91Ó°ÊÓ

The Hamiltonian for a three-dimensional system with cylindrical symmetry is given by $$ \hat{H}=\frac{\hat{\mathbf{p}}^{2}}{2 \mu}+V(\hat{\rho}) $$ where \(\rho=\sqrt{x^{2}+y^{2}}\). (a) Use symmetry arguments to establish that both \(\hat{p}_{z}\), the generator of translations in the \(z\) direction, and \(\hat{L}_{z}\), the generator of rotations about the \(z\) axis, commute with \(\hat{H}\). (b) Use the fact that \(\hat{H}, \hat{p}_{z}\), and \(\hat{L}_{z}\) have eigenstates in common to express the position-space eigenfunctions of the Hamiltonian in terms of those of \(\hat{p}_{z}\) and \(\hat{L}_{z} .\) Suggestion: Follow a strategy similar to the one that we followed in (9.94) for a spherically symmetric potential except that here we are using the eigenfunctions of \(\hat{p}_{z}\) and \(\hat{L}_{z}\) instead of the eigenfunctions \(\hat{\mathbf{L}}^{2}\) and \(\hat{L}_{z} .\) (c) What is the radial equation? Note: The Laplacian in cylindrical coordinates is given by $$ \nabla^{2} \psi=\frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho \frac{\partial \psi}{\partial \rho}\right)+\frac{1}{\rho^{2}} \frac{\partial^{2} \psi}{\partial \phi^{2}}+\frac{\partial^{2} \psi}{\partial z^{2}} $$

Step by step solution

01

Establish Commutators for Translational Symmetry

The Hamiltonian \( \hat{H} = \frac{\hat{\mathbf{p}}^{2}}{2 \mu} + V(\hat{\rho}) \) is symmetric under translations along the \(z\)-axis. The momentum operator \( \hat{p}_z = -i \hbar \frac{\partial}{\partial z} \) commutes with \( \hat{H} \) because \( V(\hat{\rho}) \) does not depend on \( z \), while the kinetic energy term only involves \( \hat{p}_z^2 \). Thus, \( [\hat{H}, \hat{p}_z] = 0 \).

02

Establish Commutators for Rotational Symmetry

The Hamiltonian also maintains rotational symmetry about the \(z\)-axis. The angular momentum operator \( \hat{L}_z = -i \hbar \frac{\partial}{\partial \phi} \) commutes with \( \hat{H} \) since the potential \( V(\hat{\rho}) \) only depends on the radial component \( \rho \), independent of the \( \phi \) angle. Therefore, \( [\hat{H}, \hat{L}_z] = 0 \).

03

Express Eigenfunctions using Symmetry

Given the commutation relations, \( \hat{H}, \hat{p}_z, \) and \( \hat{L}_z \) have common eigenstates. The position-space wave function can be written as \( \psi(\rho, \phi, z) = R(\rho) e^{i m \phi} e^{i k z} \), where \( R(\rho) \) is the radial function, \( m \) is the \(z\)-component of angular momentum, and \( k \) is the wave number in the \(z\) direction.

04

Derive the Radial Equation

Using the Laplacian in cylindrical coordinates, substitute \( \psi(\rho, \phi, z) = R(\rho) e^{i m \phi} e^{i k z} \) into the Schrödinger equation \( \hat{H} \psi = E \psi \). After separation of variables, the radial equation becomes \[-\frac{\hbar^2}{2 \mu} \left( \frac{1}{\rho} \frac{d}{d \rho} \left( \rho \frac{d R}{d \rho} \right) - \frac{m^2}{\rho^2} R \right) + V(\rho) R = E R.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hamiltonian

The Hamiltonian ( \( \hat{H} \) ) is a fundamental operator in quantum mechanics. It represents the total energy of a system, including both kinetic and potential energy. For a system with cylindrical symmetry, as given by the equation:

• \( \hat{H} = \frac{\hat{\mathbf{p}}^{2}}{2 \mu} + V(\hat{\rho}) \)

we see it composed of kinetic energy, expressed through the momentum operator, and potential energy, which is a function of the radial distance \( \rho \). This symmetry implies rotational invariance around one axis, usually the \( z \)-axis, and translational invariance along it. The independence of potential \( V(\hat{\rho}) \) from \( z \) tells us that the system behaves consistently when moved along the \( z \)-axis. This invariance leads to unique commutation properties with other operators, such as angular momentum. This behavior simplifies analysis and is a powerful tool for predicting eigenfunctions that describe a system's states.

Angular Momentum

In quantum mechanics, angular momentum is a crucial concept that reflects rotation's role in a system. The operator for the \( z \)-component of angular momentum is \( \hat{L}_z \), defined as:

• \( \hat{L}_z = -i \hbar \frac{\partial}{\partial \phi} \)

Angular momentum about the \( z \)-axis is important in a system with cylindrical symmetry. Since the system's potential \( V(\hat{\rho}) \) only depends on the radial distance \( \rho \) and is independent of the angular coordinate \( \phi \), the system is rotationally symmetric about the \( z \)-axis.
This means the operator \( \hat{L}_z \) commutes with the Hamiltonian, so their eigenstates are shared, implying that the system's states retain angular momentum consistency. These common eigenstates simplify solutions and help in constructing the wave function. The momentum around the axis, represented by \( \hat{L}_z \), aids in describing how the probability density of the quantum state behaves as it relates to rotation.

Commutators

Commutators in quantum mechanics help determine if two operators can share eigenstates, an essential feature for solving quantum systems. For the given Hamiltonian in a system with cylindrical symmetry, two key commutators are derived:

• \([\hat{H}, \hat{p}_z] = 0\)
• \([\hat{H}, \hat{L}_z] = 0\)

These results imply that both \( \hat{p}_z \), which is the momentum operator along the \( z \)-axis, and \( \hat{L}_z \), the angular momentum operator around the \( z \)-axis, share common eigenstates with the Hamiltonian \( \hat{H} \).
This commonality is due to the symmetry properties of the system. In essence, translation along the \( z \)-axis or rotation about it does not affect the overall energy described by \( \hat{H} \). These properties are powerful because they allow us to find solutions in a more straightforward manner, as the operators do not interfere when operating on the system's states.

Radial Equation

The radial equation arises naturally from solving the Schrödinger equation in systems with symmetry, where the wave function is separable. With cylindrical symmetry, the position-space wave function is expressed as:

• \( \psi(\rho, \phi, z) = R(\rho) e^{i m \phi} e^{i k z} \)

Where \( R(\rho) \) is the radial function, \( m \) is the quantum number for angular momentum, and \( k \) represents the wave number along the \( z \)-axis. Using the Laplacian in cylindrical coordinates, this separation technique simplifies the Schrödinger equation into a radial part:\[-\frac{\hbar^2}{2 \mu} \left( \frac{1}{\rho} \frac{d}{d \rho} \left( \rho \frac{d R}{d \rho} \right) - \frac{m^2}{\rho^2} R \right) + V(\rho) R = E R.\]This equation highlights how the potential energy \( V(\rho) \) and the quantum number \( m \) contribute to defining the radial behavior of the wave function. The radial equation is central in predicting how a particle behaves at different radii within the potential, integral to understanding systems like atoms and molecules where cylindrical symmetry plays a role.