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) Sketch this potential energy. (b) Show that $\psi=A e^{-B x^{2}}$ is a solution to the Schrodinger equation and that the energy of this state is $E=\frac{1}{2} \hbar \omega,$ where $\omega=\sqrt{C / m}$ (as classi- cally) and $B=m \omega / 2 \hbar .$ INote: 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 $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega_{1}$ 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) Sketch this potential energy. (b) Show that $\psi=A e^{-B x^{2}}$
is a solution to the Schrodinger equation and that the energy
of this state is $E=\frac{1}{2} \hbar \omega,$ where $\omega=\sqrt{C / m}$ (as classi-
cally) and $B=m \omega / 2 \hbar .$ INote: 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 $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega_{1}$
where $n$ is an integer.]

Key Concepts

Simple Harmonic Oscillator Potential

This concept refers to a potential energy function that is quadratic in displacement. It models systems where the force acting on a particle is proportional to the displacement and directed toward an equilibrium position. In classical mechanics, it leads to sinusoidal motion, and in quantum mechanics, it provides a model where the energy levels are discrete and equidistant.

Schrödinger Equation

The Schrödinger equation is the fundamental equation in quantum mechanics describing how the quantum state of a physical system changes over time. When applied to a particle in a potential, it provides a way to determine the allowed energy states and corresponding wave functions of the system, making it central to solving problems like the quantum harmonic oscillator.

Wave Function Solution and Eigenstates

In quantum mechanics, solutions to the Schrödinger equation are represented by wave functions, which must satisfy specific boundary conditions to be physically acceptable. For the harmonic oscillator, certain forms of wave functions, such as those with Gaussian envelopes, serve as valid eigenfunctions of the Hamiltonian operator. These eigenfunctions correspond to quantized energy levels or eigenvalues, which are critical in understanding the system’s behavior.

Zero-Point Energy

Zero-point energy is the minimum energy that a quantum mechanical physical system may have, which is non-zero due to the Heisenberg uncertainty principle. In the context of the harmonic oscillator, even the ground state (the state with the lowest possible energy) possesses this intrinsic energy, reflecting the impossibility of having a particle with zero kinetic energy.

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