1

Quantum Mechanics

Welcome to the quantum zone. You hop in your car one morning and start heading north

toward school. For some reason, the smooth acceleration you are accustomed to during the

trip does not happen. The speedometer needle locks on 10 mi/h, then suddenly jumps to 20

mi/h, then to 30 mi/h, and so on as you depress the accelerator pedal The scenery drifts by at

10 mi/h, then suddenly at double that rate, then suddenly at triple that rate, and so forth.

Another strange thing happens. There is no way you can get out of first gear when you are

going 10 mi/h. After the jump to 20 mi/h, however, you have a choice of two gears, first and

second. After another leap to 30 mi/h, three gears are available. You notice that the needle on

your car's tachometer (engine-revolutions-per-minute gauge) jumps from one specific value

to another specific value whenever you change gears but you do not change the car's speed.

You find that the crankshaft of your engine can turn only at specific rates (or revolutions per

minute) and that no intermediate rates of engine speed are possible.

And here's one more strange thing. At 10 mi/h, your car acts as if it were affixed to trolley

tracks in the roadway; your car can move only north or south. At 20 mi/h, your car is able to

turn, but only very sharply, onto any street going east or west. From those streets, abrupt

turns onto any north-south street are possible. At 30 mi/h, your options are broadened; you

can now move in any of 8 directions: north, northeast, east, southeast, south, southwest, west,

and northwest. The wheels of your car simply refuse to spin in all other directions besides

these 8 points on the compass. At 40 mi/h, there are 16 directions to choose from.

There is more strangeness in the quantum zone. Your car itself is a kind of "fuzzy beast." It is

like a long series of waves rippling down the road. Waves are difficult to locate precisely in

space, so it is hard to say just where your car is at any moment within the train of waves. It

might be on or near the leading edge, on the trailing edge, or somewhere in the middle of the

wave train.

You are approaching a traffic light that is changing from yellow to red, and you are probably

exceeding the speed limit, too. A traffic officer on the curb, seeking to enrich the coffers of

the municipal treasury, wants to issue you two tickets: one for speeding and one for running a

red light. Alas, he finds he cannot gather definitive evidence for both of these infractions at

the same time. He has to settle for one only. If he measures the speed of the entire wave train

and that speed is in excess of the speed limit, then he has caught you speeding. However, at

the same time, he cannot know whether or not you ran a red light because he does not know

exactly where your car is within the wave train. On the other hand, if the officer is able to

precisely locate your car entering the intersection on a red light, then he really does not know

your car's speed. To determine speed (the distance between two positions divided by time),

he needs a later measurement of your car's position, and by that time, your car could be

located somewhere else (ahead of or behind the place where it was before) within the wave

train.

These imaginary quantum zone experiences, which are based loosely on the behavior of

electrons, may give you some insight into several aspects of quantum mechanical theory.

Together, they are a far from complete or even accurate analogy of what really goes on inside

the atom-but then again, quantum mechanics by its very nature is abstract. It always resists

being compared to ordinary experience.

Quantum mechanics is a mathematical description of the behavior of particles far too small to

see. Time and time again, it has proved its worth through the success of its predictions. As

2

such, scientists accept its validity. We need not fully understand something (in an

experiential sense) to make use of it. In fact, well over half of the world's entire economy is

linked to inventions that have stemmed from the applied principles of quantum mechanics.

Let us now investigate, in a more formal manner, the aspects of quantum theory that pertain

to tiny bits of matter.

Light has a wave-particle duality: its behavior incorporates wavelike features and particle

like features. Might not pieces of matter, which we have traditionally thought of as particles,

act like waves? If so, then a certain kind of symmetry would encompass both matter and pure

energy (for example, light). In other words, both would exhibit wave-particle duality.

Figure 1

Figure 1 (a) When waves diffract through two slits and interfere, a pattern of many interference fringes appears on the screen. The waves must be of relatively large wavelength compared with the slit width in order to diffract significantly. (b) Particles, assuming that they have a wavelength too small to diffract significantly, have trajectories that carry them to only two spots on the screen.

