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| | The work of Tomonaga, Schwinger and Feynman in quantum electrodynamics led to the modern framework of quantum mechanics, currently applicable in quantum electrodynamics and quantum chromodynamics. | | The work of Tomonaga, Schwinger and Feynman in quantum electrodynamics led to the modern framework of quantum mechanics, currently applicable in quantum electrodynamics and quantum chromodynamics. |
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| − | ==Principles==cosine(pi/6)*200 = 100sqrt(3) | + | ==Principles== |
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| | + | *Every system can be described by a wave function, which is generally a function of the position coordinates and time. All possible predictions of the physical properties of the system can be obtained from the wave function. The wave function can be obtained by solving the [[Schrodinger equation]] (or the Klein-Gordon equation for relativistic-quantum mechanics) for the system. |
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| | + | *An observable is a property of the system which can be measured. In some systems, many observables can take only certain specific values. |
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| | + | *If we measure such an observable, generally the wave function does not predict exactly which value we will obtain. Instead, the wave function gives us the probability that a certain value will be obtained. After a measurement is made, the wave function is permanently changed in such a way that any successive measurement will certainly return the same value. This is called the collapse of the wave function. |
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| | + | ===Collapse of The Wave Function=== |
| | + | [[File:norm.png|right|150x250px|thumb|An example of a wave that could be a position function. (Actual position functions are normally much more concentrated.)]] |
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| | + | In quantum mechanics, it is meaningless to make absolute statements such as "the particle is here". This is a consequence of the Heisenberg Uncertainty Principle which (simply put) states, "particles move," in an apparently random manner. Thus, giving a definite position to a particle is meaningless. Instead, scientists use the particle's "position function," or "wave function," which gives the probability of a particle being at any point. As the function increases, the probability of finding the particle in that location increases. In the diagram, where the particle is free to move in 1 direction, we see that there is a region (close to the y-axis) where the particle is more likely to be found. However, we also notice that the wave function does not reach zero as it moves towards infinity in both directions. This means that there is a high likelihood of finding the particle around the center, but there is still a possibility that, if measured, the particle will be a long ways away. |
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| | + | When the particle is actually observed to be in a specific location, its wave function is said to have "collapsed". This means that if it is again observed immediately the probability that it will be found near the original location is almost 1. However, if it is not immediately observed, the wave function reverts to its original shape as expected. The collapsed wave function has a much narrower and sharper peak than the original wave function. |
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| | + | Collapsing of the wave function is by no means magic. In can be intuitively understood as this: You find a particle at a particular spot; if you look again immediately, it's still in the same spot. |
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| | + | ===The uncertainty principle=== |
| | + | As a result of the wave nature of a particle, certain quantities cannot be known to an arbitrary precision simultaneously. This happens when the operators for the two quantities do not [[Commutative|commute]]. An example is position and [[momentum (physics)|momentum]]. Whenever its position is measured more accurately (beyond a certain limit), its momentum becomes less certain, and vice versa. Hence, there is an inherent uncertainty that prevents precisely measuring both the position and the momentum simultaneously. This is known as the [[Heisenberg Uncertainty Principle]]:<ref>http://hyperphysics.phy-astr.gsu.edu/Hbase/uncer.html</ref> |
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| | + | :<math> \sigma_x \sigma_p \ge \frac{\hbar}{2}</math> |
| | + | where: |
| | + | * <math>\sigma_x</math>is the [[standard deviation]] (uncertainty) of position |
| | + | * <math>\sigma_p</math> is the [[standard deviation]] of momentum |
| | + | * <math>\hbar</math> is the [[Planck's Constant|reduced Planck's constant]] |
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| | + | Other examples include energy and time, as well as different components of [[angular momentum]]. |
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| | + | ===Probability=== |
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| | + | The probability a particle is found in a certain region <math>a < x < b</math> can be described as: |
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| | + | <math> |
| | + | p(a < x < b) = \int^b_a |\Psi (x,t)|^2 dx |
| | + | </math> |
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| | + | provided that the wavefunction is normalised: |
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| | + | <math> |
| | + | \int^\infty_{-\infty} |\Psi (x,t)|^2 dx = 1 |
| | + | </math> |
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| | + | In general, the expected value of a measurement of a quantity <math>Q</math> can be described as: |
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| | + | <math> |
| | + | \langle Q \rangle = \int^\infty_{-\infty} \Psi^{*} \hat{Q} \, \Psi dx |
| | + | </math> |
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| | + | where <math>\hat{Q}</math> is the [[operator]] associated with the quantity <math>Q</math>. For example the position operator is simply <math>x</math>. The [[momentum (physics)|momentum]] operator is: |
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| | + | <math>\hat{p} = \frac{\hbar}{i} \frac{\partial}{\partial x}</math> |
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| | + | so that the expected momentum is: |
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| | + | <math>\langle p \rangle = \int^\infty_{-\infty} \Psi^{*} \hat{p} \, \Psi dx = \frac{\hbar}{i} \int^\infty_{-\infty} \Psi^{*} \frac{\partial \Psi}{\partial x} dx </math> |
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| | ==Interpretations== | | ==Interpretations== |