## Quantum Indeterminacy or Uncertainty

We now know that the equations of physics are not completely deterministic. When scientists began to apply Newtonian mechanics to objects as small as electrons they ran into problems.

In the early days of atomic theory scientists thought of electrons orbiting around the nucleus like the Earth orbiting around the Sun.

Gravity attracts the Earth and Sun to each other. Electrons and nuclei also attract each other by mutual gravitation, but the mass they have is so small that the force of gravity between them has never been measured. The Earth and Sun do not have any appreciable net electric charge and therefore do not attract or repel each other by electrical forces. Electrons are negatively charged and nuclei are positively charged. Therefore electrons and the nucleus of an atom attract each other much more strongly than they would if only their mutual gravitational attraction were acting. The stronger the attraction, the greater the speed the orbiting body must have to stay in orbit.

Charges separated in space produce an electric field between them. If the charges are moving, then the electric field changes. A changing electric field radiates energy away. Another way to say this is to note that an orbiting electron constitutes a circulating current. The current would be very large if the electron moved in a circular orbit at orbital speed. As such the circulating current would send forth electromagnetic waves like a radio transmitter.

For instance, a radio station produces an electric field between its antenna and the earth. If the frequency is 1.160 megahertz that means that the electric field reverses itself and then comes back to its original value 1 160 000 times per second. The energy supplied by the radio station radiates away through the air. Some is captured by radio receivers, amplified and demodulated to reproduce the sounds the station is broadcasting.

But if a circulating electron in an atom sent out waves, the waves would carry all of the electron’s orbital energy and speed away, and the electron would end up stuck to the nucleus. The atom would collapse to a very small size. All the atoms of the universe should have run down to zero energy in a fraction of a second if the planetary model for atoms were right. The model needed improvement because under ordinary conditions atoms don’t collapse. The electrons keep on circulating around the nuclei, the atoms retain their energy, and most of the time there is no radiation.

Quantum mechanics arose to explain atomic behavior when the planetary atom failed.

Physicists knew that isolated atoms radiate away energy only at certain frequencies. Johann Jakob Balmer (Swiss mathematician and physicist, 1825–1898) discovered in 1885 that some of the frequencies for atomic hydrogen form a series that he could express with a simple formula. Neils Henrik David Bohr (Danish physicist, 1885–1962) analyzed those frequencies and discovered in 1913 that atoms have definite energy levels. These findings led Erwin Schrödinger (Austrian physicist, 1887–1961) to propose a new model of the atom.

The Schrödinger model does not try to trace the detailed motion of the electron in its orbit. A probability amplitude function gives the chances of finding the electron in any given position. The stable states for the electron are those that give equal probability for finding the electron above or below, to the left or the right, in front or in back, of the nucleus. The average position of the electron’s negative charge is always centered exactly on the center of the positive nuclear charge. The positive and negative charges therefore maintain on the average no separation. That means that there is no average electric field in a stable atom, and without an electric field the energy does not radiate away.

Some stable states have more energy than others, if the electron is on average farther from the nucleus. When an electron moves down from a higher to a lower energy level the difference in energy radiates away as a photon or wave packet. This process is called photon emission. It is reversible. Passing photons shake atoms with their electromagnetic field. In so doing a photon can suddenly disappear, giving up all its energy to move one of the atom’s electrons from a lower to a higher energy level. That process is called photon absorption.

Atoms have various series of energy levels. One of the levels has to be the lowest of all. If all the electrons are in their lowest possible level and there are no photons passing by then all the electrons stay in the same lowest energy levels. On the other hand, photon absorption may have left one of the electrons in a higher level. Later, another passing photon may shake the atom and make the electron drop to a lower level. The energy the electron gives up produces another photon. The new photon will follow the stimulating photon. We call this process stimulated emission. Suppose an atom has an electron in a higher energy level for a long time, and no photon passes by to stimulate the atom to emit? In that case the electron may fall to a lower level anyway. Its energy will go out as a photon that goes off in an arbitrary direction. We call this process spontaneous emission.

There is no way to predict when a high-energy electron will drop to a lower orbit and give up energy as a photon. Likewise one cannot predict with certainty if an atom will absorb a passing photon. All one can do is give a probability of emission or absorption.

