Modern Phys ics

<< RETURN TO THE PREVIOUS SECTION | CONTINUE TO THE NEXT SECTION >>

19.1 SPECIAL RELATIVITY

19.2 THE DISCOVERY OF THE ATOM

19.3 QUANTUM PHYSICS

19.4 NUCLEAR PHYSICS

19.5 KEY FORMULAS 19.6 PRACTICE QUESTIONS 19.7 EXPLANATIONS

Quantum PhysicsAs physicists began to probe the mysteries of the atom, they came across a number of

unexpected results along the lines of Rutherford’s gold foil experiment. Increasingly,

it became clear that things at the atomic level are totally unlike anything we find on

the level of everyday objects. Physicists had to develop a whole new set of

mechanical equations, called “quantum mechanics,” to explain the movement of

elementary particles. The physics of this “quantum” world demands that we upset

many basic assumptions—that light travels in waves, that observation has no effect

on experiments, etc.—but the results, from transistor radios to microchips, are

undeniable. Quantum physics is strange, but it works.

ElectronvoltsBefore we dive into quantum physics, we should define the unit of energy we’ll be

using in our discussion. Because the amounts of energy involved at the atomic level

are so small, it’s problematic to talk in terms of joules. Instead, we use the

electronvolt (eV), where 1 eV is the amount of energy involved in accelerating an

electron through a potential difference of one volt. Mathematically,

The Photoelectric EffectElectromagnetic radiation transmits energy, so when visible light, ultraviolet light, X

rays, or any other form of electromagnetic radiation shines on a piece of metal, the

surface of that metal absorbs some of the radiated energy. Some of the electrons in

the atoms at the surface of the metal may absorb enough energy to liberate them

from their orbits, and they will fly off. These electrons are called photoelectrons, and

this phenomenon, first noticed in 1887, is called the photoelectric effect.

The Wave Theory of Electromagnetic RadiationYoung’s double-slit experiment, which we looked at in the previous chapter, would

seem to prove conclusively that electromagnetic radiation travels in waves.

However, the wave theory of electromagnetic radiation makes a number of

predictions about the photoelectric effect that prove to be false:

Predict ions of the wave theory Observed result

Time

lap se

Elect rons need to absorb a certain

amount of wave energy before they

can be liberated, so there should be

Elect rons begin fly ing off the surface of

the metal almost instant ly after light

shines on it .

TEST PR EP C EN TER S

SAT Phy sics Test Center

SPAR K C OL L EGE

College Admissions

Financial Aid

College Lif e

Help Log in Sign Up for a Free Account

some lap se of t ime between the light

hit t ing the surface of the metal and

the first elect rons fly ing off.

Intensity The intensity of the beam of light

should determine the kinet ic energy

of the elect rons that fly off the

surface of the metal. The greater the

intensity of light , the greater the

energy of the elect rons.

The intensity of the beam of light has no

effect on the kinet ic energy of the

elect rons. The greater the intensity , the

greater the number of elect rons that fly

off, but even a very intense low-

frequency beam liberates no elect rons.

Frequency The frequency of the beam of light

should have no effect on the number

or energy of the elect rons that are

liberated.

Frequency is key : the kinet ic energy of

the liberated elect rons is direct ly

p rop ort ional to the frequency of the light

beam, and no elect rons are liberated if the

frequency is below a certain threshold.

M aterial The material the light shines up on

should not release more or fewer

elect rons dep ending on the frequency

of the light .

Each material has a certain threshold

frequency : light with a lower frequency

will release no elect rons.

Einstein Saves the DayThe young Albert Einstein accounted for these discrepancies between the wave

theory and observed results by suggesting that electromagnetic radiation exhibits a

number of particle properties. It was his work with the photoelectric effect, and not

his work on relativity, that won him his Nobel Prize in 1921.

Rather than assuming that light travels as a continuous wave, Einstein drew on

Planck’s work, suggesting that light travels in small bundles, called photons, and that

each photon has a certain amount of energy associated with it, called a quantum.

