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The double-slit experiment is one of the most important demonstrations of the wave–particle duality of quantum systems. It shows that particles such as electrons and photons exhibit interference when not observed, but behave like classical particles when measured.^[1]

Setup

A beam of particles is directed toward a barrier with two narrow slits. After passing through the slits, the particles are detected on a screen.

Interference pattern

When both slits are open and no measurement is made to determine which slit the particle passes through, the detection pattern shows interference:

$I (x) \propto | ψ_{1} (x) + ψ_{2} (x) |^{2} .$

This indicates that the particle behaves as a wave passing through both slits simultaneously.

Which-path measurement

If a measurement is made to determine through which slit the particle passes, the interference pattern disappears. The probability becomes

$I (x) \propto | ψ_{1} (x) |^{2} + | ψ_{2} (x) |^{2} .$

This demonstrates the role of measurement in quantum mechanics.

Interpretation

The experiment illustrates:

superposition of quantum states,
wave–particle duality,
the role of measurement and observation.

It is often regarded as the central experiment revealing the conceptual foundations of quantum mechanics.

Stern–Gerlach experiment

The Stern–Gerlach experiment demonstrates the quantization of angular momentum and provides direct evidence for the existence of intrinsic spin. It was first performed by Otto Stern and Walther Gerlach in 1922.^[2]

Setup

A beam of neutral atoms (typically silver atoms) is passed through a non-uniform magnetic field. The magnetic field gradient exerts a force on the magnetic moment of each atom.

Classical expectation

Classically, one would expect a continuous distribution of deflections, since the orientation of magnetic moments could take any value.

Quantum result

Experimentally, the beam splits into discrete components. For spin-½ particles, two distinct beams are observed, corresponding to:

$m_{s} = + \frac{1}{2}, m_{s} = - \frac{1}{2} .$

This shows that angular momentum is quantized.

Measurement of spin

The Stern–Gerlach apparatus effectively measures the spin component along the direction of the magnetic field:

${\hat{S}}_{z} .$

After passing through the apparatus, the system is projected into one of the eigenstates of ${\hat{S}}_{z}$ .

Sequential measurements

If particles selected in one spin state are passed through a second Stern–Gerlach apparatus oriented along a different axis, the beam splits again. This demonstrates:

the non-commutativity of spin operators,
the probabilistic nature of quantum measurements.

Physical significance

The Stern–Gerlach experiment:

provides direct evidence of spin quantization,
illustrates the role of measurement in quantum mechanics,
demonstrates the discreteness of quantum observables.

It is one of the key experiments underlying the modern understanding of quantum states and measurement.

Laser cooling

Laser cooling is a technique used to reduce the motion of atoms or ions, bringing them to extremely low temperatures close to absolute zero. It relies on the interaction between light and matter to slow down particles using photon momentum.^[3]

Principle

When an atom absorbs a photon of momentum $p = ℏ k$ , it experiences a recoil opposite to its motion. By tuning laser light slightly below an atomic resonance (red detuning), atoms moving toward the laser preferentially absorb photons, resulting in a net slowing force.

This process is often called optical molasses.

Doppler cooling

The simplest form is Doppler cooling, where counter-propagating laser beams create a velocity-dependent force:

atoms moving toward a beam absorb more photons from that direction,
repeated absorption and spontaneous emission reduce kinetic energy.

The minimum achievable temperature is the Doppler limit:

$T_{D} = \frac{ℏ γ}{2 k_{B}},$

where $γ$ is the natural linewidth of the transition.

Sub-Doppler cooling

More advanced techniques, such as Sisyphus cooling, allow temperatures below the Doppler limit by exploiting atomic internal structure and polarization effects.

Applications

Laser cooling is used to:

trap atoms in optical or magnetic traps,
prepare ultra-cold quantum gases,
improve precision in atomic clocks,
enable experiments in quantum optics and quantum information.

Physical significance

Laser cooling:

makes it possible to observe quantum behavior on macroscopic scales,
enables the creation of new states of matter such as Bose–Einstein condensates,
is a cornerstone of modern experimental quantum physics.^[4]

Bose–Einstein condensates

A Bose–Einstein condensate (BEC) is a state of matter that occurs when a dilute gas of bosons is cooled to extremely low temperatures, causing a large fraction of the particles to occupy the lowest quantum state. Under these conditions, quantum effects become apparent on a macroscopic scale.^[5]

The phenomenon was predicted by Satyendra Nath Bose and Albert Einstein in the 1920s and first realized experimentally in 1995.^[6]

Formation

A BEC is formed by cooling a gas of bosonic atoms using techniques such as laser cooling and evaporative cooling. When the temperature drops below a critical value, the de Broglie wavelengths of the atoms overlap, and the particles behave as a single quantum system.

