13.1 PHYS 1010Q © D.S. Hamilton

The Wave-Particle Duality for Light

Light behaves like a wave in traveling from a source to where it is detected. This behavior is evident from wave interference measurements and the prediction of the speed of light from the electromagnetic wave equation of Maxwell.

Light behaves like a particle when it is being emitted, absorbed, or scattered by an atom. This behavior is key to understanding the Planck radiation law, the photoelectric effect, and the spectra of atoms.

So is Light a Wave or a Particle?

Seemingly contradictory, the wave-particle duality implies that light has both particle-like and wave-like properties. This duality addresses the inability of classical concepts like "particle" and "wave" to fully describe the nature of light in all situations.

13.2 PHYS 1010Q © D.S. Hamilton

Matter as both particles and wavesIn 1924, French doctoral student Louis de Broglie proposed that particles, such as electrons, could act as waves just as waves of light could sometimes act as particles. Every particle is endowed with wave characteristics as it travels.

Planck's constant h hwavelength = ; λ = =

momentum p mv

The de Broglie wavelength of a car (mass = 1500 kg) traveling at 30 m/s (67 mph) is only 1.5x10-38 m, which is much smaller that any atom or nucleus.

What is the de Broglie wavelength of an electron (mass = 9.1x10-31 kg) traveling at 1,000 km/s? (The kinetic energy of this electron is about 3 eV.)

de Broglie also proposed that the stable Bohr orbits of the electrons in an atom are those where the electron wave closes back on itself.

stable unstable

13.3 PHYS 1010Q © D.S. Hamilton

The electron wave must fit evenly into the circumference of the circular orbit, ie C = nλ.

The radius of the n=1 Bohr orbit is 0.0531 nm. What is the wavelength and speed of an electron in this orbit?

Show that this requirement of C = nλ is equivalent to Bohr’s quantization rule for the angular momentum, L = mvr = nh/2π.

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Wave interference

An interference pattern for light (left) and electrons (right) passing by a sharp "knife-edge". Interference is conclusive evidence for wave-like behavior.

An electron microscope makes practical use of the wave nature of electrons. The de Broglie wavelength of high-speed electrons is typically thousands of times shorter than the wavelength of visible light, so the electrons are able to distinguish details not apparent with light.

13.5 PHYS 1010Q © D.S. Hamilton

What is waving??

In Schrodinger's wave equation, the thing that waves is a mathematical entity called the wave function Ψ (psi = "sigh"). The wave function represents the possibilities that can occur for a system.

This figure show the probability distribution Ψ2 of an electron cloud of the lowest energy state in the hydrogen atom. We can not tell exactly where the electron will be found in the atom at a given moment, but only the likelihood (probability) of finding it there.

The calculation of the wave function is completed by solving the Schrödinger wave equation for Ψ(x).

For a particle confined to a rigid box of length L, the solutions Ψ(x) to the Schrödinger wave equation are those standing waves that just fit inside the box. The probability distribution for finding the particles is then proportional to Ψ2.

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For an electron of mass m confined in a one-dimensional box of length L, find expressions for the de Brogliewavelength, momentum p, velocity v, and kinetic energy KE.

Compare the momentum p of an electron in the n=1 quantum ground state of a one-dimensional box of length L to the momentum when the electron is in the n=3 excited state.

13.7 PHYS 1010Q © D.S. Hamilton

Uncertainty PrincipleThere are pairs of quantities in physical systems that are linked in such a way that we cannot know them both simultaneously with infinite accuracy. For waves, it is well known that the longer we have to measure the frequency f of the wave, the smaller the uncertainty in the value of f. Mathematically, this can be stated as Δf·Δt ≥ 1.

Because matter has wave properties, there is another pair of such variables, the position of a particle and its momentum along the same direction that have an uncertainty relation, ie, Δx·Δpx ≥ h.

Show that short acoustic "click" of duration Δt=0.1 mseccontains a band of frequencies having a width Δf=10 kHz.

Show that Δf·Δt ≥ 1 implies that ΔE·Δt ≥ h.

A wave with a definite wavelength λimplies that the momentum is also precisely known since p=h/λ. But the wavefunction Ψ and the probability Ψ2

of finding the particle is now spread out over some region of space. The more certain we are about its momentum, the more uncertain we become about its position.

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Suppose you can measure the speed of an object to a precision of 10%. What is the minimum uncertainty in the position of a 0.145 kg-baseball moving at 40 m/s (90 mi/h)? For an electron in the n=1 Bohr orbit of hydrogen with a velocity of 2.2x106 m/s?

Uncertainty Principle continued

Quantum uncertainties stem from the wave nature of matter.

By adding several waves of different wavelengths λ, you can produce an interference pattern which localizes the wave. But that process involves a spread in momentum values. There is an inherent increase in the uncertainty Δp when the wave becomes more particle-like and Δx decreases. The more certain we are about its position, the more uncertain we become about its momentum.

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The wave particle duality is an underlying principle of the Universe. The complete description of an electron or a photon requires both its wave and particle aspects. If two concepts are complementary, an experiment that clearly illustrates one concept will obscure the other. For example, an experiment that illustrates the particle properties of light will not show any of the wave properties of light. Complementarity is not a compromise with the truth being somewhere in between a particle picture and a wave picture. Rather it is a statement of the dual nature of the quantum reality.

Complementarity Principle

Opposites are seen to complement each other in this view of the world. Neils Bohr was knighted for his work in quantum physics and selected the yin-yang symbol for his coat of arms.

Determinism

Newton's laws showed us how to calculate of the future motion of any particle. Is then the motion of the entire Universe completely predetermined? And does this depend on our ability to calculate the future?

Quantum mechanics suggests the future is a statistical issue. We can calculate the probabilities for certain events, but we do not know exactly where or when they will actually occur.