Matrix Theory and Linear Algebra (Несколько текстов для зачёта), страница 8

To see how particles behave in the double-slit experiment, physicists replace the water with marbles. The barrier slits are about the width of a marble, as the point of this experiment is to allow particles (in this case, marbles) to pass through the barrier. The marbles are put in motion and pass through the barrier, striking the detector at the far end of the apparatus. The results show that the marbles do not interfere with each other or with themselves like waves do. Instead, the marbles strike the detector most frequently in the two points directly opposite each slit.

When physicists perform the double-slit experiment with electrons, the detection pattern matches that produced by the waves, not the marbles. These results show that electrons do have wave properties. However, if scientists run the experiment using a barrier whose slits are much wider than the de Broglie wavelength of the electrons, the pattern resembles the one produced by the marbles. This shows that tiny particles such as electrons behave as waves in some circumstances and as particles in others.

Before the development of quantum theory, physicists assumed that, with perfect equipment in perfect conditions, measuring any physical quantity as accurately as desired was possible. Quantum mechanical equations show that accurate measurement of both the position and the momentum of a particle at the same time is impossible. This rule is called Heisenberg’s uncertainty principle after German physicist Werner Heisenberg, who derived it from other rules of quantum theory. The uncertainty principle means that as physicists measure a particle’s position with more and more accuracy, the momentum of the particle becomes less and less precise, or more and more uncertain, and vice versa.

Heisenberg formally stated his principle by describing the relationship between the uncertainty in the measurement of a particle’s position and the uncertainty in the measurement of its momentum. Heisenberg said that the uncertainty in position (represented by Δx) times the uncertainty in momentum (represented by Δp;) must be greater than a constant number equal to Planck’s constant (h) divided by 4 ( is a constant approximately equal to 3.14). Mathematically, the uncertainty principle can be written as Δx Δp > h / 4. This relationship means that as a scientist measures a particle’s position more and more accurately—so the uncertainty in its position becomes very small—the uncertainty in its momentum must become large to compensate and make this expression true. Likewise, if the uncertainty in momentum, Δp, becomes small, Δx must become large to make the expression true.

One way to understand the uncertainty principle is to consider the dual wave-particle nature of light and matter. Physicists can measure the position and momentum of an atom by bouncing light off of the atom. If they treat the light as a wave, they have to consider a property of waves called diffraction when measuring the atom’s position. Diffraction occurs when waves encounter an object—the waves bend around the object instead of traveling in a straight line. If the length of the waves is much shorter than the size of the object, the bending of the waves just at the edges of the object is not a problem. Most of the waves bounce back and give an accurate measurement of the object’s position. If the length of the waves is close to the size of the object, however, most of the waves diffract, making the measurement of the object’s position fuzzy. Physicists must bounce shorter and shorter waves off an atom to measure its position more accurately. Using shorter wavelengths of light, however, increases the uncertainty in the measurement of the atom’s momentum.

Light carries energy and momentum, because of its particle nature (described in the Compton effect). Photons that strike the atom being measured will change the atom’s energy and momentum. The fact that measuring an object also affects the object is an important principle in quantum theory. Normally the affect is so small it does not matter, but on the small scale of atoms, it becomes important. The bump to the atom increases the uncertainty in the measurement of the atom’s momentum. Light with more energy and momentum will knock the atom harder and create more uncertainty. The momentum of light is equal to Plank’s constant divided by the light’s wavelength, or p = h/λ. Physicists can increase the wavelength to decrease the light’s momentum and measure the atom’s momentum more accurately. Because of diffraction, however, increasing the light’s wavelength increases the uncertainty in the measurement of the atom’s position. Physicists most often use the uncertainty principle that describes the relationship between position and momentum, but a similar and important uncertainty relationship also exists between the measurement of energy and the measurement of time.

Quantum theory gives exact answers to many questions, but it can only give probabilities for some values. A probability is the likelihood of an answer being a certain value. Probability is often represented by a graph, with the highest point on the graph representing the most likely value and the lowest representing the least likely value. For example, the graph that shows the likelihood of finding the electron of a hydrogen atom at a certain distance from the nucleus looks like the following:

The nucleus of the atom is at the left of the graph. The probability of finding the electron very near the nucleus is very low. The probability reaches a definite peak, marking the spot at which the electron is most likely to be.

