Physlib.QuantumMechanics.OneDimension.HarmonicOscillator.TISE

For a one-dimensional quantum harmonic oscillator with angular frequency $\omega$, the $n$-th energy eigenvalue $E_n$ corresponding to the quantum number $n \in \mathbb{N}$ is defined as: $$E_n = \left(n + \frac{1}{2}\right) \hbar \omega$$ where $\hbar$ is the reduced Planck constant.

Let $\psi_0$ be the ground state eigenfunction (the eigenfunction with index $n=0$) of a one-dimensional quantum harmonic oscillator with characteristic length $\xi$. Its derivative $\psi_0'$ satisfies $$\psi_0'(x) = -\frac{x}{\xi^2} \psi_0(x).$$

Let $\psi_n$ be the $n$-th eigenfunction of a one-dimensional quantum harmonic oscillator with characteristic length $\xi$. The derivative of the ground state eigenfunction $\psi_0$ is proportional to the first excited state eigenfunction $\psi_1$ according to the relation: $$\psi_0'(x) = -\frac{\sqrt{2}}{2\xi} \psi_1(x).$$

For any natural number $n$, the derivative with respect to $x$ of the $n$-th physicist's Hermite polynomial $H_n$ evaluated at $x/\xi$ is given by $$\frac{d}{dx} H_n\left(\frac{x}{\xi}\right) = \frac{2n}{\xi} H_{n-1}\left(\frac{x}{\xi}\right),$$ where $\xi$ is the characteristic length of the quantum harmonic oscillator.

Let $\psi_n$ be the $n$-th eigenfunction of a one-dimensional quantum harmonic oscillator with characteristic length $\xi$. For any natural number $n$, the derivative of the $(n+1)$-th eigenfunction $\psi_{n+1}$ is given by $$\psi_{n+1}'(x) = \frac{1}{\xi \sqrt{2^{n+1} (n+1)!}} \left( 2(n+1) H_n\left(\frac{x}{\xi}\right) - \frac{x}{\xi} H_{n+1}\left(\frac{x}{\xi}\right) \right) \psi_0(x),$$ where $H_k$ denotes the $k$-th physicist's Hermite polynomial and $\psi_0$ is the ground state eigenfunction.

Let $\psi_0(x)$ be the ground state eigenfunction (the eigenfunction with index $n=0$) of a one-dimensional quantum harmonic oscillator with characteristic length $\xi$. For any $x \in \mathbb{R}$, the second derivative of the eigenfunction, denoted $\psi_0''(x)$, is given by: $$\psi_0''(x) = -\frac{1}{\xi^2} \left(1 - \frac{x^2}{\xi^2}\right) \psi_0(x).$$

Let $\psi_n$ be the $n$-th eigenfunction of a one-dimensional quantum harmonic oscillator with characteristic length $\xi$. For any natural number $n$ and any position $x \in \mathbb{R}$, the second derivative of the $(n+1)$-th eigenfunction $\psi_{n+1}$ is given by $$\psi_{n+1}''(x) = \frac{1}{\xi \sqrt{2^{n+1} (n+1)!}} \left[ 2(n+1) \frac{d}{dx} H_n\left(\frac{x}{\xi}\right) - \frac{4(n+1)x}{\xi^2} H_n\left(\frac{x}{\xi}\right) - \frac{1}{\xi} \left( 1 - \frac{x^2}{\xi^2} \right) H_{n+1}\left(\frac{x}{\xi}\right) \right] \psi_0(x),$$ where $H_k$ denotes the $k$-th physicist's Hermite polynomial and $\psi_0$ is the ground state eigenfunction.

Let $\psi_n$ be the $n$-th eigenfunction of a one-dimensional quantum harmonic oscillator with characteristic length $\xi$. For any natural number $n$ and any position $x \in \mathbb{R}$, the second derivative of the eigenfunction $\psi_n$ with respect to $x$ is given by $$\psi_n''(x) = -\frac{1}{\xi^2} \left( 2n + 1 - \frac{x^2}{\xi^2} \right) \psi_n(x).$$

Let $Q$ be a one-dimensional quantum harmonic oscillator. For any natural number $n \in \mathbb{N}$ and any position $x \in \mathbb{R}$, the $n$-th eigenfunction $\psi_n$ satisfies the time-independent Schrödinger equation: $$\hat{H} \psi_n(x) = E_n \psi_n(x)$$ where $\hat{H}$ is the Schrödinger operator (Hamiltonian) of the oscillator and $E_n$ is the $n$-th energy eigenvalue, defined as $E_n = (n + 1/2)\hbar\omega$.