Question

Show that the selection rule for the two-dimensional rotor in the dipole approximation is \(\Delta m_{l}=\pm 1 .\) Use \(A_{+\phi} e^{i m_{1} \phi}\) and \(A^{\prime}+\phi e^{i m_{2} \phi}\) for the initial and final states of the rotor and \(\mu \cos \phi\) as the dipole moment element.

Short answer

To show that the selection rule for the two-dimensional rotor in the dipole approximation is \(\Delta m_{l}=\pm 1\), we first write down the expression for the matrix element of the dipole moment operator and substitute the initial and final states into it. Then, we simplify the expression, use the complex exponential representation for \(\cos\phi\), and split the integral into two separate integrals. Solving the integral and determining the conditions for nonzero matrix elements, we find that the selection rule is \(\Delta m_{l}= m_{2}-m_{1}=\pm1\).

Step-by-step solution

Write down the expression for the matrix element of the dipole moment operator

The matrix element is given by: \[ \langle A'_{+\phi}e^{im_{2}\phi} | \mu \cos \phi | A_{+\phi}e^{im_{1}\phi} \rangle \]

Evaluate the matrix element using the given initial and final states

Substitute the initial and final states into the matrix element, and integrate over the azimuthal angle \(\phi\) from \(0\) to \(2 \pi\): \[ \int_{0}^{2\pi} A'_{+\phi}e^{-im_{2}\phi} \mu \cos \phi A_{+\phi}e^{im_{1}\phi} d\phi \]

Simplify the expression

Combine the exponential terms and simplify the expression: \[ \mu A'_{+\phi}A_{+\phi} \int_{0}^{2\pi} e^{i(m_{1}-m_{2})\phi}\cos \phi d\phi \]

Use the complex exponential representation for \(\cos\phi\)

Express \(\cos \phi\) using the Euler formula: \[ \cos \phi = \frac{1}{2} \left(e^{i\phi} + e^{-i\phi}\right) \] Then, the integral becomes: \[ \frac{\mu}{2} A'_{+\phi}A_{+\phi} \int_{0}^{2\pi} e^{i(m_{1}-m_{2}-1)\phi} + e^{i(m_{1}-m_{2}+1)\phi} d\phi \]

Solve the integral

We can solve this integral by splitting it into two separate integrals: \begin{align*} \frac{\mu}{2} A'_{+\phi}A_{+\phi} \left(\int_{0}^{2\pi} e^{i(m_{1}-m_{2}-1)\phi} d\phi + \int_{0}^{2\pi} e^{i(m_{1}-m_{2}+1)\phi} d\phi\right) \end{align*} Using the integral formula: \[ \int_{0}^{2 \pi} e^{in\phi} d\phi = \begin{cases} 2\pi, & \text{if \(n = 0\)} \\ 0, & \text{otherwise} \end{cases} \] The matrix element becomes: \[ \frac{\mu}{2} A'_{+\phi}A_{+\phi}(2\pi \delta_{m_{1}-m_{2}-1,0} + 2\pi \delta_{m_{1}-m_{2}+1,0}) \]

Determine the selection rules for nonzero matrix elements

The matrix element is nonzero only if either \(m_{1}-m_{2}-1 = 0\) or \(m_{1}-m_{2}+1 = 0\). Thus, the selection rule is given by: \[ \Delta m_{l} = m_{2}-m_{1} = \pm 1 \] This shows that the selection rule for the two-dimensional rotor in the dipole approximation is \(\Delta m_{l}=\pm 1\).

Key concepts

Two-dimensional Rotor

In quantum mechanics, the concept of a rotor is often used to explore rotational dynamics. A **two-dimensional rotor** is a simpler model because it only allows rotation in a plane.
It is an idealized system where a particle rotates freely in a circular path, constrained to two dimensions.
Here, each state of the rotor can be described using a wavefunction, particularly using azimuthal quantum numbers. **Azimuthal quantum numbers**, such as \(m_l\) in our exercise, determine the angular momentum of a particle around a fixed axis.
This concept is vital as it sets the stage for understanding how particle rotations differ in distinct quantum states.
In practice, such models help in elucidating phenomena in planar molecules or systems, where rotation is prominent. Calculating transitions between states in these rotors often involves understanding the selection rules, which tell us what transitions are allowed between quantum states in a physical process.

Dipole Moment

The **dipole moment** represents the separation of positive and negative charges in a system. For a two-dimensional rotor, it helps in describing how the charge distribution interacts with an external field or another dipole moment.
This property is essential in studying how molecules interact with electric fields and is heavily involved in spectroscopy.
In the context of this exercise, the dipole moment element expressed as \(\mu \cos \phi\) plays a pivotal role in determining the matrix elements between different states. These elements help us calculate the probability for a transition between initial and final states, ultimately influencing the rules governing state transitions (selection rules). Essentially, when dealing with quantum systems, knowing the dipole moment can help predict how a system will respond to electromagnetic fields.

Azimuthal Angle

The **azimuthal angle**, \(\phi\), is a critical component in polar coordinates, especially for systems involving rotational dynamics in a plane. In two-dimensional rotation problems, the azimuthal angle explains how the system or particle rotates about a central point.
In our selection rule derivation, we use \(\phi\) to integrate over the wavefunctions of the rotor states. This angle cycles from \(0\) to \(2\pi\), completing a full rotational cycle, showcasing the periodic nature of rotational states.
By integrating over \(\phi\), we effectively account for all possible orientations of the rotor, which is vital when computing matrix elements. Understanding this angle’s role in rotational dynamics is critical for further studies involving angular momentum and the behavior of molecules or quantum systems in rotational motion.

Complex Exponential Representation

**Complex exponential representation** is a mathematical tool often used in quantum mechanics to simplify trigonometric expressions. By using Euler's formula, \(e^{i\phi} = \cos \phi + i\sin \phi\), we can express oscillations and rotations cleanly without separating cosine and sine components.
This is particularly useful when dealing with periodic functions and wavefunctions.
In the exercise, we use this representation to express \(\cos \phi\) as \(\frac{1}{2}(e^{i\phi} + e^{-i\phi})\). This transformation allows us to streamline the integration process, combining exponential terms rather than trigonometric functions. This makes calculating integrals in quantum mechanics more straightforward and intuitive.Utilizing complex exponential representation simplifies computations, allowing physicists to predict behaviors and solve problems efficiently in quantum mechanics and signal processing.