Physlib.QuantumMechanics.OneDimension.HarmonicOscillator.Eigenfunction

The $n$-th eigenfunction of the one-dimensional harmonic oscillator, denoted as $\psi_n: \mathbb{R} \to \mathbb{C}$, is defined for a given quantum number $n \in \mathbb{N}$ and characteristic length scale $\xi$ as: $$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$ where $H_n$ is the $n$-th physicist's Hermite polynomial (`physHermite`). The resulting value is treated as a complex number.

For any natural number $n$, the $n$-th eigenfunction $\psi_n(x)$ of the one-dimensional harmonic oscillator is given by the formula: $$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \frac{1}{\sqrt{\sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$ where $\xi$ is the characteristic length scale and $H_n$ is the $n$-th physicist's Hermite polynomial.

The 0-th eigenfunction of the one-dimensional harmonic oscillator, denoted as $\psi_0: \mathbb{R} \to \mathbb{C}$, is given by the expression: $$\psi_0(x) = \frac{1}{\sqrt{\sqrt{\pi} \xi}} \exp\left(-\frac{x^2}{2\xi^2}\right)$$ where $\xi$ is the characteristic length scale and $x$ is the position.

For any natural number $n \in \mathbb{N}$, the $n$-th eigenfunction $\psi_n(x)$ of the one-dimensional harmonic oscillator can be expressed in terms of the ground state eigenfunction $\psi_0(x)$ as: $$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} H_n\left(\frac{x}{\xi}\right) \psi_0(x)$$ where $H_n$ is the $n$-th physicist's Hermite polynomial and $\xi$ is the characteristic length scale.

For any natural number $n \in \mathbb{N}$, the $n$-th eigenfunction of the one-dimensional harmonic oscillator, denoted as $\psi_n(x)$, is integrable. That is, the integral of the absolute value of the eigenfunction over the entire real line is finite: $$\int_{-\infty}^{\infty} |\psi_n(x)| \, dx < \infty$$ where $\psi_n(x)$ is defined in terms of the $n$-th physicist's Hermite polynomial $H_n$ and the characteristic length scale $\xi$ as: $$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$

For any natural number $n \in \mathbb{N}$ and any real number $x \in \mathbb{R}$, the $n$-th eigenfunction of the one-dimensional harmonic oscillator $\psi_n(x)$ is real-valued. Specifically, its complex conjugate (denoted by the overline) is equal to itself: $$\overline{\psi_n(x)} = \psi_n(x)$$ This holds because the explicit definition of $\psi_n(x)$ involves only real-valued components (Hermite polynomials, exponentials, and normalization constants) cast into the complex field.

For any natural number $n \in \mathbb{N}$ and any real number $x \in \mathbb{R}$, the norm of the $n$-th eigenfunction $\psi_n(x)$ of the one-dimensional harmonic oscillator is given by: $$\|\psi_n(x)\| = \left( \frac{1}{\sqrt{2^n n!}} \frac{1}{\sqrt{\sqrt{\pi} \xi}} \right) \left| H_n\left(\frac{x}{\xi}\right) \right| e^{-\frac{x^2}{2\xi^2}}$$ where $\xi$ is the characteristic length scale and $H_n$ is the $n$-th physicist's Hermite polynomial.

For any natural number $n \in \mathbb{N}$ and any real number $x \in \mathbb{R}$, the squared norm of the $n$-th eigenfunction $\psi_n(x)$ of the one-dimensional harmonic oscillator is given by: $$\|\psi_n(x)\|^2 = \left( \frac{1}{\sqrt{2^n n!}} \frac{1}{\sqrt{\sqrt{\pi} \xi}} \right)^2 H_n\left(\frac{x}{\xi}\right)^2 e^{-\frac{x^2}{\xi^2}}$$ where $\xi$ is the characteristic length scale and $H_n$ is the $n$-th physicist's Hermite polynomial.

For any natural number $n \in \mathbb{N}$, the $n$-th eigenfunction $\psi_n$ of the one-dimensional harmonic oscillator is square-integrable, meaning the function $x \mapsto \|\psi_n(x)\|^2$ is integrable over $\mathbb{R}$. The eigenfunction $\psi_n$ is defined as: $$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$ where $\xi$ is the characteristic length scale and $H_n$ is the $n$-th physicist's Hermite polynomial.

For any natural number $n \in \mathbb{N}$, the $n$-th eigenfunction of the one-dimensional harmonic oscillator, $\psi_n: \mathbb{R} \to \mathbb{C}$, is almost everywhere strongly measurable. The eigenfunction is defined as: $$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$ where $\xi$ is the characteristic length scale and $H_n$ is the $n$-th physicist's Hermite polynomial.

