- 2901 - PROBABILITY WAVE FUNCTION - The Wave Function in quantum physic is a math description if a quantum system, like an atom. It is a complex, valued probability, amplitude that lists possible results of a measurement at the atomic level and that deals with uncertainties in time and space.
--------------------------- 2901 - PROBABILITY WAVE FUNCTION
- Water waves have a wave function that is the quantity that varies periodically as the height of the water surface. In sound waves, it is pressure. In light waves, electric and magnetic fields vary in the form of waves.
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- What is it that varies in the case of “matter waves“? There is such a thing! The quantity whose variations make up matter waves is called the wave function.
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- The value of the wave function associated with a moving body at the particular point x, y, z in space at the time t is related to the probability of finding the body there at the time.
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- The wave function itself, however, has no direct physical significance , it is just math. There is a simple reason why the wave function cannot interpreted in terms of an experiment. The probability that something be in a certain place at a given time must lie between 0 (the object is definitely not there) and 1 (the object is definitely there).
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- An intermediate probability, say 0.2, means that there is a 20% chance of finding the object. But the amplitude of a wave can be negative as well as positive, and a negative probability, say 0.2, is meaningless. Hence the wavefunction cannot be an observable quantity.
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- This objection does not apply to the “square of the absolute value’” of the wave function, which is known as “probability density“. A large value wavefunction square means the strong possibility of the body's presence, while a small value means the slight possibility of its presence.
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- As long as wavefunction square is not actually 0 somewhere, however, there is a definite chance, however small, of detecting it there.
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- This interpretation was first made by Max Born in 1926.
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- There is a big difference between the probability of an event and the event itself. Although we can speak of the wave function that describes a particle as being spread out in space, this does not mean that the particle itself is thus spread out.
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- When an experiment is performed to detect electrons a whole electron is either found at a certain time and place or it is not; there is no such thing as a 20 percent of an electron. However, it is entirely possible for there to be a 20 percent chance that the electron be found at that time and place, and it is this likelihood that is specified by wavefunction square.
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- If an experiment involves a great many identical objects all described by the same wave function, the actual density (number per unit volume) of objects at x, y, z at the time t is proportional to the corresponding value of wavefunction square.
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- Let’s compare the connection between the wavefunction and the density of particles it describes. To understand this connection, let us consider the formation of a double-slit interference by photons. In the wave model, the light intensity at a place on the screen depends on E² the average over a complete cycle of the square of the instantaneous magnitude E of the em wave's electric field.
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- In the particle model, this intensity depends instead on (N *h *f), where N is the number of photons per second per unit area that reach the same place on the screen. Both descriptions must give the same value for the intensity, so N is proportional to E².
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- If N is large enough, somebody looking at the screen would see the usual double-slit interference pattern and would have no reason to doubt the wave model. If N is small perhaps so small that only one photon at a time reaches the screen the observer would find a series of apparently random flashes and would assume tha you are watching quantum behavior.
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- If the observer keeps track of the flashes for long enough, though, the pattern they form will be the same as when N is large. Thus the observer is entitled to conclude that the probability of finding a photon at a certain place and time depends on the value of E² there.
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- If we regard each photon as somehow having a wave associated with it, the intensity of this wave at a given place on the screen determines the likelihood that a photon will arrive there.
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- When it passes through the slits, light is behaving as a wave does. When it strikes the screen, light is behaving as a particle does. The linear momentum, angular momentum, and energy of the body are other quantities that can be established from the wavefunction.
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- The problem of quantum mechanics is to determine the wavefunction for a body when its freedom of motion is limited by the action of external forces. Wave functions are usually complex with both real and imaginary parts.
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- A probability must be a positive real quantity. The probability density (wavefunction square) for a complex wavefunction is therefore taken as the product wavefunction with its complex conjugate.
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- The complex conjugate of any function is obtained by replacing i(√-1) by -i wherever it appears in the function. Every complex function can be written in the form: wavefunction= A+iB, where A and B are real functions.
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--------------- Wavefunction*= A-iB.
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--------------- Wavefunction square = A²-i²B² = A²+ B².
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-------------- Wavefunction square is always a positive real quantity, as required.
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- Even before we consider the actual calculation of the wavefunction, we can establish certain requirements it must always fulfill. For one thing, since wavefunction square is proportional to the probability density P of finding the body, the integral of wavefunction over all space must be finite ,the body is somewhere, after all.
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- If the integral is zero, the particle does not exist, and the integral obviously cannot be infinity and still mean anything . Furthermore, wavefunction square cannot be negative or complex because of the way it is defined.
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- The only possibility left is that the integral be a finite quantity if the wavefunction is to describe properly a real body. It is usually convenient to have wavefunction square be equal to the probability density P of finding the particle described by the wavefunction, rather than merely be proportional to P.
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- If the wavefunction square is equal to P, then it must be true that the integral of the wavefunction square from - ∞ to + ∞ equal to 1, since if the particle exists somewhere at all time. This wavefunction is said to be normalized. Every acceptable wave function can be normalized by multiplying it by an appropriate constant.
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- We ar only considering an electron traveling in a line, in one dimension. The definite integral from a to b gives us the probability that the electron is in between points a and b. So given the wave function, the electron is most likely to be found at its tallest peak or its lowest trough, depending on which one is deeper.
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- But how does this wave function define superposition? The wave function can be broken down into many states. An electron can have only two states: spin up, and spin down, they're two completely independent states of being.
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- Let's represent the waveform, or the wave function, of an electron in spin up with A, and the waveform of an electron with spin down with B. Given this, and knowing that the electron will only be one of these two states when measured.
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- The probability that the electron is spin up is 0.5². The probability that the electron is spin down is 0.5². Knowing that the probability is 0.5 for both, they are both equal to the square root of ½. That is when the electron is in superposition.
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- However, an interesting thing happens when the electron is observed. One of the probabilities drops to zero, and the other jumps up to one. This is called the "collapse" of the superposition. This is not only for this case.
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- In a scenario where the particle is in a superposition of 15 states, all other probabilities drop to zero except for a single one, which jumps to one. The wave function becomes something called a delta function, where it has a peak at the measured value, this is why after you take a single measurement of the state of a quantum particle, it will continue to show that same state if you measure it again and again and again.
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- If you let the wave function settle back into its original waveform after a long time, it may give a different point. This means by measuring the particle, you directly alter the wave function.
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- Something else important about the wave function: the wavelength of the function is the “momentum” of the particle. A longer wavelength implies a smaller momentum. The important thing to understand here is that for a particle where we know the momentum, the wavefunction square will be the same everywhere, and we will no longer have any idea as to where the particle is.
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- When we know the “position‘, the wavelength will always be different, and we will have no idea what the “momentum’ is. This is called the “Heisenberg uncertainty principle“.
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- Heisenberg uncertainty principle is inextricably linked to the wavefunction. But what is the function itself? Nobody knows. This is a mystery that's been unsolved for nearly a century. Nobody really knows what this function is, but we use it, nonetheless.
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- This function has evaded understanding and there are definitely some theories out there as to what the function could be, but at the end of the day, nobody knows what this function is. Including yours truly.
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- November 13, 2020 2901
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