QII: III, De Broglie’s Postulate — Wavelike Properties of Particles

In this third post, we will discuss matter waves, the wave-particle duality, the uncertainty principle, property of matter waves, and some consequences of the uncertainty principle.

3.1 Matter Waves

De Broglie’s Hypothesis was that the dual (i.e. wave-particle) behavior of radiation also applied to matter; he said that a material particle (an electron for example) has an associated matter wave that regulates its motion.

For both matter and radiation, the total energy E of an entity is related to the frequency v of the wave associated with its motion by the equation:

E = hv

The momentum of an entity is related to the wavelength of the associated wave by the equation:

p = h / λ

h is the bridge to connect these two equations (Planck’s constant) and this forms a relationship through the de Broglie relation.

The evidence for de Broglie’s postulate comes from the Bragg Reflections. Classical particles can’t exhibit interference but waves can. The interference in Bragg’s reflection is the result of one wave crashing onto another wave, not one wave onto an electron.

In the Bragg relation, the first order diffraction maximum (n = 1) is usually the most intense.

3.2 Wave-Particle Duality

In classical physics, energy translocates either by waves or by particles. To understand how this energy travels requires understanding Bohr’s principle of complementarity. The principle of complementarity said it was mutually exclusive to prove either the wave nature or the particle nature of radiation or matter in the same measurement.

But was there a way to unite the two? Max Born suggested the wave function.

The wave function Ψ is a function that represents the de Broglie wave. In Born’s words: “[the wave function] gives the probability of a definite course of the mechanical process”.

At the heart of Quantum Physics is the non-deterministic nature of reality. Thus, there is a growing sense of how probability shapes our descriptions of matter and radiation.

3.3 The Uncertainty Principle

How does one measure space and time? Is there absolute simultaneity?

Classical physics, one can argue, had a hard time answering these questions before modern physics. Inherent in any measurement is uncertainty. Einstein showed that absolute simultaneity does not exist.

Can one examine at the same instant both the position and momentum of matter or radiation? According to Heisenberg, yes but with the uncertainty dictated by the uncertainty principle.

There are two parts: one is the simultaneous measurement of position and momentum and the other regards the measurement of energy and time.

Position and Momentum

The Heisenberg Uncertainty Principle states that “our precision of measurement is inherently limited by the measurement process itself” governed by …

where ΔPx is the uncertainty in momentum, Δx is the uncertainty in position, and h-bar is Planck’s constant over 2 π

Energy and Time

The measurement of energy E and the time T required for the measurement is limited by:

where ΔE is the energy spread, and Δt is the change in time

3.4 Properties of Matter Waves

The law of superposition applies to the wave function. Thus any linear combination of two wave functions give a wave function.

where the following is satisfied:

This is because the simplest solution to a wave function is considered a sinusoidal wave with frequency v and wavelength λ, expressed as:

which is equally:

Plotting

as a function of x for the initial time, t = 0, one gets the following image:

There are two waves in the above image. Each move in the positive x direction. The group velocity is the combined velocity and it is equal to the velocity of the particle whose motion they govern.

To have a wave whose amplitude changes with x or t, a superposition of several monochromatic waves is required.

When many waves combine, their sum wave function has an altered amplitude. p = h / λ and E = hv are applicable to matter and to radiation — and the uncertainty principle is the consequence of the wave-particle duality.

Single Split Experiment with a Single Electron

Diffraction phenomena still involves interference between different parts of a wave belonging to a single particle as opposed to the interference between different particles.

3.5 Consequences of The Uncertainty Principle

Radiation and matter are like coins in that they both, when flipped, be a head (= particle) or a tail (= wave). It depends on how one examines it; this is the heart of Bohr’s principle of complementarity.

The uncertainty principle demonstrates that the mechanics of quantum systems is inherently probabilistic.

Thought Questions

1. What determines if an object appears as a wave or a particle?

The wavelength of an object’s energy is often approximated as it’s size. If it is smaller than the dimensions of the measuring instruments, then it will appear as a particle otherwise it will appear as a wave.

2. What is a matter wave versus an electromagnetic wave?

They both show linearity, interference, diffraction and they carry energy and momentum. Whereas an EM wave travels at only one speed (c), the matter wave has variant travel speeds and is not massless.