9.8.B - Birth of Quantum Physics


The period 1905 to 1925 was a great time for the world's leading physicists in the race to understand the quantum nature of matter. To explain so many curious and unintuitive phenomena about radiation, atoms, molecules and solid materials, some groups worked together and sometimes compete with another. The climax happened around 1925 when the structure of quantum mechanics was finally laid down.

This started with Louis de Broglie's conjecture in 1924 that particle-like objects such as electrons should display wave properties. Indeed if light which is initially thought to be a wave can behave as a particle or quantum, why not those objects which we normally conceive of as particles display wave-like properties. Shortly after de Broglie introduced his concept of matter waves, Erwin Schrodinger proposed an answer to the question of what happens to the matter waves when a force acts on it. He came up with a wave equation now known as Schrodinger’s equation that lies at the heart of quantum mechanics. Given a particle and the force that acts on it, Schrodinger's equation gives the possible waves associated with this particle at a given position and time. And this is designated by the hardest working symbol in modern physics: the wave function.

Wolgang Pauli proposed that each electron in an atom could be described by four quantum numbers and that no two electrons in an atom could have a set of four identical quantum numbers. His exclusion principle and rules provided a system to explain the arrangement and number of electrons in each atomic orbital, thus providing an explanation for the structure of the periodic table.

In this unit you will:

  • Apply de Broglie's equation to determine the wavelength of particles
  • Describe experimental evidence to confirm the existence of matter waves
  • Explain the stability of Bohr's orbits in terms of electrons as standing waves
  • Oultine Heisenberg's uncertainty principle and how it applies to electrons in atoms
  • Outline Pauli's contribution to our understanding of electrons in atoms


De Broglie's Equation

Physics was at an interesting stage in the early 1900s. Almost everything anyone did seemed to contradict existing theory. De Broglie's idea was no exception. Quanta, or particles of EM radiation were generally accepted, as was an understanding that EM radiation could behave as though it were a wave and a particle.

If light interacted with itself, such as reflection, refraction, diffraction, etc., it would behave as a wave. However, if it interacted with matter, such as the photoelectric effect, it behaved like a particle. Scientists had accepted that waves sometimes behaved like particles, so de Broglie reasoned that particles such as electrons should also be able to behave like a waves. In 1922 he proposed that any mass that was moving had a wavelength associated with it. According to de Broglie, if a particle had momentum mv, it had an associated matter wave with wavelength, λ.

Equation - De Broglie's Equation

λ

h

m

v

wavelength of the matter wave

Planck's constant

mass of the particle

velocity of the particle

metres (m)

6.626 x 10-34 Js

kilograms (kg)

metres per second (ms-1)

You should use this equation to calculate the wavelength of particle's matter-wave.


Standing wave patterns in
allowed orbits.

De Broglie also proposed that the allowed orbits for an electron corresponded to radii where electrons formed standing waves around the nucleus. The condition for a standing wave to form would be that a whole number n of de Broglie wavelengths must fit around the circumference of an orbit of radius r. Thus the first stationary energy state (n = 1) corresponds to an allowed orbit containing one complete electron standing wave wavelength; the second stationary state corresponds to an allowed orbit containing two complete electron standing wave wavelengths; and so on.

The wavelength of an electron is in the order of 10-12 m, which means that about a thousand billion wavelengths fit into one metre. The frequency of an electron is about 1020 Hz so they are vibrating very very quickly. Given that a film shown in a movie theater is around 24 frames s-1 (Hz) and it appears as continuous motion, you then realise why we see matter as continuous motion as well.

If we now ask the simple question: are there any electron orbits for which the wave and particle descriptions are consistent? In other words, are there orbits for which the velocity of the electron (when we think of it as a particle) is appropriate to the orbit, while at the same time the electron wave (when we think of it as a wave) fits onto the orbit, given the relation between wavelength and velocity?

