Question

By substituting in the Schrödinger equation for rotation in three dimensions, show that the rotational wave func\(\operatorname{tion}(5 / 16 \pi)^{1 / 2}\left(3 \cos ^{2} \theta-1\right)\) is an eigenfunction of the total energy operator. Determine the energy eigenvalue.

Short answer

The given wave function \(\psi(\theta) = (5 / 16 \pi)^{1 / 2}(3 \cos^{2}\theta - 1)\) is indeed an eigenfunction of the total energy operator for a three-dimensional rotation problem. Upon substitution into the Schrödinger equation and simplification, the energy eigenvalue has been determined to be \(E = -\frac{\hbar^2}{2I}(6 - 18 \cos^2(\theta)) (3 \cos^2 \theta - 1)^{-1}\), where \(\hbar\) is the reduced Planck constant and \(I\) is the moment of inertia.

Step-by-step solution

Write down the Schrödinger equation for rotation in three dimensions

The Schrödinger equation for a particle with a given rotational quantum number, l, in three dimensions is represented as follows: \[ \hat{H}\psi(\theta) = E\psi(\theta) \] where \(\hat{H}\) is the total energy operator, \(\psi(\theta)\) is the wave function and E is the energy eigenvalue. For a three-dimensional rotation problem, the total energy operator, \(\hat{H}\), is given by: \[ \hat{H} = -\frac{\hbar^2}{2I}\left[\frac{1}{\sin{\theta}}\frac{\partial}{\partial \theta}\left(\sin{\theta} \frac{\partial}{\partial \theta}\right)\right] \] where \(I\) is the moment of inertia and \(\hbar\) is the reduced Planck constant. The given wave function is: \[ \psi(\theta) = (5 / 16 \pi)^{1 / 2}(3 \cos^{2}\theta - 1) \]

Substitute the given wave function into the Schrödinger equation

Now we'll substitute the given wave function, \(\psi(\theta)\), into the Schrödinger equation and compute the action of the total energy operator, \(\hat{H}\), on it. The derivatives of \(\psi(\theta)\) with respect to \(\theta\) are: \[ \frac{\partial \psi(\theta)}{\partial \theta} = (5 / 16 \pi)^{1 / 2}(-6 \cos(\theta) \sin(\theta)) \] and, \[ \frac{\partial^2 \psi(\theta)}{\partial \theta^2} = (5 / 16 \pi)^{1 / 2}(6 - 18 \cos^2(\theta)) \] Now, we'll plug in these partial derivatives into the \(\hat{H}\) expression: \[ \hat{H} \psi(\theta) = -\frac{\hbar^2}{2I}\left[\frac{1}{\sin{\theta}}\frac{\partial}{\partial \theta}\left(\sin{\theta} (5 / 16 \pi)^{1 / 2}(-6 \cos(\theta) \sin(\theta))\right)\right] \] Simplifying further, we get: \[ \hat{H} \psi(\theta) = -\frac{\hbar^2}{2I}(5 / 16 \pi)^{1 / 2}(6 - 18 \cos^2(\theta)) \]

Determine the energy eigenvalue

Comparing this result with the Schrödinger equation, we can see that the action of the total energy operator on the given wave function returns the same function multiplied by a constant, which implies that the given wave function is indeed an eigenfunction of the total energy operator. Thus, the Schrödinger equation becomes: \[ -\frac{\hbar^2}{2I}(5 / 16 \pi)^{1 / 2}(6 - 18 \cos^2(\theta)) = E (5 / 16 \pi)^{1 / 2}(3 \cos^2 \theta - 1) \] From this equation, we can determine the energy eigenvalue, E: \[ E = -\frac{\hbar^2}{2I}(6 - 18 \cos^2(\theta)) (3 \cos^2 \theta - 1)^{-1} \] Since \(\cos^2(\theta)\) takes values between 0 and 1, the energy eigenvalue, E, will be a constant. Hence, The given wave function is an eigenfunction of the total energy operator and the energy eigenvalue is \(E = -\frac{\hbar^2}{2I}(6 - 18 \cos^2(\theta)) (3 \cos^2 \theta - 1)^{-1}\).

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It challenges classical concepts such as deterministic trajectories of particles, instead describing them through probabilistic wave functions. Core to this theory is the concept of superposition, where systems exist in all possible states simultaneously, and the observer effect, which postulates that the state of a system is affected by measurement.

The Schrödinger equation is central to quantum mechanics, as it describes how the quantum state of a physical system changes over time. It plays the role that Newton's laws do in classical mechanics, defining the future evolution of a system based on its current state. Studying rotational motion in quantum mechanics, such as in the given exercise, illustrates how quantum principles can determine the behavior of a rotating object at the quantum level.

Rotational Wave Function

In quantum mechanics, a rotational wave function represents the quantum state of a rotating body, such as a molecule. It mathematically explains the probability amplitude for finding a particle with a particular orientation in space. These wave functions are solutions to the rotational part of the Schrödinger equation and depend on the angular coordinates, typically denoted as \( \theta \) and \( \phi \).

The specific form of the rotational wave function given in the exercise \( (5 / 16 \pi)^{1 / 2}(3 \cos^{2}\theta - 1) \) is a part of a group of solutions known as spherical harmonics, crucial for describing the angular part of the wave function of particles in spherical coordinate systems. These functions carry quantum numbers that define the angular momentum of the particle, reflecting the quantization of rotational energy levels.

Energy Eigenvalues

Energy eigenvalues are discrete energy levels that an electron or any quantum particle can occupy within an atom or molecule. They are derived from the solutions to the Schrödinger equation. The term 'eigenvalue' is derived from the German word 'eigen,' meaning 'own' or 'characteristic.' In the context of quantum mechanics, these values characterize the allowed energy states of a quantum system.

The determination of energy eigenvalues is akin to solving a puzzle where the waves must fit neatly inside a space in a specific, quantized manner. In the exercise, finding the energy eigenvalue of a given rotational wave function is essential to understanding the energy landscape and possible states the system can exist in. The exercise demonstrates how the rotational wave function aligns with one of these quantized energy levels.

Total Energy Operator

The total energy operator, often represented by \( \hat{H} \), also known as the Hamiltonian, is an operator corresponding to the total energy of the system, including kinetic and potential energies. In quantum mechanics, this operator is used within the Schrödinger equation to derive the system's wave functions and energy levels.

In a rotational motion problem, the Hamiltonian includes the kinetic energy associated with rotation, which depends on the moment of inertia and the angular momentum of the system. The form of the total energy operator for rotation in three dimensions, as shown in the exercise, takes the angular derivatives of the wave function, embodying the rotational kinetic energy in quantum terms. The application of this operator to a rotational wave function, as in the exercise, is integral in determining whether the function is an eigenfunction and in calculating the corresponding energy eigenvalue.