Planck’s formula determines the amount of energy in a given quantum:

where h is a very small number, J · s to be precise, called Planck’s

constant, and f is the frequency of the beam of light.

Work Function and Threshold FrequencyAs the wave theory correctly assumes, an electron needs to absorb a certain amount

of energy before it can fly off the sheet of metal. That this energy arrives all at once,

as a photon, rather than gradually, as a wave, explains why there is no time lapse

between the shining of the light and the liberation of electrons.

We say that every material has a given work function, , which tells us how much

energy an electron must absorb to be liberated. For a beam of light to liberate

electrons, the photons in the beam of light must have a higher energy than the work

function of the material. Because the energy of a photon depends on its frequency,

low-frequency light will not be able to liberate electrons. A liberated photoelectron

flies off the surface of the metal with a kinetic energy of:

EXAMPLE

Two beams of light , one blue and one red, shine up on a metal with a work funct ion of5.0 eV. The frequency of the blue light is Hz, and the frequency of the red

light is Hz. What is the energy of the elect rons liberated by the two beams

of light?

In order to solve this problem, we should translate h from units of J · s into units of

eV · s:

We know the frequencies of the beams of light, the work function of the metal, and

the value of Planck’s constant, h. Let’s see how much energy the electrons liberated

by the blue light have:

For the electrons struck by the red light:

The negative value in the sum means that , so the frequency of the red light is

too low to liberate electrons. Only electrons struck by the blue light are liberated.

The Bohr Model of the AtomLet’s now return to our discussion of the atom. In 1913, the Danish physicist Niels

Bohr proposed a model of the atom that married Planck’s and Einstein’s development

of quantum theory with Rutherford’s discovery of the atomic nucleus, thereby

bringing quantum physics permanently into the mainstream of the physical sciences.

The Problem with Rutherford’s ModelLight and other electromagnetic waves are emitted by accelerating charged particles.

In particular, the electrons being accelerated in orbit about the nucleus of an atom

release a certain amount of energy in the form of electromagnetic radiation. If we

recall the chapter on gravity, the radius of an object in orbit is a function of its

potential energy. If an electron gives off energy, then its potential energy, and hence

the radius of its orbit about the nucleus, should decrease. But according to

Rutherford’s model, any radiating electron would give off all its potential energy in a

fraction of a second, and the electron would collide with the nucleus. The fact that

most of the atoms in the universe have not yet collapsed suggests a fundamental flaw

in Rutherford’s model of electrons orbiting nuclei.

The Mystery of Atomic SpectraAnother puzzling phenomenon unexplained by Rutherford’s model, or anything else

before 1913, is the spectral lines we see when looking through a spectroscope. A

spectroscope breaks up the visible light emitted from a light source into a spectrum,

so that we can see exactly which frequencies of light are being emitted.

The puzzling thing about atomic spectra is that light seems to travel only in certain

distinct frequencies. For instance, we might expect the white light of the sun to

transmit light in an even range of all different frequencies. In fact, however, most

sunlight travels in a handful of particular frequencies, while very little or no light at

all travels at many other frequencies.

Bohr’s Hydrogen AtomNiels Bohr drew on Rutherford’s discovery of the nucleus and Einstein’s suggestion

that energy travels only in distinct quanta to develop an atomic theory that accounts

for why electrons do not collapse into nuclei and why there are only particular

frequencies for visible light.

Bohr’s model was based on the hydrogen atom, since, with just one proton and one

electron, it makes for the simplest model. As it turns out, Bohr’s model is still mostly

accurate for the hydrogen atom, but it doesn’t account for some of the complexities

of more massive atoms.

According to Bohr, the electron of a hydrogen atom can only orbit the proton at

certain distinct radii. The closest orbital radius is called the electron’s ground state.

When an electron absorbs a certain amount of energy, it will jump to a greater orbital

radius. After a while, it will drop spontaneously back down to its ground state, or

some other lesser radius, giving off a photon as it does so.

