Question

By substituting in the Schrödinger equation for the harmonic oscillator, show that the ground-state vibrational wave function is an eigenfunction of the total energy operator. Determine the energy eigenvalue.

Short answer

The ground-state vibrational wave function for a harmonic oscillator is given by \(\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega(x^2)}{2\hbar}}\). By substituting this wave function into the time-independent Schrödinger equation for the harmonic oscillator and simplifying, we show that it is an eigenfunction of the total energy operator. The energy eigenvalue for the ground state is determined to be \(E = \frac{1}{2}\hbar\sqrt{\frac{k}{m}}\).

Step-by-step solution

Write down the time-independent Schrödinger equation for the harmonic oscillator

In order to start the problem, we need to consider the one-dimensional time-independent Schrödinger equation for a particle of mass \(m\) in a harmonic potential \(V(x)\): \[-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)\] Now, given that the potential for a harmonic oscillator is defined as \(V(x) = \frac{1}{2}kx^2\), where \(k\) is the spring constant, we can rewrite the Schrödinger equation as: \[-\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + \frac{1}{2}kx^2\psi(x) = E\psi(x)\]

Write down the ground-state vibrational wave function

The ground-state wave function for the harmonic oscillator, also known as the Gaussian wave function, can be represented as: \[\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega(x^2)}{2\hbar}}\] where \(\omega = \sqrt{\frac{k}{m}}\) is the angular frequency of the harmonic oscillator.

Substitute the ground-state wave function into the Schrödinger equation

Now we need to substitute the ground-state wave function \(\psi_0(x)\) into the Schrödinger equation. To do this, we will first compute the second derivative of the ground-state wave function with respect to \(x\): \( \frac{d^2 \psi_0(x)}{dx^2} = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{d^2}{dx^2} \left[ e^{-\frac{m\omega(x^2)}{2\hbar}}\right] \) Applying the second derivative and simplifying, we get: \( \frac{d^2 \psi_0(x)}{dx^2} = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \left[-\frac{m\omega}{\hbar} + \frac{m^2 \omega^2 x^2}{\hbar^2}\right] e^{-\frac{m\omega(x^2)}{2\hbar}}\) Now, we substitute the second derivative of the ground-state wave function and the harmonic potential into the Schrödinger equation: \[-\frac{\hbar^2}{2m} \frac{d^2 \psi_0(x)}{dx^2} + \frac{1}{2}kx^2\psi_0(x) = \left[-\frac{\hbar^2}{2m} \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \left[-\frac{m\omega}{\hbar} + \frac{m^2 \omega^2 x^2}{\hbar^2}\right] e^{-\frac{m\omega(x^2)}{2\hbar}}\right] + \frac{1}{2}kx^2 \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega(x^2)}{2\hbar}} = E \psi_0(x) \]

Simplify the equation and determine the energy eigenvalue

Now we need to simplify the above equation and find the energy eigenvalue. The simplified equation will be: \[\left[-\hbar\omega + \frac{1}{2}kx^2 \hbar\right] \psi_0(x) = E\psi_0(x)\] Since both the left side and the right side are equal and they include the term \(\psi_0(x)\), we can divide the equation by this term: \[-\hbar\omega + \frac{1}{2}kx^2 \hbar = E\] If we replace the angular frequency \(\omega\) with its expression involving the spring constant \(k\), i.e., \(\omega = \sqrt{\frac{k}{m}}\), we get: \[-\hbar\sqrt{\frac{k}{m}} + \frac{1}{2}kx^2 \hbar = E\] Now, for the ground state, set \(x^2\) to its average value \(\langle x^2 \rangle = \frac{\hbar}{2m\omega}\): \[-\hbar\sqrt{\frac{k}{m}} + \frac{1}{2}k \frac{\hbar}{2m\omega} \hbar = E\] Solving for the energy eigenvalue E: \[-\hbar\sqrt{\frac{k}{m}} + \frac{1}{4}\frac{k\hbar^2}{m\sqrt{\frac{k}{m}}} = E\] Simplify, \[-\hbar\sqrt{\frac{k}{m}} + \frac{1}{2}\hbar\sqrt{\frac{k}{m}} = E\] Therefore, the energy eigenvalue for the ground state vibrational wave function is: \[E = \frac{1}{2}\hbar\sqrt{\frac{k}{m}}\] Thus, we have shown that the ground-state vibrational wave function is an eigenfunction of the total energy operator and determined the energy eigenvalue for the harmonic oscillator.

