Basic Experiments of Quantum Mechanics
Institute of Physics, Karl-Franzens-University, Steiermark, 8010 Graz, Austria
The physics of quantum mechanics are different to the physics of classic mechanics, the physics that we see around us and we understand, because the relations are completely different or not even existing. That is why the results of experiments seem strange or wrong compared to what we actually expect. Particles pass 2 slits and hit a screen behind these slits. What we expect on the screen are 2 maxima, but what we actually see is an interference pattern like we use waterwaves or electro-magnetic waves instead of particles. To describe this and other phenomena, we need the logic of quantum mechanics.
Experiments of quantum mechanics are an intresting topic, because what we would expect in some experiments with our knowledge in classic mechanics is comletely wrong to what we actually observe. Electromagnetic waves have particle characteristics, particles have wave characteristics and the energy that can be absorbed or emited is not continuous. In this article we investigate in some basic experiments, about their results and their differences between the classic mechanics to see what old relation we have to be edit or even create in quantum mechanics.
In classic mechanics the path of an particle depends on its: location, velocity, acceleration, mass, charge, energy and momentum. These quantities can be measured independendly with instruments . The difference to quantummechanics; it is not possible to research on single particles, because every information only depends on statistical predications, thus the word „path“ does not even make sence. There is a need of a wavefunktion and several quantum numbers to describe a particle in quantium mechanics.
To describe particles there is a need of a wavefunction and quantum numbers, but are measurements even necessary? The answer is yes, but there are problems by measuring two complementary variables. The problem is, the more information exist about one observable, the less information exist about the complementary variable. This is called the uncertainty principle. For example the momentum and the location of a particle. Heisenberg estimated the uncertainty of location (x) and momentum (p) to
(ħ is the reduced plank constant). Another example is the uncertainty between energy (E) and time (t).
By using the quantum hypothesis:
it is possible to rewrite the energy-time-uncertainty to a frequancy-time-uncertainty:
(ν is the frequancy; ω is the spatial frequency ω=2πν), however, this uncertainties are not logic, so we need a derivation:
ΔEΔt=FΔxΔt≈ΔpΔtΔx*Δt=ΔpΔx≥ħ2, F…force  (5)
By following back to the relation between location and momentum the derivation is complete.
The quantum mechanics have their own logic, but the reason why we still learn the classic mechanics at schools is that the classic mechanics follow the correspondence principle. That means that old theories like the classic mechanics are a limit case of new theories like the quantum mechanics. This principle prevents conflicts with older experimental results. Problems in the macroscopic mechanic eaqual a statistical overall with many states with big quantum numbers and so the statistical spread is insignificant which means we can use the classic mechanics without errors from the scale of the measurements.
By now, we know that particles have wave characteristics too and an experiment that highlights this characteristics is the double-slit experiment. The basic set-up is made of a light or particle source (electrons, neutrons, molecules (C60)), a cover with 2 slits and a screen. Normally we would expect to see 2 maxima on the screen, because light and particles only can pass those slits to hit the screen (FIG. 1), but what we actually observe is an interference pattern with more than 2 maxima and minima (FIG. 2). The question now is, what forms this interference pattern? If we use normal waves like
FIG. 1: The expected result when light or FIG. 2: The observed result when light or particles
particles pass the 2 slits. We can see pass the 2 slits. We can see many maxima
2 maxima on the screen. and minima on the screen.
water waves for experiments we get the same results, because after passing the slits, the new waves are created by small elemental waves. This principle is called the Huygen-Fresnel-Principle. We can say that we have 2 waves that interfere with each other. On bright points we have a constructive interference between waves with the same phase that means waves with an even multiple of its wavelength hit the screen and on dark points we have constructive interference between waves with the same phase which means that waves with a half multiple of its wavelength hit the screen . To get the same result with particles there is a need to add wave characteristics to particles too. If we let one particle after another through the slits, we also get this interference. We can say that a particle interferes with itself, but only if we dont measure at the slits, otherwhise we get our expected result.
On the other hand, an experiment where we underline the characteristics of particles to light is the photoelectric effect. With this effect it is possible to proof the quantization of light directly. There are 3 different types of photoelectric effects, but in all, the basic is the same (FIG. 3). We use light to remove electrons from its binding.