The limitation of only giving probabilities disturbed those physicists who believed that the laws of physics are completely deterministic.

To understand why, let’s consider the situation if we replace the electron with a ball big enough to handle.

When the masters are playing golf people may bet on whether the ball will fall in the hole or roll past. If the ball falls in the hole a physicist might say that the hole absorbed a passing ball. On a golf course one can only estimate probabilities because one can hardly take into account the power or inability of each blade of grass to deflect the ball. A pool table offers a more controlled situation. In pool the betting takes into account mainly the skill of the players. In a laboratory one might make a machine that was certain to knock a billiard ball into a given pocket. Newtonian mechanics would fully determine the ball’s path. But with objects as small as photons and electrons there is no way to control all the variables.

Newtonian mechanics required modification. The new physics of particles of very low mass is called quantum mechanics.

Quantum mechanics tells us only probabilities about the paths of the smallest and lightest particles. Indeed quantum mechanics applies to all particles large or small, but the probabilities narrow into virtual certainties when the particle becomes as large as a marble or a billiard ball or a planet.

Werner Heisenberg (German physicist, 1901–1976) developed a principle that allows us to determine approximately the range of probabilities in quantum mechanics. That is, without calculating exactly, we can know whether the range of probability in a given situation is narrow or broad.

Heisenberg based his principle on an observation about groups of waves. One can see groups of waves streaming from the bow of a speedboat as it races through relatively still water. If one is sitting in a canoe when a speedboat goes by, at first nothing happens. One can see the bow wave approaching, and then the first ripples arrive. They grow rapidly until the canoe is rocking in their troughs. Then the waves diminish again to ripples and one can see the bow waves passing on.

One cannot determine precisely the range of frequencies of a wave group and the group’s duration in time or its length, its extension in space. The problem is that a group of waves consists of a mixture of waves of different frequencies. Over a certain distance or time they reinforce each other and one can see a localized group of waves. But trying to measure the length of the group or the time it takes to pass is like trying to measure the size of a cloud. Where exactly is the starting and ending point?

For instance, we may photograph a water wave traveling in a tank with a glass side. We can measure distances on the photograph, but we have to decide how many of the leading and trailing ripples we should include in the main group of waves.

The lack of a definite starting and ending point complicates measuring the range of frequencies. One should count the number of wave crests or wave troughs between the chosen starting and ending points to find the main frequency. The spatial frequency is the number of wave crests or troughs divided by the distance between the starting and ending points. The temporal frequency is the number of wave crests or troughs divided by the time elapsed after the starting point passes and before the ending point arrives. But getting the main frequency is not enough. One has to get the frequency of the long waves and of the short waves, the general rise and fall and the little ripples, to find the range of frequencies.

The lack of definition of the starting and ending points will also produce an uncertainty in either the duration in time or the extension in space of the wave. We must express the duration or extension and the frequency range in consistent units. If we have the wave group’s duration in seconds then range of frequencies should be stated as the number of waves per second. If we have the wave train’s length or extension in meters then the frequency range should be stated as the number of waves per meter. Once we have stated the duration or extension and the frequency range in consistent units we may multiply them together to form a product. The Heisenberg uncertainty principle says that no one can reduce the product to a value much less than the number one.

When we apply this to electromagnetic waves we are talking about photon length and frequency bandwidth. Physicists define photon length in various ways. If the photon consists of many constructively interfering waves of different frequencies, there are two points where the leading and trailing edges of the photon have half the maximum amplitude. The distance between these points is the “full width at half maximum.” Similarly there are two frequencies in the photon’s frequency spectrum that have half the amplitude of the principal frequency. These define a full width at half maximum in the frequency spectrum. The product of the two full widths is always at least about equal to the number one. If we use fewer frequencies we will get a longer wave train, that is, a greater photon length. A short wave train requires many waves of a broad range of frequencies. No one can pin down a wave more precisely than the Heisenberg uncertainty principle allows.

