Simple Harmonic Oscillator. Suppose that a particle of mass...

Simple Harmonic Oscillator. Suppose that a particle of mass $m$ is trapped not in a square well, but in one whose potential energy is that of a simple harmonic oscillator: $U(x)=\frac{1}{2} C x^{2} .$ That is, if the particle is displaced from $x=0$ a restoring force $F=-C x$ acts on it, where $C$ is constant. (a) Sketch this potential energy. (b) Show that $\psi=A e^{-B x^{2}}$ is a solution to the Schrödinger equation and that the energy of this state is $E=\frac{1}{2} \hbar \omega,$ where $\omega=\sqrt{C / m}$ (as classically, Eq. $14-5$ ) and $B=m \omega / 2 \hbar$. [Note: This is the ground state, and this energy $\frac{1}{2} \hbar \omega$ is the zero-point energy for a harmonic oscillator. The energies of higher states are $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega,$ where $n$ is an integer.

Simple Harmonic Oscillator. Suppose that a particle of mass $m$ is trapped not in a square well, but in one whose potential energy is that of a simple harmonic oscillator:
$U(x)=\frac{1}{2} C x^{2} .$ That is, if the particle is displaced from $x=0$ a restoring force $F=-C x$ acts on it, where $C$ is constant.
(a) Sketch this potential energy.
(b) Show that $\psi=A e^{-B x^{2}}$ is a solution to the Schrödinger equation and that the energy of this state is $E=\frac{1}{2} \hbar \omega,$ where $\omega=\sqrt{C / m}$ (as classically, Eq. $14-5$ ) and $B=m \omega / 2 \hbar$. [Note: This is the ground state, and this energy $\frac{1}{2} \hbar \omega$ is the zero-point energy for a harmonic oscillator. The energies of higher states are $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega,$ where $n$ is an integer.

Key Concepts

Quantum Harmonic Oscillator

This model describes a particle subject to a restoring force proportional to its displacement, leading to a quadratic potential energy. It is a fundamental system in quantum mechanics, serving as an idealized approximation for various physical situations such as molecular vibrations and quantized fields.

Schrödinger Equation

The Schrödinger equation is the central equation in quantum mechanics that governs the behavior of quantum systems. For the harmonic oscillator, it determines the allowed energy eigenstates and their corresponding wave functions, highlighting the wave nature of particles in a potential.

Ground State Wave Function

In the quantum harmonic oscillator, the ground state is represented by a Gaussian function. This form of the wave function minimizes the uncertainty between position and momentum and is central to understanding the system's lowest energy configuration.

Zero-Point Energy

Zero-point energy is the lowest possible energy that a quantum system can have, which for a harmonic oscillator is (1/2)??. It reflects the quantum mechanical principle that even in the ground state, the system possesses intrinsic energy due to quantum fluctuations.

Energy Quantization

Quantum systems like the harmonic oscillator exhibit discretized energy levels, where the energy eigenvalues are given by (n + 1/2)??. This quantization arises from the boundary conditions and the mathematical structure of the Schrödinger equation in systems with confining potentials.

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