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The Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

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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.
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State Space Representation01:27

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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Entropy Change in Reversible Processes01:10

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In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
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System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
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Related Experiment Video

Updated: Jan 8, 2026

Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
09:23

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

Published on: May 30, 2014

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Learning quantum states of continuous-variable systems.

Francesco A Mele1, Antonio A Mele2, Lennart Bittel2

  • 1NEST, Scuola Normale Superiore and Istituto Nanoscienze, Pisa, Italy.

Nature Physics
|December 15, 2025
PubMed
Summary
This summary is machine-generated.

Quantum-state tomography for continuous-variable systems is highly inefficient, requiring exponentially more resources than for qubits. However, efficient methods exist for Gaussian and certain non-Gaussian states.

Keywords:
Imaging and sensingQuantum informationTheoretical physics

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Area of Science:

  • Quantum physics
  • Quantum information science
  • Quantum optics

Background:

  • Quantum measurements are inherently probabilistic, yielding partial information about quantum states.
  • Quantum-state tomography reconstructs a full quantum state description from multiple measurements.
  • Continuous-variable systems (e.g., bosonic, quantum optical) pose unique tomography challenges.

Purpose of the Study:

  • To analyze the ultimate performance limits of quantum-state tomography for continuous-variable systems.
  • To compare the efficiency of continuous-variable tomography with finite-dimensional systems.
  • To identify conditions and methods for efficient tomography of specific quantum states.

Main Methods:

  • Theoretical analysis of resource scaling in quantum-state tomography.
  • Derivation of error bounds for state reconstruction.
  • Investigation of Gaussian and non-Gaussian state preparation and tomography.

Main Results:

  • Tomography of continuous-variable systems is found to be extremely time-resource inefficient compared to qubits.
  • The number of required state copies scales exponentially with the number of modes and unfavorably with error.
  • Efficient tomography protocols are proven for Gaussian states and experimentally feasible for certain non-Gaussian states.

Conclusions:

  • Continuous-variable quantum-state tomography is generally resource-intensive.
  • Gaussian states and specific non-Gaussian states offer efficient tomography pathways.
  • These findings impact the development of quantum technologies relying on state characterization.