The Heisenberg uncertainty principle applies to particles as well as waves. Louis Victor, Prince de Broglie (French physicist, 1892–1987) showed that both matter and energy consist of waves. A particle in motion is not a point moving at a certain speed. Rather it is a packet or group of traveling waves, like the ripple that goes out when one casts a stone into still water. We can always calculate the momentum of a particle. Momentum is the result or product of multiplying the particle’s mass by its speed. The de Broglie wavelength of a particle is Planck’s constant divided by the particle’s momentum. The de Broglie frequency is the speed of light times the particle’s momentum, divided by Planck’s constant. This makes the de Broglie frequency proportional to the speed of the particle.

We may apply the Heisenberg uncertainty principle to the de Broglie waves of a particle. The uncertainty Heisenberg described hinders quantum mechanical measurements. The trouble with a group of waves is that you need a long train of waves to limit the frequency bandwidth and define the speed, but you want a small wave packet to pinpoint the location of the particle. The uncertainty we find is the uncertainty in position and momentum. Momentum is the product of mass and velocity, and a particle’s mass may be determined with high precision. Therefore the main uncertainty is in position and velocity. Because of the nature of groups of waves, the product of the uncertainties in position and momentum are of the order of Planck’s constant. This is just Planck’s constant times the uncertainty of the order of one found before.

Now let’s apply this analysis to large and small particles. If the mass is large, the uncertainty in the product of position and velocity is small. Position and velocity can be measured rather precisely for large particles. However, when a particle has as little mass as an electron, the uncertainties are large. The uncertainties are smaller for an atom and begin to approach insignificance for a molecule. This is why people who deal with objects of ordinary size never need to take into account the Heisenberg uncertainty principle.

In a stable atom, the electron is presumably in very rapid motion constantly, but the Heisenberg uncertainty principle does not allow us to know this or measure it. If we could know (that is, measure) the electron’s exact position to determine if it were at any time separated by some distance from the nucleus, we could not simultaneously know how fast the electron was going. All we can know is that the atom maintains its energy and motion. The Heisenberg uncertainty principle has proved its usefulness in the Schrödinger model of the atom. Circular orbits would radiate energy away continuously and make atoms collapse, but the Heisenberg uncertainty principle says that we can’t follow the orbits of the electrons in detail. For all we know, electrons in stable states move in some very complicated way that does not radiate energy. Heisenberg’s uncertainty principle is certain and well established.

In the early days of atomic theory scientists thought of electrons orbiting around the nucleus like the Earth orbiting around the Sun.

Gravity attracts the Earth and Sun to each other. Electrons and nuclei also attract each other by mutual gravitation, but the mass they have is so small that the force of gravity between them has never been measured. The Earth and Sun do not have any appreciable net electric charge and therefore do not attract or repel each other by electrical forces. Electrons are negatively charged and nuclei are positively charged. Therefore electrons and the nucleus of an atom attract each other much more strongly than they would if only their mutual gravitational attraction were acting. The stronger the attraction, the greater the speed the orbiting body must have to stay in orbit.

Charges separated in space produce an electric field between them. If the charges are moving, then the electric field changes. A changing electric field radiates energy away. Another way to say this is to note that an orbiting electron constitutes a circulating current. The current would be very large if the electron moved in a circular orbit at orbital speed. As such the circulating current would send forth electromagnetic waves like a radio transmitter.

For instance, a radio station produces an electric field between its antenna and the earth. If the frequency is 1.160 megahertz that means that the electric field reverses itself and then comes back to its original value 1 160 000 times per second. The energy supplied by the radio station radiates away through the air. Some is captured by radio receivers, amplified and demodulated to reproduce the sounds the station is broadcasting.

But if a circulating electron in an atom sent out waves, the waves would carry all of the electron’s orbital energy and speed away, and the electron would end up stuck to the nucleus. The atom would collapse to a very small size. All the atoms of the universe should have run down to zero energy in a fraction of a second if the planetary model for atoms were right. The model needed improvement because under ordinary conditions atoms don’t collapse. The electrons keep on circulating around the nuclei, the atoms retain their energy, and most of the time there is no radiation.

Quantum mechanics arose to explain atomic behavior when the planetary atom failed.

