Wave-Particle Duality - the Principle of Complementarity The principle of complementarity states that both the wave and particle aspects of light are fundamental to its nature. Indeed, waves and particles are just our interpretations of how light behaves. We have phenomena such as diffraction and interference that show that light is a wave, and phenomena such as the photoelectric effect that show that it is a particle. Which is it? This question has no answer; we must accept the dual wave-particle nature of light. 1

Wave Nature of Matter Just as light sometimes behaves as a particle, matter sometimes behaves like a wave. (Idea of the symmetry in nature - de Broglie 1927) The wavelength of a particle of matter (De Broglie wavelength) is: This wavelength is extraordinarily small. The wave nature of matter becomes more important for very light particles such as the electron. Electron wavelengths can easily be on the order of m; electrons can be diffracted by crystals just as X-rays can. The photon, the quantum of light Energy: Momentum: 2

Example1: What is the de Broglie wavelength of a 0.20kg ball moving with speed 15m/s? Example2: Determine the wavelength of an electron that has been accelerated through a potential difference of 100 V. Scanning electron microscopy (SEM) image 3

De Broglie’s Hypothesis Applied to Atoms De Broglie’s hypothesis is the one associating a wavelength with the momentum of a particle. He proposed that only those orbits where the wave would be a circular standing wave will occur. This yields the same relation that Bohr had proposed. The correspondence principle applies here as well – when the differences between quantum levels are small compared to the energies, they should be imperceptible. In addition, it makes more reasonable the fact that the electrons do not radiate, as one would otherwise expect from an accelerating charge. These are circular standing waves for n = 2, 3, and 5. quantization: de Broglie’s wavelength: Bohr’s quantization:

About Quantum Mechanics Quantum mechanics incorporates wave-particle duality, and successfully explains energy states in complex atoms and molecules, the relative brightness of spectral lines, and many other phenomena. It is widely accepted as being the fundamental theory underlying all physical processes. Quantum mechanics is essential to understanding atoms and molecules, but can also have effects on larger scales. Classical physics v<<c Classical relativistic physics v~c Quantum nonrelativistic physics v<<c Quantum relativistic physics v~c 5

The Wave Function and Its Interpretation Question: An electromagnetic wave has oscillating electric and magnetic fields. What is oscillating in a matter wave? Answer: This role is played by the wave function, Ψ. The square of the absolute value of the wave function at any point is proportional to the number of electrons expected to be found there. For a single electron, the wave function is the probability of finding the electron at that point. In the classical mechanics we use Newton’s equations of motion to describe particles positions and velocities, in the classical electrodynamics we use Maxwell’s equations to describe the electric and magnetic fields. In quantum mechanics we use Schrödinger equation to describe the function: Ψ= Ψ(x,y,z;t). 6

The Wave Function and the Double-Slit Experiment The interference pattern is observed after many electrons have gone through the slits. If we send the electrons through one at a time, we cannot predict the path any single electron will take, but we can predict the overall distribution. Philosophic Implications - Probability versus Determinism The world of Newtonian mechanics is a deterministic one. If you know the forces on an object and its initial velocity, you can predict where it will go. Quantum mechanics is very different – you can predict what masses of electrons will do, but have no idea what any individual one will. 7

The Heisenberg Uncertainty Principle Quantum mechanics tells us there are limits to measurement – not because of the limits of our instruments, but inherently. This is due to the wave-particle duality, and to interaction between the observing equipment and the object being observed. The Heisenberg uncertainty principle tells us that the position and momentum cannot simultaneously be measured with precision. Similar: 8

Two examples Example 1: An electron moves in a straight line with a constant speed v=1.10x10 6 m/s which has been measured to a precision of 0.10%. What is the maximum precision with which its position could be simultaneously measured? Example 2: What is the uncertainty in position, imposed by the uncertainty principle, on 150–g baseball thrown at (93 ± 2) mph = (42 ± 1) m/s? 9

The uncertainty principle for energy and time This says that if an energy state only lasts for a limited time, its energy will be uncertain. It also says that conservation of energy can be violated if the time is short enough. Example: The Z boson typically decays very quickly. Its average energy is GeV, but its short lifetime shows up as an intrinsic width of 2.49 GeV (rest energy uncertainty). What is the lifetime of this particle? 10

Imagine trying to see an electron with a powerful microscope. At least one photon must scatter off the electron and enter the microscope, but in doing so it will transfer some of its momentum to the electron. The uncertainty in the momentum of the electron is taken to be the momentum of the photon: In addition, the position can only be measured to about one wavelength of the photon: Example: Combining, we find the combination of uncertainties: 11