When you do the mathematics which are beyond the scope of this course, you find that the only orbits that satisfy these twin conditions are equivalent to the Bohr electron energy levels. That is to say, the only orbits allowed in the atom are those for which it makes no difference whether we think of the electron as a particle or a wave. In a sense then, the wave-particle duality exists in our minds, and not in nature, as nature has arranged things so that what we think doesn't matter!

The apparent wave-particle conflict has another resolution in the principle of complementarity which as first stated by Niels Bohr in 1928. The wave descriptions and particle descriptions are complementary. That is, we need both descriptions to complete our model of nature, but we will never need to use both descriptions at the same time to describe a single part of an occurence.

De Broglie succeeded in showing that Bohr's allowed orbits defined by the radii are those for which the circumference of the orbit can contain exactly an integral number of de Broglie wavelengths.

This important discovery represented the birth of quantum mechanics. This branch of science is the study of the motion of objects that come in extremely small discrete bundles, or quanta. Material inside the atom comes in tiny bundles such as electrons, protons, nuetrons or even the nucleus as a whole and they are therefore the domain of quantum mechanics. Electron transistions in atom also involve the absorption or emission of discrete energy quanta. Quantum mechanics involved sweeping revisions of our fundamental descriptions of matter. Newtonian mechanics (part of classical physics) did not work at the quantum scale so a new theory and language was needed to study these small particles and assoicated energies.

In quantum mechanics, a particle is not described as a geometric point in space but a wave-like entity that spreads out in space in three dimensions. The spatial distribution of a particle is described by a function called the wave function (ψ) which was Schrodinger's synthesis of the work of de Broglie, Planck and Einstein. The wave function is a complex mathematical equation that, when solved, contains all measurable information about the particle. If you square this function (ψ2) you obtain a probability density which for an electron and this tells us the likelyhood of finding it at a point in space and time. The larger the value of the ψ2, the higher the probability of the finding the electron there. We take up this idea again later in the section on Wolgang Pauli's contribution to the developing model of atomic structure.

Experimental Confirmation of Matter Waves



Drs Honeydew and Valder chatting about matter waves and diffraction.

Experimental confirmation of de Broglie's proposal on matter waves was achieved in 1927 by Davisson and Germer in the USA (and by George Thomson in Scotland). They were studying the scattering of electrons being a fired at a nickel surface. The vacuum tube they were using had a crack in it that let air in and formed a nickel oxide coating on the surface. They heated the nickel strongly to remove the nickel oxide, but this changed the crystalline structure on the surface of the nickel, forming nickel crystals with different interatomic spacing. The next time they used the equipment the results they obtained were different. With the changed crystalline structure, they observed diffraction patterns like those produced by X-rays in X-ray crystallography experimenst such as those pioneered by the Braggs.

Equipment used by Davisson and Germer in 1927
to confirm that electrons could indeed behave as
though they were waves.

The Davisson-Germer experiments involved streams of electrons being diffracted when they were scattered from the surface of crystals. They were able to show that electrons were diffracted from a nickel crystal in much the same way as X-rays were scattered from graphite. When they inadvertently changed the crystalline structure of the nickel, they changed the interatomic spacing of the atoms to roughly the same length as the wavelength of an electron. This then allowed diffraction of the electrons to take place. Since diffraction is a property shown only by waves, their experimental results (indicating maximum constructive interference at particular angles) agreed with the formula proposed by de Broglie. The electrons in the Davisson-Germer experiment were scattered in specific directions, which could only be explained by treating the electrons as waves with a wavelength related to their momentum (as indicated by the de Broglie relationship). Until now, electrons were accepted as particles, but diffraction is a wave property. The conclusion was that electrons (particles) were behaving as waves. These waves were called de Broglie matter waves and his idea was confirmed experimentally.

 

Heisenberg's Uncertainty Principle

Werner Heisenberg who proposed
the Heisenberg Uncertainty
Principle in 1927 (sometimes known
by students as the Heisenberg .