Physicists knew that isolated atoms radiate away energy only at certain frequencies. Johann Jakob Balmer (Swiss mathematician and physicist, 1825–1898) discovered in 1885 that some of the frequencies for atomic hydrogen form a series that he could express with a simple formula. Neils Henrik David Bohr (Danish physicist, 1885–1962) analyzed those frequencies and discovered in 1913 that atoms have definite energy levels. These findings led Erwin Schrödinger (Austrian physicist, 1887–1961) to propose a new model of the atom.

The Schrödinger model does not try to trace the detailed motion of the electron in its orbit. A probability amplitude function gives the chances of finding the electron in any given position. The stable states for the electron are those that give equal probability for finding the electron above or below, to the left or the right, in front or in back, of the nucleus. The average position of the electron’s negative charge is always centered exactly on the center of the positive nuclear charge. The positive and negative charges therefore maintain on the average no separation. That means that there is no average electric field in a stable atom, and without an electric field the energy does not radiate away.

Some stable states have more energy than others, if the electron is on average farther from the nucleus. When an electron moves down from a higher to a lower energy level the difference in energy radiates away as a photon or wave packet. This process is called photon emission. It is reversible. Passing photons shake atoms with their electromagnetic field. In so doing a photon can suddenly disappear, giving up all its energy to move one of the atom’s electrons from a lower to a higher energy level. That process is called photon absorption.

Atoms have various series of energy levels. One of the levels has to be the lowest of all. If all the electrons are in their lowest possible level and there are no photons passing by then all the electrons stay in the same lowest energy levels. On the other hand, photon absorption may have left one of the electrons in a higher level. Later, another passing photon may shake the atom and make the electron drop to a lower level. The energy the electron gives up produces another photon. The new photon will follow the stimulating photon. We call this process stimulated emission. Suppose an atom has an electron in a higher energy level for a long time, and no photon passes by to stimulate the atom to emit? In that case the electron may fall to a lower level anyway. Its energy will go out as a photon that goes off in an arbitrary direction. We call this process spontaneous emission.

There is no way to predict when a high-energy electron will drop to a lower orbit and give up energy as a photon. Likewise one cannot predict with certainty if an atom will absorb a passing photon. All one can do is give a probability of emission or absorption.

The limitation of only giving probabilities disturbed those physicists who believed that the laws of physics are completely deterministic.

To understand why, let’s consider the situation if we replace the electron with a ball big enough to handle.

When the masters are playing golf people may bet on whether the ball will fall in the hole or roll past. If the ball falls in the hole a physicist might say that the hole absorbed a passing ball. On a golf course one can only estimate probabilities because one can hardly take into account the power or inability of each blade of grass to deflect the ball. A pool table offers a more controlled situation. In pool the betting takes into account mainly the skill of the players. In a laboratory one might make a machine that was certain to knock a billiard ball into a given pocket. Newtonian mechanics would fully determine the ball’s path. But with objects as small as photons and electrons there is no way to control all the variables.

Newtonian mechanics required modification. The new physics of particles of very low mass is called quantum mechanics.

Quantum mechanics tells us only probabilities about the paths of the smallest and lightest particles. Indeed quantum mechanics applies to all particles large or small, but the probabilities narrow into virtual certainties when the particle becomes as large as a marble or a billiard ball or a planet.

Werner Heisenberg (German physicist, 1901–1976) developed a principle that allows us to determine approximately the range of probabilities in quantum mechanics. That is, without calculating exactly, we can know whether the range of probability in a given situation is narrow or broad.

Heisenberg based his principle on an observation about groups of waves. One can see groups of waves streaming from the bow of a speedboat as it races through relatively still water. If one is sitting in a canoe when a speedboat goes by, at first nothing happens. One can see the bow wave approaching, and then the first ripples arrive. They grow rapidly until the canoe is rocking in their troughs. Then the waves diminish again to ripples and one can see the bow waves passing on.

One cannot determine precisely the range of frequencies of a wave group and the group’s duration in time or its length, its extension in space. The problem is that a group of waves consists of a mixture of waves of different frequencies. Over a certain distance or time they reinforce each other and one can see a localized group of waves. But trying to measure the length of the group or the time it takes to pass is like trying to measure the size of a cloud. Where exactly is the starting and ending point?

For instance, we may photograph a water wave traveling in a tank with a glass side. We can measure distances on the photograph, but we have to decide how many of the leading and trailing ripples we should include in the main group of waves.

