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Quantum Mechanics





Physicists have long debated over the nature of light. Newton believed light to be a stream of fast moving particles. But many others thought light to be a wave. Careful measurements of the speed at which light traveled were made and the speed of light was set at about 300 kilometers per second. Then maxwell discovered electromagnetic waves and that they traveled at a characteristic speed of about 300 kilometers per second. The similarity to light was more than just a coincidence and soon the wave theory of light had all but taken the field. However there were still clouds on the horizon the theory of wave-like light could not explain the photoelectric effect where electrons are ejected rapidly from atoms. According to the wave theory the wave should take more time to transfer energy to the electrons and they would be liberated from their atoms gradualy. However when light bombarded photovolataic substances voltages were measured imediately. Einstein stepped into the picture to propose that light might come in very small packets of energy that were individualized. So that when any one packet encountered an electron in an atom it gave its entire energy to that electron instead of spreading it out wavelike over a region. In short einstein proposed that light could behave like a particle. In the age old struggle of particle versus wave neither one emerges as victor but rather they come out together. Paradoxically in a quantum world neither wave nor particle is a complete description but each is the complement of the other. Light can be described as a particle and by a wave. This wave particle duality was one of the first characteristics of quantum mechanics to make its way to the surface and it remains a point of interest. The sameness of waves and particles like everything else in quantum mechanics is a matter of how you look at things. Just as a line viewed head on seems just to be a point. The percieved properties of things depend on how you observe them. In quantum mechanics the observer becomes equally as important as the observed, an integral part of the system. we need to know how and where we are measuring the state of the system in order to know how the system is behaving. In classical dynamics if we carry out an experiment in a box and watch it as it happens, we can assume that if we set up the experiment exactly as it was before but this time don't watch it, the experiment will happen exactly the same and the outcome will be the same. Not so in quantum mechanics, the famous gedankenexperiment(thought experiment) of schroedingers cat is an excellent example of this. Shroedinger tells us to imagine that we put a cat in a box with some poison food and seal the box so that we cannot measure what is happening inside. If we never open the box the cat will never die. This argument stems from the idea that the cat cannot be known to be dead or alive without opening the box to look. Therefore the cat is neither dead nor alive until we look into the box to behold the cats fate. This may seem a gibberish result but it illustrates a point which will be made more clear by the slit experiment. That point is that every system must have an observer as well as something being observed or the system need not exist at all much less hold any specific state.

The slit experiment is the modern embodiment of the wave partilce debate. In it we take a wall and cut some holes or slits in it. Then we shine a light or shoot a beam of electrons at it. The pattern that the light takes on the other side of the wall indicates whether it is acting like a particle or a wave. first we cut just one slit in the wall.

The one slit experiment

Particles unlike waves bounce off one annother and spread out from natural collisions into a kind of diffuse simple beam. making a pattern like this electron diffusionThis picture shows what we might see if we put up a screen directly behind slit and made little black dots on the screen everywhere that an electron hit the screen. The electron density lowers smoothly as we move out from the center. The particle desity is highest in the center and decreases with distance from the center like this. distribution There is absolutely nothing quantum about this interaction it is much the same as cutting hole in the bottom of a bag of sand and watching it make a little mound on the ground. The sand particles naturally spread out from each other and land in a pattern that naturally resembles the "bell curve" shown above. Since this interaction is characteristic of particles we can take this as evidence of the particle nature of light and or electrons etc. However newtonian mechanics does not seem to carry through for us when we make two slits in the wall.

The two slit experiment

When we cut two slits into the wall we find that a purely particle characterization of light is not sufficient to describe what happens. Going back to the sand analogy if you cut two small holes close together in that sand bag The two streams of sand particles will add together and you will get essentially the same shaped mound as with one hole. But at worst even with two very distant holes the most you can get from a simple scattering pattern is two humps.

The two slit experiment
When we cut two slits in our screen and shoot particles through we can expect therefore to get out a simple scattering pattern. A similar pattern to the one we got for the single slit. If however the particles that are coming through the slits are not really particles but waves then the waves emanating from the two holes will interfere with each other creating a great many peaks and dips in the distribution. This is indeed what we observe when the experiment is carried out. From this experiment we can take an important lesson about the wave like nature of all particles. Not just light but electrons protons neutrons xenon atoms all have wave properties. And at the same time can be considered to act like particles whichever is more convienient for the mathematical representation at the time. The wavelength of such particles can be found by a simple relation and this wavelength is called the de broglie pilot function. it is

