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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.

The Dot Product

The Dot Product

Measuring how one directional quantity affects another along a specific path involves comparing their orientation and strength. When two such quantities are represented using direction and amount, a numerical result is computed to show how much one acts along the path of the other. This result comes from a rule combining both inputs' horizontal and vertical parts and adding the results.This calculation gives a single value that grows larger when both inputs point in similar directions and...

Dot Product

Dot Product

The dot product is an essential concept in mathematics and physics.
In engineering, the dot product of any two vectors is the product of the magnitudes of the vectors and the cosine of the angle between them. It is denoted by a dot symbol between the two vectors.
Consider a vehicle pulling an object along the ground using a rope. If the rope makes an angle with the horizontal axis, the work done can be calculated using the dot product of the force applied and the object's displacement.
The dot...

What is Genetic Engineering?

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Dot Product: Problem Solving

Dot Product: Problem Solving

The dot product is a powerful tool in problem-solving involving vectors, given that the dot product of two vectors is the product of their magnitudes and the cosine of the angle between them measured anti-clockwise. Solving problems involving the dot product requires understanding its properties and developing a step-by-step process to solve them. Here are the main steps to follow when solving any general problem involving the dot product:
Identify the problem: Start by reading the problem and...

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Related Experiment Video

Updated: Feb 10, 2026

Synthesis of Cd-free InP/ZnS Quantum Dots Suitable for Biomedical Applications

Synthesis of Cd-free InP/ZnS Quantum Dots Suitable for Biomedical Applications

Published on: February 6, 2016

Engineering Mn²⁺-Doped CdS/ZnS Quantum Dot Surfaces to Control Auger Upconversion Photocatalysis.

Bereket L Zekarias¹, Luis Pablo Arriaga Gonzalez¹, Olivia A De Luca²

¹Department of Chemistry, Columbia University, New York, New York 10027, United States.

Journal of the American Chemical Society

|February 9, 2026

Summary

Surface-modified manganese-doped cadmium sulfide/zinc sulfide quantum dots exhibit high stability and catalytic activity. These engineered quantum dots efficiently drive various organic transformations using minimal catalyst loadings.

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Last Updated: Feb 10, 2026

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Synthesis of Core-shell Lanthanide-doped Upconversion Nanocrystals for Cellular Applications

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Production and Targeting of Monovalent Quantum Dots

Production and Targeting of Monovalent Quantum Dots

Published on: October 23, 2014

Area of Science:

Materials Science
Photocatalysis
Quantum Dot Chemistry

Background:

Quantum dots (QDs) are semiconductor nanocrystals with tunable optical and electronic properties.
Surface ligand engineering is crucial for controlling QD stability and reactivity.
Hot electrons generated in QDs offer potential for driving chemical reactions.

Purpose of the Study:

To develop stable and highly active photocatalysts based on Mn-doped CdS/ZnS quantum dots.
To investigate the role of surface ligand modification on QD photocatalytic performance.
To explore the application of engineered QDs in challenging organic transformations.

Main Methods:

Synthesis of Mn-doped CdS/ZnS quantum dots with ion-pair forming ligands, replacing stearate.
Characterization of QD stability and properties in polar solutions.
Evaluation of photocatalytic activity in aryl chloride reduction, defluorination, and C-C bond formation reactions.

Main Results:

The modified QDs demonstrated excellent stability and catalytic activity at very low loadings (0.0005 mol %).
Efficient reduction of aryl chlorides and defluorination of fluorinated aromatics were achieved.
Facilitation of new carbon-carbon bond formation was observed, showcasing broad applicability.

Conclusions:

Engineering quantum dot surface chemistry by replacing ligands with ion-pair formers enhances photocatalytic activity.
The observed reactivity is attributed to hot electrons generated via multiphoton absorption and Auger upconversion.
These stable, electrostatically stabilized QDs represent a promising platform for advanced photocatalysis.