The lack of a definite starting and ending point complicates measuring the range of frequencies. One should count the number of wave crests or wave troughs between the chosen starting and ending points to find the main frequency. The spatial frequency is the number of wave crests or troughs divided by the distance between the starting and ending points. The temporal frequency is the number of wave crests or troughs divided by the time elapsed after the starting point passes and before the ending point arrives. But getting the main frequency is not enough. One has to get the frequency of the long waves and of the short waves, the general rise and fall and the little ripples, to find the range of frequencies.

The lack of definition of the starting and ending points will also produce an uncertainty in either the duration in time or the extension in space of the wave. We must express the duration or extension and the frequency range in consistent units. If we have the wave group’s duration in seconds then range of frequencies should be stated as the number of waves per second. If we have the wave train’s length or extension in meters then the frequency range should be stated as the number of waves per meter. Once we have stated the duration or extension and the frequency range in consistent units we may multiply them together to form a product. The Heisenberg uncertainty principle says that no one can reduce the product to a value much less than the number one.

When we apply this to electromagnetic waves we are talking about photon length and frequency bandwidth. Physicists define photon length in various ways. If the photon consists of many constructively interfering waves of different frequencies, there are two points where the leading and trailing edges of the photon have half the maximum amplitude. The distance between these points is the “full width at half maximum.” Similarly there are two frequencies in the photon’s frequency spectrum that have half the amplitude of the principal frequency. These define a full width at half maximum in the frequency spectrum. The product of the two full widths is always at least about equal to the number one. If we use fewer frequencies we will get a longer wave train, that is, a greater photon length. A short wave train requires many waves of a broad range of frequencies. No one can pin down a wave more precisely than the Heisenberg uncertainty principle allows.

The Heisenberg uncertainty principle applies to particles as well as waves. Louis Victor, Prince de Broglie (French physicist, 1892–1987) showed that both matter and energy consist of waves. A particle in motion is not a point moving at a certain speed. Rather it is a packet or group of traveling waves, like the ripple that goes out when one casts a stone into still water. We can always calculate the momentum of a particle. Momentum is the result or product of multiplying the particle’s mass by its speed. The de Broglie wavelength of a particle is Planck’s constant divided by the particle’s momentum. The de Broglie frequency is the speed of light times the particle’s momentum, divided by Planck’s constant. This makes the de Broglie frequency proportional to the speed of the particle.

We may apply the Heisenberg uncertainty principle to the de Broglie waves of a particle. The uncertainty Heisenberg described hinders quantum mechanical measurements. The trouble with a group of waves is that you need a long train of waves to limit the frequency bandwidth and define the speed, but you want a small wave packet to pinpoint the location of the particle. The uncertainty we find is the uncertainty in position and momentum. Momentum is the product of mass and velocity, and a particle’s mass may be determined with high precision. Therefore the main uncertainty is in position and velocity. Because of the nature of groups of waves, the product of the uncertainties in position and momentum are of the order of Planck’s constant. This is just Planck’s constant times the uncertainty of the order of one found before.

Now let’s apply this analysis to large and small particles. If the mass is large, the uncertainty in the product of position and velocity is small. Position and velocity can be measured rather precisely for large particles. However, when a particle has as little mass as an electron, the uncertainties are large. The uncertainties are smaller for an atom and begin to approach insignificance for a molecule. This is why people who deal with objects of ordinary size never need to take into account the Heisenberg uncertainty principle.

In a stable atom, the electron is presumably in very rapid motion constantly, but the Heisenberg uncertainty principle does not allow us to know this or measure it. If we could know (that is, measure) the electron’s exact position to determine if it were at any time separated by some distance from the nucleus, we could not simultaneously know how fast the electron was going. All we can know is that the atom maintains its energy and motion. The Heisenberg uncertainty principle has proved its usefulness in the Schrödinger model of the atom. Circular orbits would radiate energy away continuously and make atoms collapse, but the Heisenberg uncertainty principle says that we can’t follow the orbits of the electrons in detail. For all we know, electrons in stable states move in some very complicated way that does not radiate energy. Heisenberg’s uncertainty principle is certain and well established.