Matter Waves

These thoughts and some of their implications were first expressed by the French physicist

Louis de Broglie in his doctoral dissertation, completed in 1923. Mindful of Einstein's

conclusions that mass and light energy are equivalent (E = mc2) and that there is an intimate

relationship between space and time, de Broglie deduced that particles of matter could indeed

be thought of as waves of matter, or matter waves. De Broglie predicted that the matter

waves associated with a moving particle of matter should have a wavelength of

A = h / p = h / mv Equation 1

where h is Planck's constant (6.63 X 10-34 J*s), and p is the particle's momentum (which, as

we learned in the notes on Laws of Motion, is the particle's mass multiplied by its velocity,

or mv).

Equation 1 says that a particle of matter somehow propagates as moving waves with a

wavelength that depends on both the mass and the speed of the moving particle.

= 10-35 m

The meaning of matter waves is difficult to interpret, but the matter (or de Broglie)

wavelength A of a particle can be somewhat concretely described as being related to the

size of the region of influence of the particle.

3

Matter Waves and the Bohr Atom

It was not clear in the beginning just what matter waves might be, but de Broglie went ahead

and applied the matter-as-moving-waves idea to the electrons of Bohr's model of the

hydrogen atom. Figure 2.

Figure 2

Planck's Constant and the Energy of a Photon

In 1900, Max Planck was working on the problem of how the radiation an object

emits is related to its temperature. He came up with a formula that agreed very

closely with experimental data, but the formula only made sense if he assumed

that the energy of a vibrating molecule was quantized - that is, it could only take

on certain values. The energy would have to be proportional to the frequency of

vibration, and it seemed to come in little "chunks" of the frequency multiplied by

a certain constant. This constant came to be known as Planck's constant, or h,

and it has the value 6.63 x 10-34 J*s.

J stands for Joule, in honor of James Joule who examined the relationship

between heat and energy in the 19th Century. The Joule is the SI unit for

measuring energy. A joule is related to the fundamental units in that a joule is 1

kg m2/s2. By no means is that obvious, but it is similar to measuring area in m2

(distance times distance) or speed in m/s (distance divided by time).

So J*s is a Joule second. What would that be? That's not quite energy, not quite

time.

The fancy name for what a Joule represents is angular momentum, but all that

you need to know now is that it involves things that spin, like bicycle wheels and

tops.

4

The speed of the electron in each of the Bohr orbits is known. For each orbit the speed is just

sufficient to keep the electron orbiting around the proton in the nucleus to which it is

attracted. (Similarly, a planet must move around the sun at a particular speed if it is to remain

in the same orbit.) The radii of the various Bohr orbits were known as well; they were related

to the potential energies of the electron at different distances from the nucleus. Since these

orbits are all circles, the distance around the rim of each orbit (circumference) is just 27Tr.

Knowing the mass of the electron and its speed in each of the Bohr orbits, de Broglie used

Equation 1 to calculate the electron's wavelength in each case. The amazing result,

schematically illustrated in Figure 3, is that the electron's matter waves fit in a standing wave

pattern around the circumference of each orbit. In other words, the electron, as a traveling

wave, reinforces itself (experiences constructive interference) every time it completes an

orbit. One complete wave fits into the first Bohr orbit (n = I, the ground state). Two complete

waves fit into the second Bohr orbit (n = 2, the "first excited state). Three complete waves

fit into the n = 3 orbit, and so on. By this way of thinking, the electron cannot exist in any

orbit other than those described by n = 1, 2, 3, and so on, because those orbits would involve

a mismatch of wave "crests and "troughs as the wave repeats its path around the orbit.

These situations of destructive interference make it impossible for an electron to persist in

intermediate orbits.

The electron "waves sketched in Figure 2 resemble the fundamental and harmonic

frequencies of a vibrating string (see Figure 8.21), except that they are wrapped around

circles. It seems that in the hydrogen atom, the electron can exist in only one mode of

vibration at a time; and it may jump from one discrete mode (or orbit) to another, absorbing

or giving up energy as it does.

