Double Slit Experiment

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.

Physical Basis of Quantum Mechanics

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

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

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

Motion in One Dimension

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

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

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 $H_{1}$ and $H_{2}$ 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.