Physlib.QuantumMechanics.HarmonicOscillator.Eigenstates

For a quantum harmonic oscillator in $d$ dimensions, the energy eigenvalue associated with a vector of quantum numbers $n$ is given by the sum $E_n = \sum_i \hbar \omega_i (n_i + \frac{1}{2})$, where $\omega_i$ is the angular frequency of the $i$-th dimension, $n_i$ is the corresponding non-negative integer quantum number, and $\hbar$ is the reduced Planck constant.

For a quantum harmonic oscillator in $d$ dimensions with angular frequencies $\omega_i$ for each dimension $i$, the energy eigenvalue $E_n$ associated with a vector of non-negative integer quantum numbers $n = (n_1, n_2, \dots, n_d)$ is given by the sum $$E_n = \sum_{i} \hbar \omega_i \left(n_i + \frac{1}{2}\right)$$ where $\hbar$ is the reduced Planck constant.

For a quantum harmonic oscillator, the energy eigenvalue function $E_n$, which maps a vector of quantum numbers $n \in \mathbb{N}^d$ to its corresponding energy $E_n = \sum_i \hbar \omega_i (n_i + \frac{1}{2})$, is strictly monotonic. That is, if $n < m$ (meaning $n_i \leq m_i$ for all $i$ and $n_j < m_j$ for at least one $j$), then $E_n < E_m$.

For a $d$-dimensional quantum harmonic oscillator $Q$ and a given set of quantum numbers $n = (n_1, \dots, n_d)$, the normalization coefficient for the $i$-th dimension is defined as: $$\frac{1}{\sqrt{2^{n_i} n_i! \sqrt{\pi} \xi_i}}$$ where $n_i$ is the quantum number associated with the $i$-th dimension and $\xi_i$ is the characteristic length of the oscillator in that dimension.

For a $d$-dimensional quantum harmonic oscillator $Q$ and a given set of quantum numbers $n = (n_1, \dots, n_d)$, the normalization coefficient for the $i$-th dimension is given by: $$\text{eigenCoeff}(n, i) = \frac{1}{\sqrt{2^{n_i} n_i! \sqrt{\pi} \xi_i}}$$ where $n_i$ is the quantum number associated with the $i$-th dimension and $\xi_i$ is the characteristic length of the oscillator in that dimension.

For a quantum harmonic oscillator $Q$ in $d$ dimensions with characteristic lengths $\xi = (\xi_1, \dots, \xi_d)$, and given a set of integer quantum numbers $n = (n_1, \dots, n_d) \in \mathbb{N}^d$, the energy eigenfunction $\psi_n$ is a Schwartz function in $\mathcal{S}(\mathbb{R}^d, \mathbb{C})$ defined by the product: $$\psi_n(x) = \prod_{i=1}^d \frac{1}{\sqrt{2^{n_i} n_i! \sqrt{\pi} \xi_i}} H_{n_i}\left(\frac{x_i}{\xi_i}\right) \exp\left(-\frac{1}{2} \left(\frac{x_i}{\xi_i}\right)^2\right)$$ where $x = (x_1, \dots, x_d) \in \mathbb{R}^d$ are the Cartesian coordinates and $H_{n_i}$ denotes the $n_i$-th physicist's Hermite polynomial.

For a $d$-dimensional quantum harmonic oscillator $Q$ with characteristic lengths $\xi = (\xi_1, \dots, \xi_d)$ and a set of quantum numbers $n = (n_1, \dots, n_d)$, the energy eigenfunction $\psi_n$ is defined by multiplying the standard Gaussian $g(x) = \exp\left(-\frac{1}{2} \|x\|^2\right)$ by the product of normalization coefficients and physicist's Hermite polynomials $H_{n_i}$, and then pre-composing with the scaling transformation $x \mapsto (x_1/\xi_1, \dots, x_d/\xi_d)$. Explicitly, the eigenfunction is given by: $$\psi_n(x) = \prod_{i=1}^d \text{eigenCoeff}(n, i) H_{n_i}\left(\frac{x_i}{\xi_i}\right) \exp\left(-\frac{1}{2} \left(\frac{x_i}{\xi_i}\right)^2\right)$$ where the normalization constant for the $i$-th component is $\text{eigenCoeff}(n, i) = \frac{1}{\sqrt{2^{n_i} n_i! \sqrt{\pi} \xi_i}}$.

For a quantum harmonic oscillator $Q$ in $d$ dimensions with characteristic lengths $\xi = (\xi_1, \dots, \xi_d)$, and given a vector of integer quantum numbers $n = (n_1, \dots, n_d) \in \mathbb{N}^d$, the value of the energy eigenfunction $\psi_n$ at a position $x = (x_1, \dots, x_d) \in \mathbb{R}^d$ is given by the product: $$\psi_n(x) = \prod_{i=1}^d C_{n,i} H_{n_i}\left(\frac{x_i}{\xi_i}\right) \exp\left(-\frac{1}{2} \left(\frac{x_i}{\xi_i}\right)^2\right)$$ where $C_{n,i}$ are the normalization coefficients (`eigenCoeff`), $H_{n_i}$ are the $n_i$-th physicist's Hermite polynomials, and $\xi_i$ are the characteristic lengths in each dimension.

For a quantum harmonic oscillator $Q$ in $d$ dimensions and a vector of quantum numbers $n = (n_1, \dots, n_d) \in \mathbb{N}^d$, the energy eigenstate is the element of the Schwartz submodule of the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ corresponding to the energy eigenfunction $\psi_n$. It is defined by mapping the Schwartz function $\psi_n \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ into the Hilbert space via the canonical linear isomorphism.

For a $d$-dimensional quantum harmonic oscillator $Q$ and a vector of quantum numbers $n = (n_1, \dots, n_d) \in \mathbb{N}^d$, the energy eigenstate $|n\rangle$ in the Hilbert space is equal to the image of the energy eigenfunction $\psi_n \in \mathcal{S}(\mathbb{R}^d, \mathbb{C})$ under the canonical linear isomorphism between the Schwartz space and its corresponding subspace in the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$.

For a $d$-dimensional quantum harmonic oscillator $Q$, the inner product of two energy eigenstates with multi-indices of quantum numbers $n, n' \in \mathbb{N}^d$ in the Hilbert space $L^2(\mathbb{R}^d, \mathbb{C})$ is equal to the Kronecker delta $\delta_{n, n'}$. That is, $$\langle \psi_n, \psi_{n'} \rangle = \delta_{n, n'}$$ where $\delta_{n, n'}$ is $1$ if $n = n'$ and $0$ otherwise.