PortLab

Schrödinger equation and dimensionless form

The stationary problem for the one-dimensional harmonic oscillator is

\[ -\frac{\hbar^2}{2m}\,\psi''(x) + \tfrac12 m\omega^2 x^2\,\psi(x) = E\,\psi(x). \]

Introducing the dimensionless variable \(\xi = x/x_0\) with characteristic length \(x_0=\sqrt{\hbar/m\omega}\) and reduced energy \(\varepsilon = E/(\hbar\omega)\), one gets

\[ \psi''(\xi) + \big(2\varepsilon - \xi^2\big)\,\psi(\xi) = 0. \tag{1} \]

Asymptotic behaviour → Hermite equation

As \(\xi\to\pm\infty\) the term \(-\xi^2\psi\) dominates, whose normalizable solutions behave like \(e^{-\xi^2/2}\). Factoring \(\psi(\xi)=e^{-\xi^2/2}\,u(\xi)\) and substituting into (1) leaves the Hermite equation:

\[ u'' - 2\xi\,u' + (2\varepsilon-1)\,u = 0. \tag{2} \]

Parity separation → Kummer equation

The potential is even, so the eigenfunctions have definite parity. With \(t=\xi^2\) the two sectors separate. For the even solutions, \(u(\xi)=f(t)\):

\[ t\,f'' + \big(\tfrac12 - t\big)f' + \tfrac{n}{2}\,f = 0, \]

and for the odd ones, \(u(\xi)=\xi\,f(t)\):

\[ t\,f'' + \big(\tfrac32 - t\big)f' + \tfrac{n-1}{2}\,f = 0. \]

Both have the form of Kummer's equation \(t\,f'' + (b-t)f' - a\,f = 0\), with

\[ a = -\Big\lfloor \tfrac{n}{2}\Big\rfloor, \qquad b = \begin{cases}\tfrac12 & n \text{ even}\\[2pt] \tfrac32 & n \text{ odd.}\end{cases} \]

Riemann's P-symbol

Gauss's hypergeometric equation \(z(1-z)w'' + [c-(a+b+1)z]w' - ab\,w = 0\) has three Fuchsian singular points at \(0,1,\infty\), summarized by Riemann's P-symbol:

\[ w = P\!\begin{Bmatrix} 0 & 1 & \infty & \\ 0 & 0 & a & ;\,z \\ 1-c & c-a-b & b & \end{Bmatrix}. \]

Each column collects the indicial exponents at the corresponding singular point; their sum equals \(1\) (Fuchs relation):

\[ (0 + 1-c) + (0 + c-a-b) + (a+b) = 1. \]

Confluence → Kummer equation

Setting \(z\to x/b\) and letting \(b\to\infty\), the singular points \(1\) and \(\infty\) merge into a single irregular singular point at infinity, while the Fuchsian one at \(0\) remains. The ODE coefficients tend to

\[ x\,w'' + (c-x)\,w' - a\,w = 0, \]

i.e. Kummer's equation, and Gauss's series reduces term by term to \(_1F_1\):

\[ {}_2F_1\!\big(a,b;c;\tfrac{x}{b}\big)\;\xrightarrow[\;b\to\infty\;]{}\;{}_1F_1(a;c;x). \]

Quantization from series termination

As \(t\to+\infty\), \(_1F_1(a;b;t)\sim \dfrac{\Gamma(b)}{\Gamma(a)}e^{t}t^{a-b}\); multiplied by \(e^{-\xi^2/2}\) this would give \(\psi\sim e^{+\xi^2/2}\), not normalizable. The only way out is for the series to terminate, i.e.

\[ a = -\Big\lfloor\tfrac n2\Big\rfloor \in \{0,-1,-2,\dots\}. \]

Eigenfunctions

In unified form, with \(m=\lfloor n/2\rfloor\):

\[ \psi_n(\xi) = N_n\,\xi^{\,(n\bmod 2)}\,e^{-\xi^2/2}\,{}_1F_1\!\Big(-m;\;b;\;\xi^2\Big), \]

equivalent to the closed form \(\psi_n(\xi)=\big(2^n n!\sqrt\pi\big)^{-1/2}H_n(\xi)\,e^{-\xi^2/2}\) with the Hermite polynomials \(H_n\). The \(\psi_n\) form an orthonormal basis: \(\langle\psi_m|\psi_n\rangle=\delta_{mn}\).

Time evolution

A generic state expands on the eigenstates, \(\Psi(\xi,0)=\sum_n c_n\psi_n(\xi)\), and evolves as

\[ \Psi(\xi,\tau) = \sum_n c_n\,\psi_n(\xi)\,e^{-i\,\varepsilon_n\,\tau}, \qquad \tau=\omega t,\ \varepsilon_n=n+\tfrac12. \]

• Single eigenstate: \(\Psi=\psi_n\,e^{-i\varepsilon_n\tau}\). The density \(|\Psi|^2\) is stationary; in the complex plane the wave function rotates rigidly about the \(\xi\) axis — the spiral shown in the 3D panel.
• Coherent state \(|\alpha\rangle\), with \(c_n=e^{-|\alpha|^2/2}\alpha^n/\sqrt{n!}\): a Gaussian packet oscillating as \(\langle\xi\rangle(\tau)=\sqrt2\,\mathrm{Re}\!\big(\alpha e^{-i\tau}\big)\), reproducing the classical motion.
• Periodicity: since the levels are equally spaced, every state is exactly periodic with \(T=2\pi/\omega\).

Eigenvalues and eigenfunctions are computed by the Python engine (via gw2py); this page prepares its input and visualization.