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Chapter 4: Problem 11

The Schrödinger equation for a rigid body that is constrained to rotate about a fixed axis and that has a moment of inertia \(I\) about this axis is $$ i \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 I} \frac{\partial^{2} \psi}{\partial \phi^{2}} $$ where \(\psi(\phi, t)\) is a function of the time \(t\) and of the angle of rotation \(\phi\) about the axis. What boundary conditions must be applied to the solutions of this equation? Find the normalized energy eigenfunctions and eigenvalues. Is there any degeneracy?

Short Answer

Expert verified
The boundary conditions are that The normalized energy eigenfunctions are and the energy eigenvalues are The eigenvalues are doubly degenerate for each non-zero eigenvalue.

Step by step solution

01

Understand the Schrödinger Equation

The given Schrödinger equation is:
02

Identify the Boundary Condition Requirements

Boundary conditions for The function must be periodic.Therefore, we require
03

Propose a Solution of the Form

Try a solution of the form Putting this into the Schrödinger equation and separating variables, we obtain two separate differential equations: one for for and one for
04

Solve the Time-Independent Equation

The differential equation for is: The general solution to this equation is
05

Apply the Boundary conditions

Using the boundary condition from Step 2, we find that This implies that
06

Normalize the Eigenfunctions

The eigenfunctions must be normalized so that: The normalization condition leads to: This gives the normalized eigenfunctions as:
07

Determine the Energy Eigenvalues

Putting in the time-independent equation, the energy eigenvalues are found to be:
08

Check for Degeneracy

Since the eigenfunctions are , there are two degenerate states for each eigenvalue except for the ground state (

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Key Concepts

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

Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the smallest scales of energy levels of atoms and subatomic particles. One of its core principles is wave-particle duality, which suggests that particles like electrons exhibit both wave-like and particle-like properties. The Schrödinger equation, central to quantum mechanics, describes how the quantum state of a physical system changes over time. In our exercise, it is applied to a rigid body rotating about a fixed axis. This equation helps us determine the allowed energy levels and the corresponding wave functions (eigenstates) of the system.
Moment of Inertia
The moment of inertia (I) is a measure of an object's resistance to changes in its rotation. In simpler terms, it indicates how difficult it is to change the rotational speed of an object. It depends on the mass distribution of the object relative to the axis of rotation. For a rigid body with a rotational axis, the Schrödinger equation takes this concept into account by incorporating I into the differential operator. This influences the behavior of the system's wave functions and energy levels.
Boundary Conditions
Boundary conditions are constraints necessary for solving differential equations. They ensure the solutions (wave functions) are physically meaningful. For our rotating rigid body, the wave function teq teq approaches. This is because the rotation angle teq complete turns. Therefore, the boundary condition is teq.Applying this constraint allows us to find valid solutions to the Schrödinger equation for the system.
Normalized Eigenfunctions
Eigenfunctions of a system describe the possible states the system can be in. Normalizing these eigenfunctions means scaling them so that their total probability is one. Mathematically, this is expressed as:teq.For the given problem, we derived the normalized solutions of the form teq separated variables approach. This ensures the corresponding probabilities are physically consistent and meaningful.
Energy Eigenvalues
Energy eigenvalues represent the quantized energy levels that the system can occupy. In the given problem, by solving the time-independent Schrödinger equation, we find these energy levels arise from the relationship:teq.These values correspond to the stationary states of the system, indicating the allowed rotational energy levels based on the system's moment of inertia teq.
Degeneracy in Quantum States
Degeneracy in quantum states refers to multiple different states (eigenfunctions) having the same energy eigenvalue. For the rotating rigid body problem, each energy level (except the ground state) has a degeneracy of two. This means there are two distinct states corresponding to the same energy level. Mathematically, these states are represented by the sine and cosine functions that satisfy the boundary conditions.

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Most popular questions from this chapter

Use the generating function for the Hermite polynomials to evaluate $$ \int_{-\infty}^{\infty} \bar{u}_{n}(x) x^{2} u_{m}(x) d x $$ where the \(u\) 's are normalized harmonic-oscillator wave functions.

Find the energy levels of a three-dimensional isotropic harmonic oscillator \(\left(V(r)=\frac{1}{2} K r^{2}\right)\), by solving the wave equation in cartesian coordinates. What is the degeneracy of each level? Show that this equation can also be separated in spherical and in cylindrical coordinates.

Obtain an approximate analytic expression for the energy level in a square well potential \((l=0)\) when \(V_{0} a^{2}\) is slightly greater than \(\pi^{2} \hbar^{2} / 8 m\).

Consider Eq. (14.17) with \(l=0\) and \(V(r)=-V_{0} e^{-\frac{r}{a}}\). Change variables from \(r\) to \(z=e^{-\frac{r}{2 a}}\), and show that Bessel's equation results. What boundary conditions are to be imposed on \(\chi\) as a function of \(z\), and how can these be used to determine the energy levels? What is the lower limit to \(V_{0}\) for which a bound state exists?
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