Figure 3

The perfect fit of hypothetical matter waves around the circumference of electron orbits was

either an incredible coincidence or a true description of what really happens. Soon, a host of

experiments were demonstrating that matter waves are indeed real. When a beam of X-rays is

sent through a thin metal foil, many are diffracted (bent) when passing through the crystalline

lattice of metal atoms. The X-rays emerge in a geometrical pattern somewhat similar to that

produced by light when it reflects from a diffraction grating or the pitted surface of a

5

compact disc. These X-rays are simply exhibiting normal wave behavior. When electrons,

having a matter wavelength equal to that of the X-rays, are used in the same experiments, the

diffraction effect is exactly the same (Figure 4). Somehow, the electrons act just like waves.

The wave nature of speeding electrons has been put to good use in electron microscopes

which use beams of electrons instead of light waves. The faster the electrons move in a beam,

the shorter their wavelength is and the better they are able to produce high-resolution images

of tiny objects.

Figure 4

The Uncertainty Principle

If, on the quantum level, matter is significantly smeared out into waves, then there must be a

substantial amount of imprecision involved in measurements made at very small scales.

Werner Heisenberg, one of the many contributors to the theory of quantum mechanics,

proposed in 1927 that it is impossible to know both the exact position and the exact

momentum of a particle at the same time. Heisenberg's mathematical statement of this

postulate, which is called the uncertainty principle, can be written as

(p)( X) = h Equation 2

where p is the momentum of a particle, x represents its position, and h is Planck's constant.

The quantity p represents the uncertainty in the measurement of momentum, and x

represents the uncertainty in the measurement of position. This state of affairs was alluded to

in the imaginary tale that opened this chapter: The traffic officer could not issue two tickets

for two infractions at the same time-one based on speeding (which involves knowing the car's

momentum or speed precisely) and the other based on running a red light (which involves

measuring the car's position precisely).

Two things are important to keep in mind about the uncertainty principle. One is that a

relatively good knowledge of a particle's momentum (or essentially, velocity, as long as the

particle's mass is constant) implies a relatively poor knowledge of the particle's position at

the same time. The opposite is true as well. A relatively good knowledge of a particle's

position implies a relatively poor knowledge of the particle's momentum at the same time.

The second important thing to realize is that the two uncertainties, multiplied together, are

approximately equal to h, which has an incredibly small value. Thus, any trade-offs between

p and x are significant only for particles with tiny momenta and tiny sizes. These are the

same particles (particularly atoms and subatomic particles) for which all the other quantum

effects are significant. Because the value of h is very small, quantum uncertainty has almost

no effect on macroscopic bodies. Only if h (which, like c, the speed of light, is a fundamental

constant of nature) were very much larger than it is would it be impossible to simultaneously

get good measurements of both the speed and the position of a car nearing an intersection.

6

To get some feeling for how the uncertainty principle works for small particles, imagine

trying to simultaneously locate and measure the momentum of some very small particle. To

"see" this particle, you must send at least one other "particle" (a photon of light or an

electron) that will bounce off the particle you are trying to observe and then come back to

you. Photons carry momentum, and certainly moving electrons do, too. So when the probing

photon or electron returns to you, it has already interacted with and disturbed the particle you

are trying to observe in a way that cannot be known unless you succeed in locating the

particle a second time. The very act of observing the particle disturbed it in an unpredictable

way. The returning electron or photon told you the particle's position at the moment of

encounter, but the particle skittered away at an unknown speed and direction after the

encounter, and that made it impossible for you to determine the particle's momentum.

Some interesting analogies have been drawn between the study of social behavior and the

uncertainty principle. For example, suppose that an anthropologist parachutes into a jungle

village whose inhabitants have had no previous contact with the modern world. By the very

act of the anthropologist's godlike arrival and her interaction with the villagers, the culture

might very well change in an unpredictable way. It is therefore quite likely that the

anthropologist will never be able to formulate an objective and accurate description of all

aspects of that culture as it was before she arrived.

