Gino Segre Faust in
The development of the theory of QM reads like a long unfolding detective story on “reality.” It shows how partial truths can acquire advocacy, until the inadequacy of the partial truth can no longer be ignored. What unfolded in the development of quantum mechanics is still unfolding in our current attempts to explain the universe.
Notes on the book:
Before 1869, each element had been labeled by the weight of its representative atom; taking hydrogens mass as one unit, helium’s mass was 4, liuthium’s 7, etc.
In 1869, Dmitry Mendeleev used the atomic numbers and observed chemical properties to group the elements into families. His work was ignored at first, prophesied the existence of three new elements.
More accurate measurements showed that atomic weights were
only approximately integers. With
Niels Bohr took the next step by asking if quantum theory could determine the possible paths of the electrons circling the nucleus. His explanation of the hydrogen atom, Atomic Weight 1, proposed in 1913, and with subsequent refinements, explained a great deal of experimental data so well that by the early 1920s, physicists felt there had to be some truth to it. P.29-30.
He assumed that the electrons orbit the nucleus, much like planets orbit the sun.
Applying quantum principles, Bohr assumed the hydrogen atom radiates energy in the form of quanta, as Planck and Einstein had shown, and that a quantum’s energy was just planck’s constant multiplied by the quantum’s frequency.
In 1885 Johann Balmer found that hydrogen atom spectral frequencies were proportional to a constant, multiplied by the difference of inverse squares of integers; for example ½ squared minus 1/3 squared.
This provided Bohr’s first crucial insight into the quantum theory of the atom. Balmer’s formula led Borh to a construction of a hydrogen atom model, with electron orbit radii and therefore electron energy values fixed by Balmer’s integers, which were now called quantum numbers.
Sommerfeld imagined that, analogous to Kepler’s generalization of Copernicus’ data, the three numbers specified electron motion: size, eccentricity, inclination in orbit.
Bohr’s theory successfully predicted the details of ionized
helium and the atomic weight of helium as 4.00163, and successfully predicted the “missing
element” of atomic weight 72. The new element was named hafnium, from Hafniae, the Latin name for
There were problems; for example, his theory would not work for non-ionized helium, (AW 2, with both electrons), or for any other element.
In January 1924, Bohr, is assistant Hendrik Kramers, and American John Slater published a manifesto with no equations, refered to as the BKS paper. It asserted that conservation of energy and momentum is only true in the average, and not for specific electrons; that strict causality only held over the average, and that photons do not exist. Einstein was totally opposed to the statements in this paper. This was the beginning of titanic disputed between Bohr and Einstein. Within a year experiments showed that photons do exist and behaved exactly as Einstein had predicted.
In 1924, Wolfgang Pauli added a fourth quantum number, which could have only a value of +1/2 or -1/2, and developed the idea that no two electrons could have the same quantum numbers, called the Pauli exclusion principle.
Two of Ehrenfest’s students extended the planetary analogy and called Pauli’s number “spin”, representing an assumed rotation of the electron about its axis. Quantum theory would constrain the spin to take on only 2 possible values, up or down orientation.
Beginning in 1925, Heisenberg and Pauli were both convinced that the picture of electrons orbiting the nucleus could not be the literal truth of what atoms looked like. Electrons may behave as if they move in orbits, but one can not trace their paths. P. 131
The key was to discover a rule for multiplying together the amplitudes of the electrons oscillatory motion.
While alone on the small grassless
He sent a copy of the manuscript to Max Born, who found the curious rule Heisenberg found was strangely familiar. Born wrote “Heisenberg’s symbolic multiplication was nothing but the matrix calculation well known to me since my student days.” P. 133
Heisenberg’s physics was original, but the mathematical techniques had already been explored. This meant that quantum theorists did not have to invent a new branch of mathematics.
Similarly, when Einstein formulated his general theory, relating the curvature of space to the presence of energy, he used the theory of differential geometry, developed over decades by mathematicians. P. 133
Finding the off diagonal matrix values was a challenge, but Born working with Heisenberg and Pascual Jordon found the essence of this new rule. 23 year old Paul Dirac independently made the same discovery.
With A as particles position and B as its momentum (mass times velocity), they discovered that AB minus BA was proportional to Planck’s constant.