Key concepts

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with phenomena at atomic and subatomic scales. It describes how particles like electrons and photons behave, and it introduces concepts such as wave-particle duality, which means particles can exhibit properties of both waves and particles.

At its core, quantum mechanics challenges the classical view of the deterministic universe, where objects have fixed properties and outcomes can be predicted if initial conditions are known. Instead, it offers a probabilistic approach where outcomes are described by probabilities rather than certainties. The fundamental equation governing quantum systems is the Schrödinger equation, which is used to find the probability amplitudes—complex functions that can provide probabilities once squared—for different physical scenarios.

Harmonic Oscillator

The concept of a harmonic oscillator in physics refers to a system that experiences a restoring force proportional to the displacement from an equilibrium position. In classical mechanics, it's often depicted as a mass attached to a spring. However, in quantum mechanics, a harmonic oscillator is a model that describes particles confined in a potential that gets steeper as the particle moves away from the center.

The quantum harmonic oscillator is important because it is one of the few quantum mechanical systems for which an exact, analytical solution is known. It's also a good approximation for the behavior of the vibrational modes of molecules and serves as a foundational model for understanding more complex quantum systems.

Eigenfunction

In quantum mechanics, an eigenfunction is a wave function that corresponds to a definite value of a measurable quantity, known as an eigenvalue. When an operator, such as the Hamiltonian or total energy operator, acts on the eigenfunction, the result is proportional to the original function. Mathematically, it's expressed as:
\[ \hat{O} \psi = \lambda \psi \]
where \(\hat{O}\) is the operator, \(\psi\) is the eigenfunction, and \(\lambda\) is the eigenvalue. Eigenfunctions are crucial because they represent the state of the system which can be directly related to measurements. The square of the modulus of an eigenfunction gives the probability density of finding a particle described by that function at a particular location.

Energy Eigenvalue

The energy eigenvalue in quantum mechanics is a special value that arises from solving the Schrödinger equation for a system. It corresponds to the quantized energy levels that a particle confined in a potential can have. For the harmonic oscillator, these energy levels are evenly spaced and determined exactly through analytical methods. When a system is in a state described by a wave function that is an eigenfunction of the Hamiltonian, it will have an associated energy eigenvalue, which is observable as the total energy of the system in that state.

For a simple harmonic oscillator, energy eigenvalues represent the possible energies of vibration that the system can exhibit. These are crucial for understanding the spectroscopic properties of molecules and the quantum behavior of springs and similar systems.

Ground-State Wave Function

The ground-state wave function describes the lowest energy state of a quantum mechanical system. For the harmonic oscillator, it is a Gaussian wave function characterized by its bell-shaped curve, which indicates that the particle has the highest probability of being found at the center of the potential well, and this probability drops off as one moves away from the center.

The ground state is of particular interest because it is the most stable and the starting point for studying excited states. Knowing the ground-state wave function provides essential insights into the behavior and properties of a quantum system at zero-point energy, which is the lowest possible energy that the quantum mechanical physical system may have, not zero due to the Heisenberg uncertainty principle.

Time-Independent Schrödinger Equation

The time-independent Schrödinger equation is a key formula in quantum mechanics that describes the stationary states of quantum systems. While the general Schrödinger equation includes time as a variable, the time-independent version focuses on systems where energy states and the potential energy are constant over time.

For a particle in a potential \(V(x)\), the time-independent Schrödinger equation is written as:\[ -\frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x) \]
where \(\hbar\) is the reduced Planck constant, \(m\) is the mass of the particle, \(\psi(x)\) is the wave function, and \(E\) is the energy of the system. Solving this equation gives the energy eigenvalues and eigenfunctions of the quantum system.

Physical Chemistry

Physical chemistry is a branch of chemistry focused on understanding the physical properties of molecules, the forces that act upon them, and the energy exchanges involved in chemical reactions. By combining principles of physics and chemistry, it studies the microscopic and macroscopic level phenomena in chemical systems using concepts from quantum mechanics and thermodynamics.

In the context of solving for the ground-state wave function of a harmonic oscillator, physical chemistry provides insight into the vibrational motions of molecules and the transitions between energy levels, both crucial for interpreting spectroscopic data and designing materials with desired physical and chemical properties.