What is Rigid Rotator? Write Schrodinger's Equation for it and obtain Eigen values and Eigen functions.
1. Rigid Rotator:
In quantum mechanics, the rigid rotator is a simplified model used to describe the rotation of a particle around a fixed axis. It assumes that the particle's distance from the axis remains constant, so it only considers the angular motion. This is applicable to diatomic molecules, such as H2, and other systems with rotational symmetry.
2. Hamiltonian Operator:
The Hamiltonian operator (\(\hat{H}\)) represents the total energy of the system. For the rigid rotator, it is given by:
\[ \hat{H} = \frac{\hat{L}^2}{2I} \]
where:
- \(\hat{L}\) is the angular momentum operator, given by \(\hat{L} = -i\hbar \frac{\partial}{\partial \theta}\). This operator measures the angular momentum of the particle.
- \(I\) is the moment of inertia of the particle with respect to the rotation axis. It quantifies the particle's resistance to rotational motion.
3. Schrödinger's Equation:
Schrödinger's equation describes how the wave function (\(\Psi\)) of a quantum system evolves over time. For the rigid rotator, it is given by:
\[ \hat{H} \Psi(\theta) = E \Psi(\theta) \]
where:
- \(\Psi(\theta)\) is the wave function, representing the state of the particle as a function of the angle \(\theta\).
- \(E\) is the energy eigenvalue associated with the state \(\Psi(\theta)\).
4. Solving the Schrödinger's Equation:
To find the eigenvalues and eigenfunctions of the rigid rotator, we need to solve the Schrödinger's equation. First, we express the angular momentum operator in terms of the wave function \(\Psi(\theta)\):
\[ \hat{L}^2 = -\hbar^2 \frac{\partial^2}{\partial \theta^2} \]
Now, we substitute this back into the Schrödinger's equation:
\[ -\frac{\hbar^2}{2I} \frac{\partial^2}{\partial \theta^2} \Psi(\theta) = E \Psi(\theta) \]
Next, we rearrange the equation to separate variables:
\[ \frac{1}{\Psi(\theta)} \frac{\partial^2 \Psi(\theta)}{\partial \theta^2} = -\frac{2IE}{\hbar^2} \]
Now, the left side of the equation only depends on \(\theta\), while the right side is a constant. To balance both sides of the equation, we set them equal to a constant, let's say \(l(l+1)\):
\[ \frac{1}{\Psi(\theta)} \frac{\partial^2 \Psi(\theta)}{\partial \theta^2} = -\frac{2IE}{\hbar^2} = l(l+1) \]
This results in the following differential equation:
\[ \frac{\partial^2 \Psi(\theta)}{\partial \theta^2} + l(l+1) \Psi(\theta) = 0 \]
5. Solving the Differential Equation:
The differential equation \(\frac{\partial^2 \Psi(\theta)}{\partial \theta^2} + l(l+1) \Psi(\theta) = 0\) is a standard second-order differential equation. Its solutions are the eigenfunctions of the rigid rotator. To solve this, we propose a solution of the form \(\Psi(\theta) = A \cos^n(\theta) \), where \(A\) and \(n\) are constants.
Substituting this solution into the differential equation:
\[ -A n(n-1) \cos^{n-2}(\theta) + l(l+1) A \cos^n(\theta) = 0 \]
To make this equation valid for all values of \(\theta\), we set \(n\) equal to \(l\) and obtain:
\[ n(n-1) = l(l+1) \]
Solving this quadratic equation for \(n\), we get two solutions \(n = l\) and \(n = -(l+1)\). However, the solution \(n = -(l+1)\) leads to singularities, so we discard it.
Thus, the eigenvalues \(l\) represent the quantized angular momentum values, and they are given by:
\[ l = 0, 1, 2, 3, \ldots \]
6. Eigenfunctions:
Now that we have the eigenvalues, we can find the corresponding eigenfunctions. The eigenfunctions \(\Psi_l(\theta)\) are the associated Legendre polynomials of order \(l\):
\[ \Psi_l(\theta) = P_l(\cos(\theta)) \]
where \(P_l(\cos(\theta))\) is the \(l\)th-order Legendre polynomial, and \(\cos(\theta)\) is the cosine of the angle of rotation.
7. Energy Eigenvalues:
The energy eigenvalues (\(E_l\)) are related to the quantized angular momentum values (\(l\)) and the moment of inertia (\(I\)):
\[ E_l = \frac{\hbar^2}{2I} l(l+1) \]
The energy levels are proportional to \(l(l+1)\), where \(l\) takes the values \(0, 1, 2, 3, \ldots\).
In summary, the rigid rotator model in quantum mechanics allows us to describe the rotational motion of a particle around a fixed axis. By solving Schrödinger's equation, we find that the angular momentum is quantized, and the associated eigenfunctions are the Legendre polynomials. The energy levels are determined by the quantized angular momentum values and the moment of inertia. This analysis provides valuable insights into the quantum behavior of rotational motion.
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