Pauli succeeded in calculating the hydrogen energy spectrum using Heisenberg’s new matrix based formalism. The result agreed with Bohr’s 1913 result, the Balmer formula, without reference to a mechanical-kinematic visualization of the motion of electrons. P 137
Almost at the same time, 38 year old Austrian Erwin Schrodinger found a totally different way to solve the same problem. Schrodinger’s approach was called wave mechanics, because the electrons seemed to be guided by waves.
His approach was based on the discovery by Louis-Victor, Prince de Broglie of the wave nature of electrons.
De Broglie knew that in 1905 Einstein asserted that electromagnetic radiation travels as discrete packets of energy, or quanta, with each quantum having a particle like nature. De Broglie just turned this argument around. If light is made of photons and therefore waves are particles, then particles should be waves. In particular, electrons should display wave like behavior. Einstein used his theory of relativity to calculate the photon’s momentum in terms of its wave length. De Broglie just reversed this, finding the electron’s wavelength in terms of its momentum.
De Broglie then found he could predict Bohr’s quantization of electron orbits by thinking of the orbits as vibrating cords. Pythagoras had shown that stretched cords could only sustain wavelengths that bear a simple relation to the cords length. Thus for example, the wavelength cannot be 2.5. but if wavelengths are inversely proportional to momentum, then only certain momenta are possible on the cord/orbit. This led to a new derivation of Bohr’s quantization rule.
Early in 1926, Schrodinger returned from a vacation with an equation that forms the basis of wave mechanics. In the next few months, he wrote four papers, applying solutions of his equation to many problems: the hydrogen atom; the diatomic molecule, the harmonic oscilltorEM perturbations, and absorption and emission of radiation. The equation demanded that certain parameters have a constrained set of values, or eigenvalues, which were quickly shown to be the ame as Bohr’s quantum numbers. He also wrote a paper displaying the equivalence of his wave QM and the Heisenberg Born Jordan derived matrix QM.
Question: if matrix qm has the term Planck’s constant in it, how could wave and matrix QM be equivalent?
For scientists, matrix QM seemed ad hoc, obscure and hard to grasp, while wave QM used techniques that physicists were familiar with; ones they had learned in school. In a few months, Schrodinger had solved many of the key problems in quantum theory P. 140 f.
Of course neither Schrodinger nor Heisenberg liked each other’s theory.
Schrodinger felt that matrix QM involved difficult appearing transcendental algebra, and lacked visualizability. He wanted to stay away from the notion of electrons jumping from one orbit to another, Heisenberg was convinced that Schrodinger’s approach was wrong. An electron could not have its motion be determined by a guiding wave, the object Schrodinger called a wave function or psi function. To Heisenberg, the only observable was the jump.
Despite Heisenberg’s objections, Schrodinger’s approach was preferred by academia, including Einstein. When Heisenberg objected that wave mechanics involved unobservable quantities, he was scolded by a senior professor, saying “you must understand that we are now finished with all that nonsense about quantum jumps”.
P. 142 f.
Bohr, Pali, and Heisenberg joined forces as the key opponents to the growing acceptance of Schrodinger’s views.
In October 1926, Schrodinger came to
There was general agreement in
How does one measure where an electron is and how it moves? What does measurement even mean in an atomic environment? Is an electron a particle or a wave or both?
But how to proceed? Heisenberg was opposed to the idea that electrons move according to a guiding wave,
And believed the matrix equations should be their guide. Bohr felt that the idea of a guiding wave was not completely correct, but that it had to be part of the synthesis. First they must obtain an overall understanding of the physical situation, and the mathematics would come later. Heisenberg and Bohr argued continuously.
They separated to think independently. Heisenberg then arrived at the “uncertainty principle”; that one cannot know simultaneously both the position and velocity of an electron. This undercut schrodinger’s premise that the position and velocity of an electron could be specified simultaneously.
The 1927 Solvay conference was the watershed in QM. At the
beginning, only Bohr, Heisenberg, and Pauli were confident that a consistent
formulation, called the
Dirac wrote that he was more interested in getting the correct equations than taking sides on the Bohr Einstein debate. 50 years later Dirac wrote that he thought Einstein might in the long run turn out to be correct, and the debate has continued to this day. P. 158 f.
Shortly after the 1927 Solvey conference, Dirac, at age 25, did get the correct equations and in doing so joined Bohr’s QM to Einstein’s relativity theory. This is the equation which predicted negative energy, or mass, and anticipated the positron. Dirac’s equation put into place the last of quantum theory’s building blocks. P. 165 f.
Eugene Wigner and John von Neumann were classmates. In 1926,
the American J. Robert Oppenheimer came to