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Quantum Numbers

Quantum Numbers

It is said that the energy of an electron in an atom is quantized; that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels.

The Quantum-Mechanical Model of an Atom

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra.

Oscillations In An LC Circuit

Oscillations In An LC Circuit

An idealized LC circuit of zero resistance can oscillate without any source of emf by shifting the energy stored in the circuit between the electric and magnetic fields. In such an LC circuit, if the capacitor contains a charge q before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the capacitor. This energy is given by

Forced Oscillations

Forced Oscillations

When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.

Damped Oscillations

Damped Oscillations

In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...

Oscillations about an Equilibrium Position

Oscillations about an Equilibrium Position

Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so...

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Related Experiment Video

Updated: Jan 31, 2026

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators

Published on: May 30, 2014

Quantum magnetic J-oscillators.

Jingyan Xu^1,2,3, Raphael Kircher^1,2,3, Oleg Tretiak^1,2,3

¹Helmholtz Institute Mainz, Mainz, Germany.

Nature Communications

|January 29, 2026

Summary

This summary is machine-generated.

Quantum J-oscillators leverage molecular J-couplings for magnet-free, high-resolution spectroscopy. This breakthrough enables precise measurements and molecule discrimination, offering a novel platform for quantum dynamics exploration.

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Area of Science:

Quantum physics
Spectroscopy
Molecular dynamics

Background:

Zero-field nuclear magnetic resonance (NMR) provides magnet-free access to scalar J-couplings, crucial for molecular characterization.
Conventional zero-field NMR faces limitations in spectral resolution and frequency stability due to transient signals.

Purpose of the Study:

To introduce quantum J-oscillators for generating phase-coherent oscillations using molecular J-couplings.
To achieve ultra-high resolution and frequency stability in magnet-free spectroscopy.
To explore nonlinear spin dynamics and quantum chaos in a compact tabletop platform.

Main Methods:

Development of quantum J-oscillators exploiting J-couplings in molecules.
Operation in zero magnetic field with digital feedback control.
Proof-of-principle experiment using [15N]-acetonitrile.

Main Results:

Achieved phase-coherent continuous oscillations in zero magnetic field.
Demonstrated a linewidth of 340 μHz over 3600 s for [15N]-acetonitrile, over two orders of magnitude narrower than conventional zero-field NMR.
Established a compact, magnet-free platform for precision spectroscopy and quantum dynamics.

Conclusions:

Quantum J-oscillators offer a significant advancement in high-resolution, magnet-free spectroscopy.
The technology facilitates precise J-coupling measurements and molecular discrimination.
This platform opens new avenues for applications requiring ultraprecise frequency references and molecular fingerprints.