When you look at a piece of matter such as this book, you can see it because light bounces off the book and comes to your eye, a very sophisticated detector. When you examine a piece of fruit, you apply energy by squeezing it to detect if it feels too ripe. Many professions employ sophisticated devices to make their measurements. Air traffic controllers reflect microwaves off airplanes to determine their positions, oceanographers bounce sound waves off deep-ocean sediments to map the seafloor and dentists pass X-rays through your teeth and gums to look for decay. In our everyday world we assume that such interactions of matter and energy do not change the objects being measured in any significant way. Microwaves don't alter an airplane's flight path, nor do sound waves disturb the topography of the ocean's bottom. And while prolonged exposure to X-rays can be harmful, the dentist's brief exploratory X-ray photograph has no obvious immediate effects on the tooth. Our experience tells us that a measurement can usually be made on a macroscopic object (something large enough to be seen without a microscope) without altering that object, because the energy of the probe is much less than the energy of the object.

The situation is rather different in the quantum world. If you want to 'see' an electron, you have to bounce energy off it so that the information can be carried to your detectors. But nothing at your disposal can interact with the electron without simultaneously affecting it. You can bounce a photon off it, but in the process the electron's energy will change. You can bounce another particle off it, but the electron will recoil like a billiard ball. No matter what you try, the energy of the probe is too close to the energy of the thing being measured. The electron cannot fail to be altered by the interaction.

Many everyday analogies illustrate the process of measurement in the quantum world. It's like trying to detect bowling balls by bouncing other bowling balls off them. The act of measurement in the quantum world poses a dilemma analogous to trying to discover if there is a car in a tunnel when the only means of finding out is to
send another car into the tunnel and listen for a crash. With this technique you can certainly discover whether the first car is there. You can probably even find out where it is by measuring the time it takes the probe car to crash. What you cannot do, however, is assume that the first car is the same after the interaction as it was before. In the same way, nothing in the quantum world can be the same after the interaction associated with a measurement as it was before.

In principle, this argument would apply to any interaction, whether it involves photons and electrons or photons and bowling balls. The effects of the interaction for large scale objects are so tiny that they can simply be ignored, while in the case of interactions at the atomic level, they cannot. This fundamental difference between the quantum and macroscopic worlds is what makes quantum mechanics quite different
from the classical mechanics of Isaac Newton. Remember that every experiment, be it on planets or fruit or quantum objects, involves interactions of one sort or another. The consequences of small-scale interactions make the quantum world different, not the fact that a measurement is being made.

In 1927, a young German physicist, Werner Heisenberg (1901-1976), put the idea of limitations on quantum-scale measurements into precise mathematical form. His work, which was one of the first results to come from the new science of quantum mechanics, is called the Heisenberg uncertainty principle in his honor. The central concept of the uncertainty principle is simple:

At the quantum scale, any measurement significantly alters the object being measured.

Suppose, for example, you have a particle such as an electron in an atom and want to know where it is and how fast it's moving. The uncertainty principle tells us that it is impossible to measure both the position and the velocity with infinite accuracy at the same time. The reason for this state of affairs, of course, is that every measurement changes the object being measured. Just as the car in the tunnel could not be the same after the first measurement was made on it, so too will the quantum object change. The result is that as you measure one property such as position more and more exactly, your knowledge of a property such as velocity gets fuzzier and fuzzier.

The uncertainty principle doesn't say that we cannot know an electron's location with great precision. It is possible, at least in principle, for the uncertainty in position to be zero, which would mean that we know the exact location of the electron. In this case, however, the uncertainty in the velocity has to be infinite. Thus, at the point in time when we know exactly where the electron is, we have no idea whatsoever how fast it is moving. By the same token, if we know exactly how fast the electron is moving, we cannot know where it is. It could, quite literally, be in the room with us or in China.

In practice, every quantum measurement involves trade-offs. We accept some fuzziness in the location of the particle and some fuzziness in the knowledge of the velocity, playing the two off against each other to get the best solution to whatever problem it is we're working on. We cannot have precise knowledge of both at the same time, but we can know either one as accurately as we like at any time.

Adapted from:
Trefil, J. and Hazen, R. (2010) The Sciences: An Integrated Approach, John Wiley and Sons, pp.183 - 184.