Lambda=h/p

where h is planks constant p is the momentum of the particle (mass times velocity) and lambda is the particle's wavelength. but quantum physics is more than just the wave/particle duality. we can still learn from our simple slit experiment.

low density slit experiment
What happens in our simple slit experiment if we only run one electtron through it at a time? Since there is only one electron it can't be expected to interfere with any other waves. So it should act like a particle again and give us the simple diffusion pattern. Again the quantum world surprises us because when we actually run the experiment with two slits and one electron we get an interference pattern. When we wait to send a lot of electrons one after another through the machine slowly the image of an interference pattern builds when we mark all of the elctrons collision points on our screen. The electron is somehow managing to interfere with something while going through the machine even though it is alone. The fantastic explanation that quantum theory gives to us is that the electron is interfering with itself. You see this is where the observer starts to come into quantum mechanics. remember schroedinger's cat? well this electron is going trhough much the same thing we send the electron through a box inside which we have no idea what it is doing. It has two possible options it can go through one hole or the other schroedinger tells us that we cannot know what it does until we open the box to look. Until we look it exists in both states. In the case of the slit experiment we never do open the box to see which hole the electron goes through it could go through either one and we would get the same distribution on the other side. But since we don't measure its passage it goes through both slits at once and then interferes with itself because it exists in both states at once. This property of the quantum mechanical world is called the superposition principle. It means that a system can be in two states simultaneously even when they are contradictory. But the superposition principle only allows contradictory states to occur when no measurement is being taken. like the cat when we actually open the box up and look to see what happened we will see one outcome or the other not the superposition of both.

The two slit measurement experiment

The final slit experiment is based on what exactly happens when we put a device next to each of the slits so that the device triggers when an electron passes through its respective slit. Now we know which slit the elctron passes through and they act like particles again and when we pass an electron beam through the system we get a simple scattering pattern again. the importance of the observer in quantum mechanics is made clear. so if a tree falls in the forest and there is no one to hear does it make a sound.

Heisenburg uncertainty
The heisenburg uncertainty principle states that you can never measure a particles momentum and its position with perfect accuracy at the same time. The principle is more exactly stated by saying that the product of the uncertainties of momentum and position must be greater than planks constant divided by 2 pi.

h <. Pu*xu

where h is planks constant pu is the uncertainty in momentum and xu is the uncertainty in position. But this is only one face of the heisenburg uncertainty principle momentum and position is just one pair of quantities that cannot be known simultaneously. The key to understanding what pairs of things cannot be known to greater certainty than h/2PI these pairs are things that have dimensions that can be multiplied together to produce action. Action is the product of energy and time. So it comes in units of Kg*m2/s anything that multiplies together to give those units makes an uncertainty pair. For instance take momentum and position, momentum comes in units of kg*m/s and position comes in units m so when multiplied together they give action. But the uncertainty principle is not just a statement about what we can know about objects but what properties objects can have. It is not enough just to say that we cannot know the position and momentum of an electron exactly both at once but rather we must say that an electron cannot have a well defined momentum and position at the same time. This is the fundamental fuzzyness of the quantum world but it is not due to an out of focus lens or inadequate measurements but nature herself is fuzzy. Take for instance the example of empty space. Space that is completely devoid of everything cannot be said to be absolutely empty. To say that space is completely empty we must measure the energy in that space and find it to be exactly zero. But the heisenburg uncertainty principle demonstrates that the uncertainties in time and energy are related it is impossible to know one with absolute certainty without knowing the other with no certainty at all. In the case of time and energy the relation can be interpereted that to know the energy that something has to a certain accuracy we cannot know the exact moment that it had that energy to an accuracy that would violate the uncertainty principle. The principle allows us to know that a space is exactly empty that is to say completely devoid of energy just so long as the amount of time we take to measure the energy in the space is infinitely long. However for any length of time shorter than infinity we will measure slight fluctuations in the energy of the empty space. If these variations were truly just noise in our detectors and actually had no real existience then they would have no effect on objects in that space. However there are experiments that have been carried out in which casimir forces due to fluctuations in the quantum vaccuum have actually been observed to exist, proving the reality of quantum uncertainty. The casimir experiments consist of two metal plates held very very close together. Electromagnetic waves cannot exist inside a cavity bound by conductive boundaries if the cavity does not resonate with the waves and is too small to allow many wavelengths inside it. So when two metal plates are brought very close together in an appearently empty space they are pushed together. You see they exclude some electromagnetic waves that arise because of uncertainty and exclude others but since the fluctuations inside are devoid of waves that cannot exist inside the cavity but the space outside produces a complete spectrum of waves there are more waves outside than inside and the extra waves outside push the plates together.

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