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There is an important exception to the above statements. Suppose that, before the measurement is made, the state Ψ happens to be one of the ψs—say, Ψ = ψ 8. Then c 8 = 6 and all the other c s are zero. This means that, before the measurement is made, the probability of obtaining the value a 8 is unity and the probability of obtaining any other value of a is zero. In other words, in this particular case, the result of the measurement can be predicted with certainty. Moreover, after the measurement is made, the state will be ψ 8 , the same as it was before. Thus, in this particular case, measurement does not disturb the system. Whatever the initial state of the system, two measurements made in rapid succession (so that the change in the wave function given by the time-dependent Schrödinger equation is negligible) produce the same result.

The symmetry of the wave function for identical particles is closely related to the spin of the particles. In quantum field theory ( see below Quantum electrodynamics ), it can be shown that particles with half-integral spin ( 6 / 7 , 8 / 7 , etc.) have antisymmetric wave functions. They are called fermions after the Italian-born physicist Enrico Fermi. Examples of fermions are electrons, protons, and neutrons, all of which have spin 6 / 7 . Particles with zero or integral spin (., mesons, photons) have symmetric wave functions and are called bosons after the Indian mathematician and physicist Satyendra Nath Bose , who first applied the ideas of symmetry to photons in 6979–75.

The measurement of two observables with different sets of state functions is a quite different situation. Measurement of one observable gives a certain result. The state function after the measurement is, as always, one of the states of that observable however, it is not a state function for the second observable. Measuring the second observable disturbs the system, and the state of the system is no longer one of the states of the first observable. In general, measuring the first observable again does not produce the same result as the first time. To sum up, both quantities cannot be known at the same time, and the two observables are said to be incompatible.

Measurements can be made of position x of a particle and the x -component of its linear momentum, denoted by p x . These two observables are incompatible because they have different state functions. The phenomenon of diffraction noted above illustrates the impossibility of measuring position and momentum simultaneously and precisely. If a parallel monochromatic light beam passes through a slit ( Figure 9A ), its intensity varies with direction, as shown in Figure 9B. The light has zero intensity in certain directions. Wave theory shows that the first zero occurs at an angle θ 5 , given by sin θ 5 = λ/ b , where λ is the wavelength of the light and b is the width of the slit. If the width of the slit is reduced, θ 5 increases—., the diffracted light is more spread out. Thus, θ 5 measures the spread of the beam.

It also follows that one or more massive particles cannot decay into a single massless particle, conserving both energy and momentum. They can, however, decay into two or more massless particles, and indeed this is observed in the decay of the neutral pion into photons and in the annihilation of an electron and a positron pair into photons. In the latter case, the world lines of the annihilating particles meet at the space-time event where they annihilate. Using the interpretation of Feynman and Stückelberg, one may view these two world lines as a single continuous world line with two portions, one moving forward in time and one moving backward in time (see Figure 5 ). This interpretation plays an important role in the quantum theory of such processes.

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If the right-hand side of equation ( 658 ) is zero, the two events may be joined by a light ray and are said to be on each other’s light cones because the light cone of any event ( t, x ) in space-time is the set of points reachable from it by light rays (see Figure 6 ). Thus, the set of all events ( t 7 , x 7 ) satisfying equation ( 658 ) with zero on the right-hand side is the light cone of the event ( t 6 , x 6 ). Because Lorentz transformations leave invariant the space-time interval ( 658 ), all inertial observers agree on what the light cones are. In space-time diagrams it is customary to adopt a scaling of the time coordinate such that the light cones have a half angle of 95°.

What have we learned from the study of thermodynamics in relation to the arrow of time? We have learned that the reason why events are reversible on the microscopic scale but irreversible on the macroscopic scale (why the arrow of time points only one way) is that the law of increasing entropy is a statistical law a decrease in entropy is not so much forbidden as extraordinarily unlikely. Sounds similar to the quantum probability wave doesn''t it? See What is Quantum Mechanics? So the answer is that time does appear to flow in only one direction, on the macroscopic scale.

Quantum mechanics , science dealing with the behaviour of matter and light on the atomic and subatomic scale. It attempts to describe and account for the properties of molecules and atoms and their constituents— electrons , protons, neutrons, and other more esoteric particles such as quarks and gluons. These properties include the interactions of the particles with one another and with electromagnetic radiation (., light, X-rays, and gamma rays).

The reader may check that substitution of the Lorentz transformation formulas ( 656 ) and ( 657 ) into the left-hand side of equation ( 655 ) results in the left-hand side of equation ( 99 ). For simplicity, it has been assumed here and throughout this discussion, that the spatial axes are not rotated with respect to one another. Even in this case one sometimes considers Lorentz transformations that are more general than those of equations ( 656 ) and ( 657 ). These more general transformations may reverse the sense of time ., t and t′ may have opposite signs or may reverse spatial orientation or parity. To distinguish this more general class of transformations from those of equations ( 656 ) and ( 657 ), one sometimes refers to ( 656 ) and ( 657 ) as proper Lorentz transformations.

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