For any natural number $n \in \mathbb{N}$, the $n$-th eigenfunction $\psi_n: \mathbb{R} \to \mathbb{C}$ of the one-dimensional harmonic oscillator belongs to the Hilbert space (denoted as `MemHS`, typically corresponding to the space of square-integrable functions $L^2(\mathbb{R})$). The eigenfunction is defined as: $$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$ where $\xi$ is the characteristic length scale and $H_n$ is the $n$-th physicist's Hermite polynomial.

For any natural number $n$ and any real number $x$, the $n$-th eigenfunction $\psi_n: \mathbb{R} \to \mathbb{C}$ of the one-dimensional harmonic oscillator is differentiable at $x$. The eigenfunction is defined as: $$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$ where $\xi$ is the characteristic length scale and $H_n$ is the $n$-th physicist's Hermite polynomial.

For any natural number $n$, the $n$-th eigenfunction $\psi_n: \mathbb{R} \to \mathbb{C}$ of the one-dimensional harmonic oscillator is a continuous function. The eigenfunction is defined as: $$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$ where $\xi$ is the characteristic length scale and $H_n$ is the $n$-th physicist's Hermite polynomial.

For any natural number $n$, the $n$-th eigenfunction $\psi_n: \mathbb{R} \to \mathbb{C}$ of the one-dimensional harmonic oscillator is an eigenfunction of the parity operator $\hat{P}$ with eigenvalue $(-1)^n$. That is, $$\hat{P} \psi_n = (-1)^n \psi_n$$ where the parity operator $\hat{P}$ is defined as $(\hat{P} \psi)(x) = \psi(-x)$, and the eigenfunction $\psi_n$ is given by: $$\psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi} \xi}} H_n\left(\frac{x}{\xi}\right) e^{-\frac{x^2}{2\xi^2}}$$ with $\xi$ being the characteristic length scale and $H_n$ being the $n$-th physicist's Hermite polynomial.

For any natural numbers $n$ and $p$, the product of the $n$-th and $p$-th eigenfunctions $\psi_n(x)$ and $\psi_p(x)$ of the one-dimensional harmonic oscillator at position $x \in \mathbb{R}$ is given by: $$\psi_n(x) \psi_p(x) = \frac{1}{\sqrt{2^n n!}} \frac{1}{\sqrt{2^p p!}} \frac{1}{\sqrt{\pi} \xi} H_n\left(\frac{x}{\xi}\right) H_p\left(\frac{x}{\xi}\right) \exp\left(-\frac{x^2}{\xi^2}\right)$$ where $\xi$ is the characteristic length scale and $H_k$ denotes the $k$-th physicist's Hermite polynomial.

For any natural number $n$, the square of the $n$-th eigenfunction $\psi_n(x)$ of the one-dimensional harmonic oscillator at position $x \in \mathbb{R}$ is given by: $$\psi_n(x)^2 = \frac{1}{2^n n! \sqrt{\pi} \xi} H_n\left(\frac{x}{\xi}\right)^2 \exp\left(-\frac{x^2}{\xi^2}\right)$$ where $\xi$ is the characteristic length scale and $H_n$ denotes the $n$-th physicist's Hermite polynomial.

For any natural number $n \in \mathbb{N}$, the $n$-th eigenfunction $\psi_n$ of the one-dimensional harmonic oscillator is normalized. That is, the inner product of $\psi_n$ with itself in the Hilbert space is equal to 1: $$\langle \psi_n, \psi_n \rangle = 1$$ Here, $\psi_n$ is the $n$-th eigenfunction defined in terms of Hermite polynomials and the characteristic length scale $\xi$.

For any distinct natural numbers $n$ and $p$ ($n \neq p$), the $n$-th and $p$-th eigenfunctions $\psi_n$ and $\psi_p$ of the one-dimensional quantum harmonic oscillator are orthogonal. That is, their inner product in the Hilbert space is zero: $$\langle \psi_n, \psi_p \rangle = 0$$ where $\psi_k$ is the $k$-th eigenfunction defined using Hermite polynomials and a Gaussian envelope.

The eigenfunctions $(\psi_n)_{n \in \mathbb{N}}$ of the one-dimensional quantum harmonic oscillator form an orthonormal set in the Hilbert space. That is, for any natural numbers $n, m \in \mathbb{N}$, their complex inner product satisfies: $$\langle \psi_n, \psi_m \rangle = \delta_{nm}$$ where $\delta_{nm}$ is the Kronecker delta, which is equal to $1$ if $n = m$ and $0$ otherwise.

The sequence of eigenfunctions $(\psi_n)_{n \in \mathbb{N}}$ of the one-dimensional quantum harmonic oscillator is linearly independent over the field of complex numbers $\mathbb{C}$. Each eigenfunction $\psi_n$ is an element of the Hilbert space of the system.