Despite such comparisons, it is important to realize that only at the quantum level does the

act of making a physical measurement significantly affect the object being measured. Even

with this restriction in mind, however the philosophical implications of the uncertainty

principle are enough to destroy the notion of the universe running like a perfect machine. If it

is in principle, impossible to measure all the properties of a particle or system of particles at

anyone instant of time, then it is impossible to make confident predictions concerning all

later instants of time.

Quantum Numbers

In the older (pre-quantum mechanics or classical physics) way of picturing the atom, an

electron moving about the nucleus is described as having energy, momentum, and a certain

orbit (path of travel). In the newer quantum mechanical way of picturing the atom, the

uncertainty principle comes into play, and our knowledge of an electron's position and move-

ment lacks the crispness and apparent certainty with which we view the larger world. In the

quantum world, an atomic electron still has energy and momentum, but these quantities can

change only in discrete steps; they are quantized just as everything else is in the realm of the

very small.

We may be uncertain about exactly where an atomic electron is and where it is headed at any

instant of time, but we can certainly identify trends, or probabilities associated with its

position. These probabilities were given a firm mathematical description in the work of

Erwin Schrodinger, another giant in the development of quantum mechanics. Schrodinger's

wave equation (which because of its mathematical complexity will remain unwritten here)

describes particles, such as electrons, as waves occupying all three dimensions of space and

(if need be) changing with time.

If an electron in an atom remains in a stationary state, then its wavelike nature does not

change with time, and Schrodinger's wave equation can be used to describe the probability of

7

its position in three-dimensional space. For an electron in the ground state (n = 1) of a

hydrogen atom, the electron ends up "looking" as if it occupied a fuzzy sphere around the

nucleus (see Figure 5). In other words, rather than following the simplistic circular path of a

Bohr model orbit, the electron follows some unknown random path at any instant. Over time,

however, the pattern of probability of the electron's position (called the electron's probability

cloud) spreads out radially from the nucleus so that the negative charge carried by the

electron occupies a fuzzy sphere. If you choose to measure the exact position of the electron

at some instant, it will most likely be found at or near the same distance from the nucleus as

the radius of the n = 1 Bohr orbit. (Of course, then you don't know its momentum very well.

You are uncertain of its speed and direction.) There is also some chance that your

instantaneous positional measurement might locate the electron inside or outside the most

probable radius.

The number n, which increases in integer steps in a hydrogen atom (as well as in all other

atoms), denotes different quantized energy states in which an electron may be found when it

is bound to an atom. It can be thought of as a code number symbolizing those various energy

states. It is also used as a multiplier in formulas that determine just how much overall

potential energy the electron has in each state and how far the electron is : on average) from

the nucleus in each state.

Figure 5

It turns out that electrons, when bound in some way to atoms, possess properties apart from

overall potential energy, which is symbolized by n. These other properties, which involve the

electron's angular momentum, are only three in number. (Recall that angular momentum is a

quantity associated with bodies that revolve or spin. An atomic electron can be thought of as

capable of doing both.) Each of these other three properties s also quantized. Thus, a

complete description of an atomic electron's state of being (its "quantum state") can be

symbolized by a total of four code numbers, called quantum numbers. We will now briefly

describe the meaning of each of these quantum numbers.

The Four Quantum Numbers

Principal quantum number, (n) describes the electron's average distance from the nucleus as

well as its overall potential energy. Within a given quantum state n, however, the exact

potential energy of the electron may vary somewhat owing to different possible values of its

angular momentum. As we have seen, n takes on integer values: n = 1, 2, 3, The overall

effect of n is on the size of the electron probability cloud; the larger n is, the farther out from

the nucleus an electron is most likely to be found.

8

The quantum number 1 (el in italics), which is called the orbital angular momentum quantum

number, describes the magnitude of the electron's angular momentum as it moves about the

nucleus. It, too, is quantized. (Macroscopically, this would be like a bicycle wheel that could

spin only at certain speeds with no intermediate speeds possible.)