Because the electron can only make certain jumps in its energy level, it can only emit

photons of certain frequencies. Because it makes these jumps, and does not emit a

steady flow of energy, the electron will never spiral into the proton, as Rutherford’s

model suggests.

Also, because an atom can only emit photons of certain frequencies, a spectroscopic

image of the light emanating from a particular element will only carry the frequencies

of photon that element can emit. For instance, the sun is mostly made of hydrogen, so

most of the light we see coming from the sun is in one of the allowed frequencies for

energy jumps in hydrogen atoms.

Analogies with the Planetary ModelBecause the electron of a hydrogen atom orbits the proton, there are some analogies

between the nature of this orbit and the nature of planetary orbits. The first is that

the centripetal force in both cases is . That means that the centripetal

force on the electron is directly proportional to its mass and to the square of its

orbital velocity and is inversely proportional to the radius of its orbit.

The second is that this centripetal force is related to the electric force in the same

way that the centripetal force on planets is related to the gravitational force:

where e is the electric charge of the electron, and Ze is the electric charge of the

nucleus. Z is a variable for the number of protons in the nucleus, so in the hydrogen

atom, Z = 1.

The third analogy is that of potential energy. If we recall, the gravitational potential

energy of a body in orbit is . Analogously, the potential energy of an

electron in orbit is:

Differences from the Planetary ModelHowever, the planetary model places no restriction on the radius at which planets

may orbit the sun. One of Bohr’s fundamental insights was that the angular

momentum of the electron, L, must be an integer multiple of . The constant

is so common in quantum physics that it has its own symbol, . If we take n to

be an integer, we get:

Consequently, . By equating the formula for centripetal force and the

formula for electric force, we can now solve for r:

Don’t worry: you don’t need to memorize this equation. What’s worth noting for the

purposes of SAT II Physics is that there are certain constant values for r, for

different integer values of n. Note also that r is proportional to , so that each

successive radius is farther from the nucleus than the one before.

Electron Potential EnergyThe importance of the complicated equation above for the radius of an orbiting

electron is that, when we know the radius of an electron, we can calculate its potential

energy. Remember that the potential energy of an electron is . If

you plug in the above values for r, you’ll find that the energy of an electron in a

hydrogen atom at its ground state (where n = 1 and Z = 1) is –13.6 eV. This is a

negative number because we’re dealing with potential energy: this is the amount of

energy it would take to free the electron from its orbit.

When the electron jumps from its ground state to a higher energy level, it jumps by

multiples of n. The potential energy of an electron in a hydrogen atom for any value

of n is:

Frequency and Wavelength of Emitted PhotonsAs we said earlier, an excited hydrogen atom emits photons when the electron jumps

to a lower energy state. For instance, a photon at n = 2 returning to the ground state

of n = 1 will emit a photon with energy . Using

Planck’s formula, which relates energy and frequency, we can determine the

frequency of the emitted photon:

Knowing the frequency means we can also determine the wavelength:

As it turns out, this photon is of slightly higher frequency than the spectrum of visible

light: we won’t see it, but it will come across to us as ultraviolet radiation. Whenever

an electron in a hydrogen atom returns from an excited energy state to its ground

state it lets off an ultraviolet photon.

EXAMPLE

A hy drogen atom is energized so that it s elect ron is excited to the n = 3 energy state.How many different frequencies of elect romagnet ic radiat ion could it emit in returningto it s ground state?

Electromagnetic radiation is emitted whenever an electron drops to a lower energy

state, and the frequency of that radiation depends on the amount of energy the

electron emits while dropping to this lower energy state. An electron in the n = 3

energy state can either drop to n = 2 or drop immediately to n = 1. If it drops to n = 2,

it can then drop once more to n = 1. There is a different amount of energy associated

with the drop from n = 3 to n = 2, the drop from n = 3 to n = 1, and the drop from n = 2

to n = 1, so there is a different frequency of radiation emitted with each drop.

Therefore, there are three different possible frequencies at which this hydrogen atom

can emit electromagnetic radiation.

