This state of existing in all possible states at once is called an object's coherent superposition. The total of all possible states in which an object can exist — for example, in a wave or particle form for photons that travel in both directions at once makes up the object's wave function. When we observe an object, the superposition collapses and the object is forced into one of the states of its wave function.
In the late 1920s and early 1930s, Heisenberg, Dirac, and John von Neumann, codified the formalism of quantum mechanics as a two-step process. One part involves the continous evolution of states via the deterministic Schrodinger equation. Map out a system's potential energy distribution in the form of a well, for example and the spectrum of possible quantum states is set. If the states are time-dependent, then they predictably transform. That could set out, for instance, a superposition of states that spreads out in position space over time, like an expanding puddle of water. Yet experiments show that if an apparatus is designed to measure a particular quantity, such as the position, momentum or spin-state of a particle, quantum measurements yield specific values of that respective physical parameter. Such specificity requires a second type of quantum operation that is instantaneous and discrete, rather than gradual and continuous: the process of collapse. Collapse occurs when a measurement of a certain physical parameter position, let's say precipitates a sudden transformation into one of the "eigenstates" (solution states) of the operator (mathematical function) corresponding to that parameter the position operator, in that case.
Then the measured value of that quantity is the "eigenvalue" associated with that eigenstate the specific position of the particle, for instance. Eigenstates represent the spectrum of possible states and eigenvalues the measurements associated with those states. We can imagine the situation of quantum collapse as being something like a slot machine with a mixture of dollar coins and quarters; some old enough to be valuable, others shining new. Its front panel has two buttons: one red and the othe blue. Press the red button and the coins instantly become sorted according to denomination. A number of dollar coins drop out (a mixture of old and new). Press the blue button and the sorting is instantly done by date. A bunch of old coins (of both denominations) are released. While someone seeking quick buck-might press red, a coin collector might push blue. The machine is set that you are not permitted to press both buttons. Similarly, in quantum physics, according to Heisenberg's famous uncertainty principle certain quantities such as position and momentum are not measurable at once with any degree of precision. 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. When an experiment is performed to detect electrons, for instance, 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 wave function square.
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. If you let the wave function settle back into it's original waveform after a long time, it may give a different point. This means by measuring the particle, you directly alter the wave function. 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. 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. Heisenberg uncertainty principle is inextricably linked to the wave function.
For any physically measurable quantity “a”, which we call a dynamical variable, there is associated an operator A. The Eigenvalues of A are the only possible results of measurement of the quantity a. The eigenfunctions of A represent the states of the particle in which the value of the quantity is precisely known. We say that “a” has a definite value in such a state. If the wave function of the system before the measurement of “a” happens to be an eigenfunction of A, the result of the measurement is sure to be the corresponding eigenvalue. If the wave function of the particle before the measurement is not an eigenfunction of A, the result of measurement cannot be predicted beforehand with certainty. If the measurement gives a result, the wave function just after the measurement becomes an eigenfunction of A corresponding to the eigenvalue, irrespective of what it was before the measurement. This principle is called 'collapse of wave function in a measurement'. Any wave function |Δ> can be written as a linear combination of the eigenfunctions of A. Eigenvalues of a Hermitian operator are real:
- Let A be a Hermitian operator and |Δ> be an eigenfunction of A with eigenvalue “a”. This means A|Δ> = a|Δ>. Operation of A on |Δ> gives the same function |Δ> except for an overall multiplication by a pure number “a”. Take scalar product of each side of this equation with |Δ>. This gives, <Δ|A|Δ> = a<Δ|Δ> . I have written “a” outside the scalar product <Δ|A|Δ>. Now take complex conjugate of each side of <Δ|A|Δ>* = a*<Δ|Δ>*, but <Δ|Δ>= real & positive, hence <Δ|Δ>*= <Δ|Δ>. Also A is a Hermitian operator so A*= A, thus <Δ|A|Δ>*=<Δ|A|Δ>. So <Δ|A|Δ> = a*<Δ|Δ>. Comparing, a<Δ|Δ>=a*<Δ|Δ>. So (a-a*) <Δ|Δ>= 0, so a= a*. Hence “a” is real, so all eigenvalues of Hermitian operator are real.
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