Experimental background of Quantum Mechanics

Electron Discovery (1897)

Cathode rays

Cathode rays are normally generated in vacuum tubes. The experimental set-up is—evacuated glass tubes equipped with usually two metal electrodes (cathode and the anode) to which a voltage is applied. When the cathode was heated (thermal emission) in a vacuum with a large potential difference applied between the two electrodes, a beam were emitted at the cathode that can travel between the two electrodes. Before J. J. Thompson this cathode rays were believed to be composed of electromagnetic waves. (The Cathode rays could pass through thin sheets of gold)[1]

J. J. ‘s Experiment and Discovery

Thompson conducted the experiment and found that he could deflect the cathode rays in an electric field [2](later on magnet field) produced by a pair of metal plates. The cathode rays were repelled from the negatively charged metal plate and attracted to the positively charged one. Then he refined his experiment by creating better vacuum tube (remove more gas) and repeated the experiment. Again when the cathode ray passed through the electrical field it did bend to positively charged plate. Then he realized the rays must be negatively charged. He also repeated his experiments with different cathode metals including iron and platinum, and found that the specific charge did not change. He argued that the charges carried rays were actually particles; therefore, he named it electron. He also realized that this kind of particles existed in many types of atoms since he used many different cathodes. (He won the Nobel Prize for Physics for this discovery) He measured the charge-to-mass ratio of electrons as well. He believed that electrons were a universal constituent of all matter - they form part of all the atoms in the universe. It is interesting to note that his son, G.P. Thompson, won the Nobel Prize a few decades later for proving that electrons were waves!

Blackbody Radiation(1900)

Thermal Radiation

It is our common sense that hot objects would emit light. As a matter of fact every object emits light at any temperature, of which the phenomenon is called thermal radiation. (Also electromagnetic radiation) However, the light emitted from objects at room temperature is in the infrared region therefore we cannot see it[3].


An object which gives off a certain distribution of light at the lowest temperature possible is called a blackbody. At a given temperature, all objects give off a "redder" light distribution than a blackbody at the same temperature, and so appear to be "cooler". A large, heated cavity with a tiny hole in the side is often presented as being an excellent approximation to such an ideal blackbody. The spectral distribution is measured by "looking into" the box through the small hole (or more correctly, by allowing the light inside to escape out through the hole and then measuring it. The hole needs to be small compared to the size of the box so that the amount of energy escaping is negligible compared to all the energy inside the box. Then the measurement doesn't disturb the distribution.)

Ultraviolet Catastrophe and Plank’s Postulate

Lord Rayleigh method was like this: he assumed that blackbody consists a collection of oscillators that could absorb and emit electromagnetic radiation at any frequency. Then he used this model to attempted to describe this distribution. The problem came, this model fit the data very well at low energies however it fails towards the ultraviolet end of the spectrum. This is so called the "Ultraviolet Catastrophe"
Max Planck played around with the equations and came across a mathematical form that exactly predicted the observed blackbody spectra. He came up with only possible conclusion was that the oscillators could only absorb and emit it in only discrete quantities - that the energy of the oscillators had to be quantized. This was the birth of the quantum. Plank could not explain why they were quantized however it predicted exactly the observed blackbody emission spectra. It is interesting to note that Planck himself never believed it was physically real. (Please refer to satyesh for detailed mathematical procedures)


The Photoelectric Effect (1905)

Experiment of Photoelectric effect

The experimental arrangement was very similar to that of the cathode ray tube. It was found that electrons could be ejected from the cathode when light impinged upon the surface of the cathode[2,4]. It was also found that for certain cathode (certain material), the electron current depended on the intensity of the light. However, the onset of the current did not depend on the intensity but the wavelength (frequency) of the light. And different material has different onset frequency.