Wave-Particle DualityThe photoelectric effect shows that electromagnetic waves exhibit particle

properties when they are absorbed or emitted as photons. In 1923, a French graduate

student named Louis de Broglie (pronounced “duh BRO-lee”) suggested that the

converse is also true: particles can exhibit wave properties. The formula for the so-

called de Broglie wavelength applies to all matter, whether an electron or a planet:

De Broglie’s hypothesis is an odd one, to say the least. What on earth is a wavelength

when associated with matter? How can we possibly talk about planets or humans

having a wavelength? The second question, at least, can be easily answered. Imagine

a person of mass 60 kg, running at a speed of 5 m/s. That person’s de Broglie

wavelength would be:

We cannot detect any “wavelength” associated with human beings because this

wavelength has such an infinitesimally small value. Because h is so small, only objects

with a very small mass will have a de Broglie wavelength that is at all noticeable.

De Broglie Wavelength and ElectronsThe de Broglie wavelength is more evident on the atomic level. If we recall, the

angular momentum of an electron is . According to de Broglie’s

formula, mv = h/ . Therefore,

The de Broglie wavelength of an electron is an integer multiple of , which is the

length of a single orbit. In other words, an electron can only orbit the nucleus at a

radius where it will complete a whole number of wavelengths. The electron in the

figure below completes four cycles in its orbit around the nucleus, and so represents

an electron in the n = 4 energy state.

The de Broglie wavelength, then, serves to explain why electrons can orbit the

nucleus only at certain radii.

EXAMPLE

Which of the following exp lains why no one has ever managed to observe and measurea de Broglie wavelength of the Earth?

(A) The Earth is t raveling too slowly . It would only have an observable de Brogliewavelength if it were moving at near light sp eed.

(B) The Earth is too massive. Only objects of very small mass have not iceablewavelengths.

(C) The Earth has no de Broglie wavelength. Only objects on the atomic level havewavelengths associated with them.

(D) “Wavelength” is only a theoret ical term in reference to mat ter. There is noobservable effect associated with wavelength.

(E) The individual atoms that const itute the Earth all have different wavelengths thatdest ruct ively interfere and cancel each other out . As a result , the net wavelengthof the Earth is zero.

This is the sort of question you’re most likely to find regarding quantum physics on

SAT II Physics: the test writers want to make sure you understand the theoretical

principles that underlie the difficult concepts in this area. The answer to this question

is B. As we discussed above, the wavelength of an object is given by the formula =

h/mv. Since h is such a small number, mv must also be very small if an object is going to

have a noticeable wavelength. Contrary to A, the object must be moving relatively

slowly, and must have a very small mass. The Earth weighs kg, which is

anything but a small mass. In fact, the de Broglie wavelength for the Earth is

m, which is about as small a value as you will find in this book.

Heisenberg’s Uncertainty PrincipleIn 1927, a young physicist named Werner Heisenberg proposed a counterintuitive

and startling theory: the more precisely we measure the position of a particle, the less

precisely we can measure the momentum of that particle. This principle can be

expressed mathematically as:

where is the uncertainty in a particle’s position and is the uncertainty in its

momentum.

According to the uncertainty principle, if you know exactly where a particle is, you

have no idea how fast it is moving, and if you know exactly how fast it is moving, you

have no idea where it is. This principle has profound effects on the way we can think

about the world. It casts a shadow of doubt on many long-held assumptions: that

every cause has a clearly defined effect, that observation has no influence upon

experimental results, and so on. For SAT II Physics, however, you needn’t be aware

of the philosophical conundrum Heisenberg posed—you just need to know the name

of the principle, its meaning, and the formula associated with it.

<< RETURN TO THE PREVIOUS SECTION | CONTINUE TO THE NEXT SECTION >>

Help | Feedback | Make a request | Report an error

Modern Phys ics

When your books and teachers don’t make sense, we do.

Contact Us | Privacy Policy | Terms and Conditions | About | Sitemap | Advertise

Fiction Books | Textbooks | eTextbooks | Classic Books | Used Books | Teen Books | nook | eReader

©2011 SparkNotes LLC, All Rights Reserved