essential skills for students navigating the quantum mechanics landscape.