Everything about Quantum Superposition totally explained
Quantum superposition is the fundamental law of quantum kinematics. It defines the allowed state space of a quantum mechanical system.
In probability theory, every possible event has a positive number associated to it, the probability, which gives the chance that it happens. If you want to know the probability of two independent events, like the probability that it'll rain and also that the stock market will go up, you multiply the probability for each event. If there's a .3 chance that it'll rain and a .6 chance that the market will go up, there's a .18 chance that both will happen.
If the market can go up because of two exclusive events, and you want to know the total probability that the market will go up, you add the probability for the two events. For example, if the market will go up only if the fed lowers interest rates by 2 points or by 3 points, the probability that it'll go up is the sum of the probability that the fed lowers the interest rate by 2 points or by 3 points.
Quantum mechanics has the exact same rules for multiplying and adding numbers associated with events, except that the quantities are complex numbers called amplitudes instead of positive real numbers called probabilities. The superposition principle says that the way to describe the world is to assign such a complex number to every possible situation, and that the way to describe how things change is to treat these numbers mathematically as if they were probabilities. Because these numbers can be positive or negative, quantum mechanics allows the counterintuitive phenomenon that sometimes when there are more ways for a thing to happen, the chance that it happens goes down. An event with a negative amplitude can cancel with an event with a positive amplitude.
For example, if a photon in an plus spin state has a .1 amplitude to be absorbed and take an atom to the second energy level, and if the photon in a minus spin state has a -.1 amplitude to do the same thing, a photon which has an equal amplitude to be plus or minus would have zero amplitude to take the atom to the second excited state and the atom won't be excited. If the photon's spin is measured before it reaches the atom, whatever the answer, plus or minus, it'll have a .1 amplitude to excite the atom.
The probability in quantum mechanics is equal to the square of the absolute value of the amplitude. The further the amplitude is from zero, the bigger the probability. In the example above, the probability that the atom will be excited is .01. But the only time probability enters the picture is when an observer gets involved. If you look to see which way the atom is, the different amplitudes become probabilities for seeing different things. So if you check to see whether the atom is excited immediately after the photon reaches it, you've a .01 chance of seeing the atom excited.
Observations have different outcomes described by probabilities, while microscopic events are described by amplitudes. This difference leads many people to wonder what the correct
interpretation of the amplitude is.
Superposition Principle
The principle of superposition states that if the world can be in any configuration, any possible arrangement of particles or fields, and if the world could also be in another configuration, then the world can also be in a state which is a mixture of the two, where the amount of each configuration that's in the mixture is specified by a complex number.
For example, if a particle can be in position A and position B, it can also be in a state where it's an amount "3i/5" in position A and an amount "4/5" in position B. To write this, people usually say:
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where the sum extends over all possible values of q. This matrix is necessarily symmetric because it's formed from the orthogonal states, and has eigenvalues q. The matrix A is called the observable associated to the physical quantity. It has the property that the eigenvalues and eigenvectors determine the physical quantity and the states which have definite values for this quantity.
Every physical quantity has a
Hermitian linear operator associated to it, and the states where the value of this physical quantity is definite are the
eigenstates of this linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the value of the physical quantity will be random, with a probability equal to the square of the coefficient of the superposition in the linear combination. Immediately after the measurement, the state will be given by the eigenvector corresponding to the measured eigenvalue.
It is natural to ask why "real" (macroscopic, Newtonian) objects and events don't seem to display quantum mechanical features such as superposition. In 1935,
Erwin Schrödinger devised a well-known thought experiment, now known as
Schrödinger's cat, which highlighted the dissonance between quantum mechanics and
Newtonian physics, where only one configuration occurs, although a configuration for a particle in Newtonian physics specifies both position and momentum.
In fact, quantum superposition results in many directly observable effects, such as
interference peaks from an
electron wave in a
double-slit experiment, although it can be shown that these effects are small for cats. The superpositions, however, persist at all scales, absent a mechanism for removing them. This mechanism can be philosophical as in the Copenhagen interpretation, or physical.
If the operators corresponding to two observables don't
commute, they've no simultaneous eigenstates and they obey an
uncertainty principle. A state where one observable has a definite value corresponds to a superposition of many states for the other observable.
Further Information
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