양자역학에서 조화진동자
Introduction
Time-independent Schrödinger equation is as follows.
$$ -\frac{\hbar^{2}}{2m} \frac{ d^{2}\psi }{ dx^{2} } + V\psi = E\psi $$
The potential of the harmonic oscillator is given by $V=\frac{1}{2}kx^{2}$ for the spring constant $k$. For the angular frequency $\omega$ we have $k = m\omega^{2}$, and substituting this yields the following Schrödinger equation for the harmonic oscillator.
Equation
In quantum mechanics, the Schrödinger equation for the harmonic oscillator is as follows.
$$ -\frac{\hbar^{2}}{2m} \frac{ d^{2}\psi }{ dx^{2} } + \dfrac{1}{2} m \omega^{2} x^{2} \psi = E\psi $$
Explanation
As in classical mechanics, the harmonic oscillator is a fundamental and important system in quantum mechanics as well. There are broadly two methods of solution.
- Algebraic solution
- Analytic solution