 

Wolfgang Pauli's Exclusion Principle

Much of the material presented here
on quantum numbers is beyond the
scope of the syllabus. The material
on Pauli's contribution to the model
of the atom is, however, assessable.
The final section on Pauli's
contribution and the Pauli exclusion
principle is mandated content.

Bohr's model of the atom added to our understanding of the properties of electrons in motion around the nucleus. De Broglie enhanced this understanding by thinking about the electrons as standing waves around the nucleus with discrete energies and this entrenched the wave-particle duality. Heisenberg added further to our understanding by telling us that knowing the exact position and momentum of electrons at the same time was problematic because measuring their position changes their momentum. Schrodinger's wave equation then provided a way of determining how likely it was that we might find an electron in a certain distance from the nucleus at a certain time if it was observed and the wave nature of the electron collapsed into its particle state. It was at this point in the history of the development of a quantum mechanical model of the atom that Wolgang Pauli provided the tools for telling us even more about the properties of electrons in atoms.

Developing Model of the Atom

You will remember that the Schrodinger equation (and the wave function) provides us with information about an electron's position in an atom. For a particular atom, because the energy is quantised into energy levels, the equation has solutions for specific energy values corresponding to the energy levels in that atom. For each quantised energy value (energy level), the equation generates a wavefunction that describes how the electrons are distributed in three dimensional space around the nucleus. These wave functions are called orbitals.

During the decade after 1913, physicists focused on the role of quantum numbers. Each quantised property of an electron can be represented or indexed using a quantum number. The question being asked at the time was about how many quantum numbers were needed to descrine an electron completely. Pauli responded by presenting a solution to address the puzzle. He found that by assigning four quantum numbers to each electron in an atom, combined with a set of ruless, he was able to produce a system that explained a number of outstanding issues including the sturcture of the periodic table. His ideas also considerably advanced atomic theory.

Quantum Numbers

Each electron in an atom has three quantum numbers that specify its three variable properties: energy; angular momentum; and orbital orientation and shape. A fourth quantum number describes the spin of an electron. To describe an electron completely, scientists specify a value for each of its four quantum numbers.

Principal Quantum Number

The most important quantised property of an atomic electron is its energy. The quantum number that indexes the energy for any atom containing only a single electron is the principal quantum number (n). Each electron in a multi-electron atom can be assigned a value of n that is a positive integer and which broadly correlates with the energy of the electron.

The lowest energy for an atomic electron corresponds to n = 1, and each successively higher value of n describes a higher energy state. The principal quantum number also tells us something about the size of an atomic orbital. Because the energy of an electron is correlated with its distribution in space. The higher the principal quantum number, the more energy the electron has and the greater its average distance from the nucleus.

Azimuthal Quantum Number

A second quantum number indexes the angular momentum of an atomic orbital. This quantum number is the azimuthal numher (l). The solutions for the Schrodinger equation and experimental evidence show that the distribution of an electron associated with a particular orbital can be described by a variety of shapes. Note that it is technically incorrect to talk about the 'shape' of an orbital itself. As we have seen, an orbital is merely a mathematical function. We can, however, talk about the shape of the electron distribution associated with a particular orbital and this tells us where we are most likely to find an electron within this orbital.

The value of l correlates with the number of axes an orbital shape can have and therefore identifies the shape of an electron distribution for the orbital. The value of the principle quantum number (n) also limits the possible values of l. The smaller n is, the more compact the orbital and the more restricted its possible electron distributions.



The electron distribution shapes for electrons indexed with azimmuthal quantum numbers 0 (s orbital), 1 (p orbital),
2 (d orbital) and 3 (f orbital).

The azimuthal quantum number (l) can be zero or any postive integer smaller than n. Historically, orbitals have been idetnified with letters rather than numbers such that l = 0 is the s orbital, l = 1 is the p orbital, l = 2 is the d orbital, l = 3 is the f orbital and l = 4 is the g orbital. Thus a 3s orbital has quantum numbers n = 3 and l = 0 and a 5f orbital has quantum numbers n = 5 and l = 3. Note that the restrictions on l mean that 1s orbitals exist but there are no 1p, 1d, 1f or 1g orbitals. Similarly, there are 2s and 2p orbitals but no 2f or 2g orbital.



