Double Slit Experiment

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

Bullet

Double slit thought experiment with classical bullets

Imagine a gun which is shooting bullets randomly toward a wall with two slits in it separated by a distance,  d \! . The slits are about the size of a bullet. A histogram of the bullet's location after it passes through the two slits is plotted. If slit 2 is closed, but the slit 1 is open, then the green peak is observed which is given by the distribution function p_1\!. Similarly, if the slit 1 is closed, but he slit 2 is open, the pink peak is observed which is given by the distribution function p_2\!. When both slits are open, p_{12}\! (purple) is observed. This agrees with the classical view, where the bullet is the particle and p_{12}\! is simply a sum of p_1\! and p_2\!. The bullets do not follow purely linear trajectories because they are allowed to hit the edges of the slits they pass through and be deflected. It is because the bullets can be deflected that the result of this experiment is a probability distribution rather than the bullets going to just the two locations that are along straight line trajectories from the gun through the slits.

The equation describing the probability of the bullet arrival if both of the slit are open is therefore

p_{12}=p_1+p_2.\!


Classical Waves

Double slit thought experiment with water waves

As waves are passed through the double slit, they are diffracted so that the waves emerge from the slit as circular waves. This effect can only occur when the size of the slits is comparable to the wavelength. The intensities H1 and H2 of the waves, which are proportional to the squares of their amplitudes, are observed when only slit 1 and slit 2 are open, respectively. These intensities are similar to the histograms for the bullets in the previous demonstration. However, an interference pattern of intensity H_{12}\! is observed when both slits are opened. This is due to constructive (peaks) and destructive (troughs) interference of the two waves.

Hot Tungsten Wire (thermal emission of electrons)

A high current is passed through a tungsten wire, resulting in electrons being emitted from the wire which then enter the double slits one at a time, arriving in the same manner as the bullet arrives from the gun. However, after plotting a histogram of the locations where the electron landed, it looks like  H_{12}\! for the double slit wave experiment. This shows that electrons exhibit both the wave-like and the particle-like character. The probability distribution of the electron's landing on the screen thus exhibits the interference patterns. It is the laws obeyed by these probability "amplitudes" that quantum mechanics describes.

Reference

[1] Feynman, R.B. Leighton and M.L.Sands The Feynman Lectures on Physics, vol 3, Addison-Wesley, (1989), Chapter 1.

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