FIG. 3: Basic principle of the photoelectric effect, light hits a binding and gives electrons energy. The electrons can either go to a higher energy level or get kicked comletely out of the binding. The bindings can be on a metallic surface, in an electronic semiconductor or even a single atom.
If the energy of the photon is greater than the binding energy (EBind), the electron leaves the binding with a cinetic energy.
In this case we call this outer photoelectric effect (FIG. 4).
FIG. 4: Example for an experiment for the outer photoelectric effect. Putting a voltage between 2 metall surfaces and connect them to measure the electric current .
If the added energy is less then the binding energy, the electron will not leave the binding, but it could be enough to put it to a higher energy level. If the new energy level is in a conduction band, we get en electric current. This is called the inner photoelectric effect  (FIG. 5).
FIG. 5: schematic principle of the inner photoelectric effect. In the valence band the electrons are bound to their atoms; in the conduction band electrons are not bound to their atoms and move around in the contuction band which equals and electric current.
If our binding is only an atom and we use light to kick electrons aways from it and ionize the particles, we can call it photoionization. Normally we would expect that the higher the intensity, the higher the cinetic energy of the ejected electrons and the frequency of the light is not so important. The electrons also eject after an amount of time. But the observation shows us that the more intensity, the more electrons and the higher the frequency, the higher the energy of the ejected electrons and the electrons start intstantly when the photons hit them.
If we shine ligh on a black body, we will not measure emited electrons, but the radiation of it is quiet intresting. First off, a black body is a hypothetical body thas absorbs every kind of light with any intensity or wavelength and send out electromagnetic waves (EM-waves). The maxima of the frequency of the emited EM-waves depends on the temprature of the body. The molecules/atoms in the body start to oscillate with added heat, this means many charges start to move and a charge with motion send out EM-waves. Without the knowledge about quantum mechanics, there are 2 law that would describe the intensity of the emited EM-waves: Wien’s radiation law and rayleigh jeans radiation law. The basic idea of wien’s radiation law is using the bolzman distribution to get a relation:
where u is the intensity, λ the wavelength, KB the Bolzmann factor, c the ligh speed, T the temprature and A a contstant. This law fits good for small wavelengths.
On the other hand there is the Rayleigh Jeans law, which fits good for big wavelengths. The base idea here is counting fourier modes:
The main problem in both laws is that they expect the energy of an oscillating object is continuous. Max Plank finded out that the energy is not continuous like in the classic mechanics. By assuming the energy only can absorbed or emited in packages (quantum hypothesis; (3)) and connecting wien’s law and rayleigh jeans law he got a relation that fits to big and small wavelengths:
and we can see, if we assume ω is very high, we get a proportion to (7) and for very small wavelengths (8)  (FIG. 6).
FIG. 6: Differences between Wien’s law (blue), Rayleigh Jeans radiation law (red) and Plank’s radiation law (green).
What we do not meantion by now is the difference of magnetism between classic and quantum mechanics. In classic mechanics we expect the magnetic momentum is continuous, but the Stern Gerlach experiment shows that it is not. The basic set-up of this experiment is made of an Furnace, a magnetic constellation where the magnetic field only points in z-direction and a screen (FIG. 7). Silver atoms come from the Furnace and pass the megnetic field. (The silver atomes are perfect for this experiment, because every energy level is occupied except the 5s and the angular momentum of this electron is zero so the only component of the total angular momentum of the atom depends on the spin of the 5s electron. The whole atom reacts like a single electron with spin ½ in the experiment. This sircumstances are necessary, because the silver atom has no charge in total that means there are no effects of stray fields or the lorentz force, not as we would use electrons directly. ) This field force only has a z-direction and forces the magnetic momentums of the atoms to be parallel to it, which means a force spread up the atoms depending on their magnetic momentum. We expected a continuous line, but what we actually get are 2 points. That means particles with the spin of ½ only can have 2 states.
FIG. 7: Basic set-up of the Stern-Gerlach experiment. We can see the difference between what we expect from the classic mechanics and what we observe.
To sum up, we have investigeted the basic experiments of quantum mechanics and which basic differences between classic and quantum mechanics exist by comparing what we expect and what we observe at the experiments. A path is not really declaired in quantum mechanics, because every information depends on a statistical distribution, particles have wave characteristics and light particle characteristics and the spin of particles only can have 2 states. In addition, we showed that the best fitting relation to describe the distribution of the intensity of a black body is plank‘ radiation law, because he expected the energy as not continuous.
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