Macroscopic wavefunction

In a Bose–Einstein condensate, the system can be described by a single macroscopic wavefunction

$Ψ (𝐫, t),$

which represents the collective quantum state of the condensate.

This leads to phenomena such as coherence and interference at macroscopic scales.

Properties

BECs exhibit several remarkable properties:

Superfluidity — flow without viscosity
Coherence — phase coherence across the entire system
Quantized vortices — circulation occurs in discrete units

These properties reflect the underlying quantum nature of the condensate.

Applications

Bose–Einstein condensates are used to:

study quantum many-body systems,
simulate condensed matter phenomena,
investigate quantum coherence and interference,
develop precision measurement devices.

Physical significance

Bose–Einstein condensation demonstrates that quantum mechanics can govern the behavior of macroscopic systems. It provides a bridge between microscopic quantum physics and emergent collective phenomena, making it a central topic in modern experimental and theoretical physics.

Table of contents (138 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Matter (by scale)

Matter (by scale)

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

Methods and tools

Methods and tools

Full contents

1. Foundations (7)↑

2. Conceptual and interpretations (10)↑

3. Mathematical structure and systems (11)↑

4. Atomic and spectroscopy (11)↑

Physics:Quantum Atomic structure and spectroscopy
Physics:Quantum Hydrogen atom
Physics:Quantum Multi-electron atoms
Physics:Quantum Fine structure
Physics:Quantum Hyperfine structure
Physics:Quantum Isotopic shift
Physics:Quantum Zeeman effect
Physics:Quantum Stark effect
Physics:Quantum Spectral lines and series
Physics:Quantum Selection rules
Physics:Quantum Fermi's golden rule

5. Wavefunctions and modes (8)↑

Physics:Quantum Wavefunction
Physics:Quantum Superposition principle
Physics:Quantum Eigenstates and eigenvalues
Physics:Quantum Boundary conditions and quantization
Physics:Quantum Standing waves and modes
Physics:Quantum Normal modes and field quantization
Physics:Number of independent spatial modes in a spherical volume
Physics:Quantum Density of states

6. Quantum dynamics and evolution (9)↑

7. Measurement and information (6)↑

8. Quantum information and computing (6)↑

9. Quantum optics and experiments (4)↑

Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
Physics:Quantum optics beam splitter experiments
Physics:Quantum Ultra fast lasers
Physics:Quantum Experimental quantum physics

Template:Quantum optics operators

10. Open quantum systems (7)↑

11. Quantum field theory (18)↑

12. Statistical mechanics and kinetic theory (10)↑

13. Condensed matter and solid-state physics (7)↑

14. Plasma and fusion physics (9)↑

Physics:Quantum Plasma (fusion context)
Physics:Quantum Fusion reactions and Lawson criterion
Physics:Quantum Magnetic confinement fusion
Physics:Quantum Inertial confinement fusion
Physics:Quantum Plasma instabilities and turbulence
Physics:Quantum Tokamak
Physics:Quantum Tokamak core plasma
Physics:Quantum Tokamak edge physics and recycling asymmetries
Physics:Quantum Stellarator

15. Timeline (8)↑

16. Advanced and frontier topics (6)↑

References

↑ Feynman, Richard P. (1965). The Feynman Lectures on Physics, Vol. 3. Addison-Wesley.
↑ Stern, Otto; Gerlach, Walther (1922). "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld". Zeitschrift für Physik 9.
↑ Metcalf, Harold J.; van der Straten, Peter (1999). Laser Cooling and Trapping. Springer.
↑ Foot, Christopher J. (2005). Atomic Physics. Oxford University Press.
↑ Pethick, C. J.; Smith, H. (2008). Bose–Einstein Condensation in Dilute Gases. Cambridge University Press.
↑ Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A. (1995). "Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor". Science 269.

Author: Harold Foppele

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[1] Feynman, Richard P. (1965). The Feynman Lectures on Physics, Vol. 3. Addison-Wesley.

[2] Stern, Otto; Gerlach, Walther (1922). "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld". Zeitschrift für Physik 9.

[3] Metcalf, Harold J.; van der Straten, Peter (1999). Laser Cooling and Trapping. Springer.

[4] Foot, Christopher J. (2005). Atomic Physics. Oxford University Press.

[5] Pethick, C. J.; Smith, H. (2008). Bose–Einstein Condensation in Dilute Gases. Cambridge University Press.

[6] Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A. (1995). "Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor". Science 269.

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Physics:Quantum Experimental quantum physics

Setup

Interference pattern

Which-path measurement

Interpretation

Stern–Gerlach experiment

Setup

Classical expectation

Quantum result

Measurement of spin

Sequential measurements

Physical significance

Laser cooling

Principle

Doppler cooling

Sub-Doppler cooling

Applications

Physical significance

Bose–Einstein condensates

Formation

Macroscopic wavefunction

Properties

Applications

Physical significance

See also

Table of contents (138 articles)

Index

Full contents

References

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