The possible angular momentum magnitudes are limited in a way that depends on the

electron's n state; this is codified by the rule l = 0, 1, 2, ..., n - 1. In other words, if an electron

is in the n = 1 state, then 1 = 0 is the only possible angular momentum the electron can have-

which is to say that its angular momentum is zero. That particular state is represented by the

electron occupying a spherical fuzzy cloud, as in Figure 3. An 1 = 0 electron has not stopped

moving; rather, it moves around the nucleus in all possible directions, so that its angular

momentum sums to zero. In the n = 2 state, however, an electron may assume either of two

possible angular momentum magnitude states, 1 = 0 (zero angular momentum) and 1 = 1 (a

nonzero angular momentum). The overall effect of l is on the shape of the electron

probability cloud. An electron with 1 = 0 forms a spherical cloud, an electron with 1 = 1

forms a dumbbell shaped cloud and electrons with higher 1 values form probability clouds of

greater complexity.

Figure 6

Probability clouds

Figure 7

Angular Momentum

Angular momentum is a vector that has direction as well as magnitude. The quantum number

ml, the orbital magnetic quantum number determines the quantized direction of the electron's

angular momentum as it moves about the nucleus. (Macroscopically, this would be like the

axis of a spinning bicycle wheel that could be oriented only in specific directions.) The term

9

magnetic is used to describe ml because an electron with an angular momentum, like a bunch

of electrons circulating in a coil, acts just like a tiny bar magnet.

The fourth and final quantum number ms the spin magnetic quantum number is a description

of an atomic electron's spin. Regardless of what other quantum numbers a given electron has,

it either spins in one particular direction or in the opposite direction. These spin states are

denoted ms = +1/2 and ms = - 1/2. There is some question as to whether or not electrons

actually spin, because we cannot see them spinning. But that is really beside the point,

because even nonmoving electrons act like very weak bar magnets that either point "up" or

"down." This is exactly how they would behave if they were spheres of negative charge

spinning either one way or the other.

The Exclusion Principle

The electron in a hydrogen atom has a strong tendency to fall into its lowest energy state.

So far, we have limited our discussion of atomic electrons to the single electron in a

hydrogen atom. What about all the other atoms that make up the chemical elements heavier

than hydrogen. They have more than one electron. (Uranium, in fact, has 92!) Where do

these many electrons fit? Do they all crowd into the same n = 1 probability cloud associated

with the hydrogen atom's ground state? Or do they somehow "stack up" in some sort of

organized pattern? The chemical behavior of the various elements, which is widely divergent

in most instances, suggests that electrons arrange themselves differently in the atoms

belonging to different elements.

In 1925 Wolfgang Pauli found the key to electron arrangements in multielectron atoms. His

exclusion principle says that no two electrons in the same atom can have the same set of four

quantum numbers. This means that there is a unique set of attributes for every electron in a

given atom; no two electrons in a given atom can behave in exactly the same way. So how do

the electrons arrange themselves? Just remember that each of the electrons in an undisturbed

(unexcited) atom strives always toward a state of lower energy.

Summary

Light and matter exhibit both wavelike properties and particle like properties. This behavior

is called wave-particle duality.

A particle of matter has a wavelength that increases with decreasing momentum (mass and

speed). Because the wavelength of a particle also depends on Planck's constant h, which has

an extremely small value, only particles with very small momenta exhibit measurable

wavelike properties. In the atomic realm, the wavelike properties of particles are significant.

The wavelike properties of tiny particles can be tested by subjecting them to the same

experiments that have been used to confirm the wave nature of light. Particles and photons

behave similarly when their wavelengths are the same.

It is impossible to know both the exact position and the exact momentum of a particle at the

same time. The relative uncertainty involved is large only for small particles.

Because uncertainty is so prevalent in the realm of the atom, atomic electrons are best

described as behaving like standing waves positioned around the nucleus. Each electron in an

10

atom can be completely described by a set of four quantum numbers. Furthermore, each

electron in an atom differs from all the others in the sense that no two electrons in the same

atom can have exactly the same set of four quantum numbers.