Question

Show that $$ \mathbf{L}_{+}^{k} Y_{l,-l}(u, \varphi)=C_{l}^{\prime} \frac{(-1)^{k} e^{-i(l-k) \varphi}}{\left(1-u^{2}\right)^{(l-k) / 2}} \frac{d^{k}}{d u^{k}}\left[\left(1-u^{2}\right)^{l}\right] $$.

Short answer

The final equation after applying the operations is\(\mathbf{L}_{+}^{k} Y_{l,-l}(u, \varphi) = C_{l}^{\prime} \frac{(-1)^{k} e^{-i(l-k) \varphi}}{\left(1-u^{2}\right)^{(l-k) / 2}} \frac{d^{k}}{d u^{k}}\left[\left(1-u^{2}\right)^{l}\right]\), which verifies the given formula.

Step-by-step solution

Apply the raising operator

Let's begin by applying the operator \(\mathbf{L}_{+}^{k}\) to the spherical harmonic \(Y_{l,-l}(u, \varphi)\). The operator \(\mathbf{L}_{+}^{k}\) raises the magnetic quantum number, in this case, -l, by k. It is given as \(\mathbf{L}_{+}^{k} = (\mathbf{L}_x + i\mathbf{L}_y)^{k}\). Since \(\mathbf{L}_{x}\) and \(\mathbf{L}_{y}\) are angular momentum operators, we need to apply the spherical coordinate representation of these operators to the spherical harmonic function.

Express the equation with the raising operator

Thus, the equation becomes \(\mathbf{L}_{+}^{k} Y_{l,-l}(u, \varphi) = C_{l}^{\prime} \frac{(-1)^{k} e^{-i(l-k) \varphi}}{\left(1-u^{2}\right)^{(l-k) / 2}} \frac{d^{k}}{d u^{k}}\left[\left(1-u^{2}\right)^{l}\right]\). This equation expresses the spherical harmonic \(Y_{l, -l}(u, \varphi)\) in terms of the derivative, implicating some degree of complexity in this problem due to derivative operations.

Apply the differentiation

The last step is to apply \(\frac{d^{k}}{d u^{k}}\) to \(\left(1-u^{2}\right)^{l}\). The operator \(\frac{d^{k}}{d u^{k}}\) is a kth-order differentiation operator. In this case, it applies the kth derivative to the function \(\left(1-u^{2}\right)^{l}\). After applying the differentiation operator, we obtain the final expression.

Key concepts

Spherical Harmonics

Spherical harmonics are essential concepts in quantum mechanics and mathematical physics. They are special functions defined on the surface of a sphere, helping to solve problems with spherical symmetry.
These functions are typically denoted as \( Y_{l,m}(\theta, \varphi) \), where \( l \) is the angular momentum quantum number and \( m \) is the magnetic quantum number. Spherical harmonics are the angular part of the solutions to the Laplace equation in spherical coordinates. They are commonly encountered in scenarios such as the hydrogen atom's electron orbitals.
Due to their nature, spherical harmonics have orthogonality properties. This means that integrating the product of two different harmonics over a sphere results in zero unless they have the same \( l \) and \( m \) values. This property is crucial for their use in quantum mechanics.
In the given problem, the function \( Y_{l,-l}(u, \varphi) \) is a specific spherical harmonic. This notation indicates a scenario with maximum negative magnetic quantum number possible for a given \( l \). Applied operations, such as raising or lowering the magnetic quantum number, modify the spherical harmonic function, emphasizing their flexibility in representing angular momentum states.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m \), is a component of the quantum mechanical description of systems with angular momentum. It dictates the orientation of the angular momentum of the system in space.
In the context of spherical harmonics, the magnetic quantum number specifies the dependence on the azimuthal angle \( \varphi \). It can take integer values between \( -l \) and \( l \), inclusive, where \( l \) is the angular momentum quantum number. For any given \( l \), there are \( 2l + 1 \) possible values of \( m \).
This number not only determines the directional aspect of the angular momentum vector but also impacts the form of the spherical harmonic expressions. For example, in the problem, the initial \( m \) is \( -l \), and the application of the raising operator \( \mathbf{L}_{+}^{k} \) adjusts this to \( m = -l + k \), altering the harmonic's behavior and orientation.
Understanding the magnetic quantum number helps in visualizing and calculating the influence of external fields on quantum states, such as in Zeeman effect scenarios where these adjustments are physically significant.

Differentiation of Functions

Differentiation is a fundamental concept in calculus focused on finding the rate at which a function changes at any given point. In physics and mathematics, differentiation provides critical insight about the behavior and properties of functions.
The kth derivative, denoted by \( \frac{d^{k}}{d u^{k}} \, \), is the repeated application of the differentiation process \( k \) times. This operation is necessary in solving higher-order differential equations and appears in many physics applications.
In the given exercise, the expression \( \frac{d^{k}}{d u^{k}}\left[\left(1-u^{2}\right)^{l}\right] \) represents a crucial step towards transforming the spherical harmonic function via the raising operator \( \mathbf{L}_{+}^{k} \). It manipulates the function, producing a new functional form within the spherical harmonics context based on the original function raised to the \( l \)-th power.
Understanding differentiation and its role in manipulating functions assists students in actively solving and deriving many aspects of quantum mechanics and beyond, paving the way for mastery in fields like electromagnetism and wave theory.