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and the lack of complete determinism. It states that for a pair of conjugate variables such as position / momentum and time/energy (including the rest mass energy mc2), it is impossible to have a precisely determined value of each member of the pair at the same time. This statement is illustrated with a schematic diagram in Figure 12-02. The corresponding formula is: xpx > , where denotes the uncertainty, x is the position of the point mass m along the x-axis, px = m vx is the momentum along the x-axis, vx is the velocity along the x-axis, and = h/2 = 1.054x10-27 erg-sec. A similar relation exists for the uncertainty of the time t and energy E, e.g., | |
Figure 12-02 Uncertainty Principle [view large image] |
t E . In case of heavy mass (such as a macroscopic object), the uncertainties and thus the quantum effect becomes very small, classical physics is applicable once more. See "A Derivation of the Uncertainty Principle" for more detail. |
Figure 12-03a illustrates a very simple superposition of two spin states - one parallel and the other anti-parallel to the direction of a magnetic field, where |a|2 and |b|2 are probabilities of finding the particle in the corresponding state. It is only when the state of the particle, e.g. the spin in this case, is measured that it settles into a definite state. But as soon as we stop monitoring its behavior, the particle dissolves into a superposition again. | |
Figure 12-03a Superposition |
Note : Probability = |Probability Amplitude|2, e.g., a and b in Figure 12-03a are the probability amplitudes, the corresponding probabilities are |a|2 and |b|2. |
packet, which is the combination from waves of different wavelengths and amplitudes (Figure 12-03c). If an object's wavelength is of a similar order to the size of the objects around it, the wave nature comes to the fore. The wavelength of a macroscopic object such as a moving car is something around 10-36 cm; thus it takes some pretty tiny objects to expose the car's wavelike properties. Only microscopic objects | ||
Figure 12-03b Probability Pattern [view large image] |
Figure 12-03c de Broglie Wave [view large image] |
such as electron has a large enough wavelength to show its wavelike property with objects in manageable size. For example, the de Broglie wavelength for a 75 ev electron is 2 x 10-8 cm, thus the spacing between atoms in a crystal is a good diffraction grating for such electron. |
Figure 12-03d Vacuum Fluctuation [view large image] |
of as being surrounded by a cloud of virtual particles (and pairs) of various kinds, which are being created and (usually) reabsorbed by the parent particle. The lifetime of each of these virtual particles (and therefore the distance it can travel from its parent particle) depends on its mass-energy and the leeway allowed by the uncertainty principle. Interactions occur when a real particle come close enough (as in high energy collision) for one or more of the virtual particles in the cloud to be absorbed by the other real particle. | |
Figure 12-03e Feynman Diagram |
Since the virtual photons in between two parallel metal plates placed a short distance apart can exist only when they can form a standing wave, there are fewer photons in each cubic centimeter of vacuum between the plates than there are in the vacuum outside. So, in effect, there is an excess pressure from outside pushing the plates together. This is known as Casimir effect (see Figure 12-03f). The resulting force is very small, but it has been measured (for plates separated by gaps of a few nanometers), proving that quantum fluctuations of the vacuum are a real phenomenon. | |
Figure 12-03f Casimir Effect [view large image] |