Scientists use a mathematical expression called a wave function to describe the characteristics of a particle that are related to time and space—such as position and velocity. The wave function helps determine the probability of these aspects being certain values. The wave function of a particle is not the same as the wave suggested by wave-particle duality. A wave function is strictly a mathematical way of expressing the characteristics of a particle. Physicists usually represent these types of wave functions with the Greek letter psi, Ψ. The wave function of the electron in a hydrogen atom is:

The symbol  and the letter e in this equation represent constant numbers derived from mathematics. The letter a is also a constant number called the Bohr radius for the hydrogen atom. The square of a wave function, or a wave function multiplied by itself, is equal to the probability density of the particle that the wave function describes. The probability density of a particle gives the likelihood of finding the particle at a certain point.

The wave function described above does not depend on time. An isolated hydrogen atom does not change over time, so leaving time out of the atom’s wave function is acceptable. For particles in systems that change over time, physicists use wave functions that depend on time. This lets them calculate how the system and the particle’s properties change over time. Physicists represent a time-dependent wave function with Ψ(t), where t represents time.

The wave function for a single atom can only reveal the probability that an atom will have a certain set of characteristics at a certain time. Physicists call the set of characteristics describing an atom the state of the atom. The wave function cannot describe the actual state of the atom, just the probability that the atom will be in a certain state.

The wave functions of individual particles can be added together to create a wave function for a system, so quantum theory allows physicists to examine many particles at once. The rules of probability state that probabilities and actual values match better and better as the number of particles (or dice thrown, or coins tossed, whatever the probability refers to) increases. Therefore, if physicists consider a large group of atoms, the wave function for the group of atoms provides information that is more definite and useful than that provided by the wave function of a single atom.

An example of a wave function for a single atom is one that describes an atom that has absorbed some energy. The energy has boosted the atom’s electrons to a higher energy level, and the atom is said to be in an excited state. It can return to its normal ground state (or lowest energy state) by emitting energy in the form of a photon. Scientists call the wave function of the initial exited state Ψ_i (with “i” indicating it is the initial state) and the wave function of the final ground state Ψ_f (with “f” representing the final state). To describe the atom’s state over time, they multiply Ψ_i by some function, a(t), that decreases with time, because the chances of the atom being in this excited state decrease with time. They multiply Ψ_f by some function, b(t), that increases with time, because the chances of the atom being in this state increase with time. The physicist completing the calculation chooses a(t) and b(t) based on the characteristics of the system. The complete wave equation for the transition is the following:

Ψ = a(t) Ψ_i + b(t) Ψ_f.

At any time, the rules of probability state that the probability of finding a single atom in either state is found by multiplying the coefficient of its wave function (a(t) or b(t)) by itself. For one atom, this does not give a very satisfactory answer. Even though the physicist knows the probability of finding the atom in one state or the other, whether or not reality will match probability is still far from certain. This idea is similar to rolling a pair of dice. There is a 1 in 6 chance that the roll will add up to seven, which is the most likely sum. Each roll is random, however, and not connected to the rolls before it. It would not be surprising if ten rolls of the dice failed to yield a sum of seven. However, the actual number of times that seven appears matches probability better as the number of rolls increases. For one million or one billion rolls of the dice, one of every six rolls would almost certainly add up to seven.

Similarly, for a large number of atoms, the probabilities become approximate percentages of atoms in each state, and these percentages become more accurate as the number of atoms observed increases. For example, if the square of a(t) at a certain time is 0.2, then in a very large sample of atoms, 20 percent (0.2) of the atoms will be in the initial state and 80 percent (0.8) will be in the final state.

One of the most puzzling results of quantum mechanics is the effect of measurement on a quantum system. Before a scientist measures the characteristics of a particle, its characteristics do not have definite values. Instead, they are described by a wave function, which gives the characteristics only as fuzzy probabilities. In effect, the particle does not exist in an exact location until a scientist measures its position. Measuring the particle fixes its characteristics at specific values, effectively “collapsing” the particle’s wave function. The particle’s position is no longer fuzzy, so the wave function that describes it has one high, sharp peak of probability. If the position of a particle has just been measured, the graph of its probability density looks like the following:

In the 1930s physicists proposed an imaginary experiment to demonstrate how measurement causes complications in quantum mechanics. They imagined a system that contained two particles with opposite values of spin, a property of particles that is analogous to angular momentum. The physicists can know that the two particles have opposite spins by setting the total spin of the system to be zero. They can measure the total spin without directly measuring the spin of either particle. Because they have not yet measured the spin of either particle, the spins do not actually have defined values. They exist only as fuzzy probabilities. The spins only take on definite values when the scientists measure them.