Einstein’s Explanation

There are several questions raised if we tried to explain this using classic wave theory of light. First, we would expect a time lag, which would increase as the intensity of the light decreased. Because the total energy would be spread across a wave front striking the entire cathode and nothing could happen until sufficient energy was absorbed. Also, we could expect the amplitude of the wave of light not the frequency matters since its amplitude carried the amount of energy in a wave.
Einstein who considered light as particles, in 1905, took Planck's quantum hypothesis and applied it the photoelectric effect and showed how the consideration of the structure of matter has having quantized energy levels accounted precisely for these observations. This was a tremendous boost to the integrity of the quantum concept. The equation for the photoelectric effect is
Energy of photon = Energy needed to remove an electron + Kinetic energy of the emitted electron

Nuclear Scattering (1911)

“Plum Pudding Model”

Originally Thompson believed that the hydrogen atom must be made up of more than two thousand electrons since he found that the hydrogen atom was two thousand times heavier than one electron. (Particles this small could pass between atoms in a solid like gold foil) However this postulate was not in accord with the observation that atoms were usually uncharged. Thomson then proposed a new model named “plum pudding model” [2,5]of the atom, with a sphere of positive charge containing electrons.

Experiment of Ernest Rutherford

Rutherford conducted his experiment by fire positively charged alpha particles (originally from Becquerel experiment of radioactivity later on we know it is He atom nuclei) into a thin foil of gold and watched where they bounced. Originally he thought that the alpha particles would move through the gold foil with little resistance, and only slightly slowed down by positively charged bulk of gold as well as would be deflected slightly along the path by the electrons.
However, many alpha particles passed through the foil unaffected. To their surprise that they had also found some particles even bounced off the foil and left a spot on their scintillation screen. The force that could cause such large deflections had to be large and a reasonable explanation would be that the positive charge had to be collected in a much smaller sphere than the atom size. And that would be the nuclei of the atoms, which would be not only small but also extremely dense as well, which were positively charged. It was also apparent that the bulk of the mass of an atom had to reside in the nucleus and not in the electrons. Therefore Rutherford proposed a new model atom which contained a massive nucleus containing all of its positive charge, and with much lighter equally negatively charged electrons outside this nucleus and orbited around it.

The Bohr Atoms (1913)

In accordance with classic EM theory the electrons orbit the nucleus would lose its energy by radiation of EM waves and hence this configuration would not be stable. Niels Bohr adapted Rutherford's model but he abandoned the classic notion of orbits instead he asserted that orbits of the electrons were quantized, meaning that they could exist only at certain distances from the nucleus.

The Bragg’s X-Ray Diffraction (1912)

In the autumn of 1912, William Lawrence Bragg realised that X-rays could be used to detect the arrangement of individual atoms inside solid crystals. With his father's help he created a new science of X-ray crystallography.[2]

Röntgen's 'X'-Rays

In November 1895, Wilhelm Röntgen discovered X-rays while working at the University of Wurzburg, Germany. Later on Wilhelm Wien, calculated that the wavelength should be around one hundredth of a nanometre, which would be ten thousand times shorter than the wavelength of visible light.

Bragg's Law

The father and son worked out the so called Bragg’s Law together.

When n is an integer the reflected waves from different layers are perfectly in phase with each other therefore they are constructive and produce a bright point on a piece of photographic film. Otherwise it is either be missing or feint. The most important application of this law is to explaining crystal structures (Detail refer to jialan)

Specific Heat (1912)

Dulong-Petit Law

During the early part of the nineteenth century, studies on the heat capacity of materials tended to indicate that it was independent of temperature, which gave rise to the Dulong-Petit law. The constant volume heat capacity is even more nearly the same for all elements. The equipartition theorem would assign a total energy of 3kT to each atom in a solid. For N atoms, this would give 3NkT for the total energy. The molar internal energy awould be
From here the molar constant volume heat capacity is easily predicted to be

This is precisely the Dulong-Petit Law and readily shows its independence from temperature.

Einstein’s Model

However when measurements were made for lower temperatures it became apparent that the heat capacity actually drops with decreasing temperature and in fact approaches zero at absolute zero temperature. Then Einstein using solved the problem again using Planck's quantum ideas. He suggested that each atom was an oscillator of frequency ν and then asserted that any oscillation would have to be an integer multiple nhν.


However, when at temperature T->0, Einstein’s model was not fit with experimental results, the specific heat approached zero exponentially while much faster than T3[6]

Debye’s Model

Peter Debye produced a further correction by not assuming that all atoms had oscillated at the same frequency, but rather averaged over all frequencies present. This Debye Equation nicely predicts the observed experimental results, lending further credence to Planck's Quantum Hypothesis. (Detail refer to liang)

[1] http://en.wikipedia.org/wiki/Electron_beam
[6]Omar, M. Ali. “Elementary Solid State Physics”, Chapter 3. Pearson Education, (2007).

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