The s orbitals on the left shown as spheres and increasing in size as the principal quantum number increases. The graphs show how the probability of finding an electron changes with distance from the nucleus. The p orbital is
shaped like a peanut shell and is shown in the diagrams on the right. The third qunatum number, the magnetic
quantum number, dictates the the p orbital is oriented along three axes in the three dimensions of space.


Magnetic Quantum Number

A sphere has no prefened axis, so it has no directionality in space. When there is a preferred axis,
as for the p orbital, the diagrams above show that the axis can point in many different directions relative to
an xyz coordinate system. Thus, objects with preferred axes have directionality as well as shape. The electron distribution within an s orbital is spherical and has no directionality. Electron distributions in other orbitals are non-spherical and therefore have a directional dependence. Like energy and orbital electron distribution, this directional dependence is quantised. The electron distributions within p, d and f orbitals have restricted numbers of possible orientations. The magnetic quantum number (mf) indexes these restrictions.

Just as orbital size (n) limits the number of prefened axes (l), the number of preferred axes (l) limits the possible orientations of the preferred axes (mf). When l = 0, there is no preferred axis and therefore no prefened orientation, so mf = 0 for the s orbitals. One preferred axis (l = 1) can orient in any of three directions, giving three possible values for mf: +1, 0 and -1. Two preferred axes (l = 2) can orient in any of five directions, giving five possible values for mf (+2, +1, 0, -1 and -2). Each time l increases in value by one unit, two additional values of mf become possible and the number of possible orientations increases by two. The magnetic qunatum number (mf) can he any positive integer hetween 0 and l: mf = 0, ±1, ±2 ... ±l.

Spin Quantum Number

Electrons have a property called spin, which means they can behave in two different ways when placed in a magnetic field. When a beam of silver atoms is passed through a magnetic field, the atom beam is split. Some atoms are deflected in one direction while others are deflected in the opposite direction. Since classical physics associates a spinning electric charge with a magnetic field, the experimental observation was explained by using a property called electron spin. Given that only two responses are possible, spin is quantised. The spin quantum number (ms) indexes this behaviour and there are only two posible values, +½ and -½.

The Pauli Exclusion Principle

A complete description of an atomic electron requires a set of four quantum numbers: n, l, mf and ms which must meet all of the restrictions summarised in the table below. Each electron in an atom has a unique set of quantum numbers and therefore no two electrons in an atom can have the same set of quantum numbers. This was first postulated by the Austrian physicist Wolfgang Pauli (1900 - 1958) who won the Nobel Prize in Physics in 1945 and it is known as the Pauli exclusion principle.

As a direct consequence of the Pauli exclusion principle, any orbital can only have a maximum of two electrons. In this state it is full and the two electrons must have opposite spin. Considering the 2p orbital for a neon atom with six pairs of electrons, there are six valid sets of quantum numbers, one set for each electron as shown below.



The six unique sets of quantum numbers for the six electrons found in the 2p orbital of a neon atom. The arrows
represent the opposite spin of electrons in the pair.


In summary, Pauli made several important contributions to atomic theory:

  1. He proposed that each electron in an atom would be described by four quantum numbers.

  2. He proposed, through his exclusion principle, that no two electrons in an atom could have a set of four identical quantum numbers.

  3. His exclusion principle and rules provided a system to explain the arrangement and number of electrons in each atomic orbital, thus providing an explanation for the structure of the periodic table.


Quantum Mechanics Explained: This video is based on the book by renowned physicist and author Brian Greene, taking us to the frontiers of physics to see how scientists are piecing together the most complete picture yet of space, time, and the universe. The video covers the early physicists contribution to our new understanding of the atom including those studied in this module such as Bohr, Heisenberg, Schrodinger and Pauli.