## PHY 1020: Homework set 13

If two fair coin flips come up heads in a succession, what is the % chance of receiving two heads in a row? Is there a chance of obtaining five heads in a row? How would you go about putting your prediction about receiving five heads in a row to the test in an experimental setting? Answer: To make the question more obvious and less unclear, it should really be phrased somewhat differently, as follows: While two coins are tossed in a fair coin toss, what is the likelihood of receiving two heads when throwing both coins?

(This is exactly what the book’s author intended when he wrote it.) The solution is as follows: The chance of getting a head with one unbiased (“fair”) coin is = 1/2 = 50 percent when using one unbiased (“fair”) coin.

This is because the coins are independent, hence the probabilities compound.

You should expect to see around three all-heads outcomes for every hundred tosses of five coins.

QUESTIONS FOR REVIEW 1.Answer the review questions 1, 2, 3, 5, 7, 10, 11, 12, 13, 15, 19, 25, 29, and 30 on pages 578-579 of Hewitt. 2.Answer the review questions 1, 2, 3, 5, 7 on pages 578-579 of Hewitt. On page 593 of Hewitt, answer the review questions 7 and 9 through 15, as well as 17 and 20. Exercise 1. The photoelectric effect is the most effective demonstration ofa.the wave nature of light, b.the particle nature of light, or c.both of these characteristics. d.none of the preceding. (2) The more intense the illuminating light on the surface, the greatera.the number of electrons released from the surface in the photoelectric effect A.the ejection velocity of electrons; B.both of these; C.neither of these; d.neither of the preceding options.

- When electrons are used in the double slit experiment, the pattern shown on the screen can be described asa.wave-like, b.particle-like, or c.neither of these.
- a baseball, a spitball, or an electron.
- Moreover, the uncertainty principle holds true not only for momentum and position, but also for energy and time.
- 7 According to the uncertainty principle, the greater the amount of information we have about a particle’s momentum, the less information we have about its kinetic energy.
- c.velocity and d.location e.none of the options.
- In order to produce the photon, there must be a difference in atomic energy states.
- d.neither of the preceding options.

b.energy; c.a combination of the two.

10.The speed of an electron is the same as the speed of a baseball.

1.The electron 2.The baseball 3.The baseball 4.The baseball 5.

The longer wavelength of a proton is due to the fact that its momenta are similar to those of its electron counterpart.

c.they are both the same.

According to the uncertainty principle, the more the amount of information we have about a particle’s position, the less information we have about the particle’s velocity.

c.kinetic energy; d.all of the above-mentioned.

The measurement of the velocity of a tiny particle with an electromagneta., according to quantum physics, has an effect on the velocity of the tiny particle.

The photoelectric effect is characterized by electrons being expelled from bound states in the photosensitive material having a lower kinetic energy than the photon’s energy being absorbed.

a.light b.sound c.electrons d.electricity D.all of them e.none of these are acceptable.

hasa.wave characteristics b.particle characteristics c.both of the foregoing.

In spite of the fact that the nuclei of heavy atoms have greater mass than light atoms, they are not noticeably bigger in size than light atoms.

c.nucleons.

e.none of the options.

Theories of macroscopic particles are discussed in section b.

a new theory adheres to the correspondence principle if and only if ita.corresponds to all of the theories that exist in nature b.improves on the substance of the previous hypothesis.

d.takes into consideration the outcomes of the old theory that have been validated.

21, the quantum mechanical probability cloud for the electron in the hydrogen atom has an average radiusa.that is significantly different from the radius anticipated by Bohr.

Twenty-seconds after being stimulated, an atom returns to its ground state and produces a photon of green light.

b.violet.

d.any of the following.

a.The frequency of the matter wave associated with a particle determines the likelihood of detecting the particle at a particular position in space; b.

c.neither of the preceding options.

In accordance with the Uncertainty Principle,a.everything is fully unpredictable.

In the case of tiny particles, the location and momentum are not known.

25.Complementary qualities are in close proximity to one another.

c.describe the many characteristics of an object.

b.Photons possessed particle-like characteristics.

d.none of the preceding.

Nodes in the de Broglie waves region (number 27).

b.positions where there is a zero percent chance of detecting particles connected with the de Broglie wave.

e.every one of these.

What evidence does this experiment provide that light is composed of waves?

What does it reveal about the nature of physical reality, and how can we interpret it? 4.How does the classical mechanical view of physical reality vary from the quantum mechanical picture in terms of its underlying assumptions?

## Modern Physics Exam 2 Flashcards

De Broglie waves:(1) are a form of electromagnetic radiation.(2) are a basic property of all particles, whether at rest or in motion.(3) travel at the speed of light.(4) describe the wave-type behavior of moving particles. | ||

The deBroglie wave of a particle can best be described as:(1) a form of electromagnetic wave.(2) a characteristic of the oscillation of the particle.(3) a probability wave.(4) none of the above. | ||

Which of the following is NOT true about the deBroglie wavelength?(1) It is larger for an electron than for a baseball moving at the same speed.(2) It applies only to charged particles.(3) It describes the wave properties of particles such as electrons.(4) It is a property of waves of probability. | ||

The uncertainty relationships:(1) apply to all types of waves.(2) apply only to de Broglie waves.(3) apply only to classical waves.(4) apply only to light waves. | ||

In the following situations, choose which particle has the larger de Broglie wavelength:(1) the electron(2) the proton(3) they are both the same(a) An electron and a proton moving with the same momentum.(b) An electron and a proton moving at the same speed.(c) An electron and a proton with the same nonrelativistic kinetic energy.(d) An electron and a proton with the same kinetic energy, in both cases much larger than the rest energy | ||

Why is it not possible to observe double-slit interference with baseballs?(1) The de Broglie wavelength of a baseball is too large.(2) The de Broglie wavelength of a baseball is too small.(3) Baseballs are too large to fit through a double-slit apparatus.(4) Baseballs can’t be accelerated to the speed of light. | ||

A beam of electrons moving with speed v passes through a single slit and strikes a screen, where it forms a diffraction pattern with a bright central maximum and some less intense maxima on either side of center.(a) If the speed of the electrons is increased to 2v, what happens to the width of the central maximum?(1) Increases (2) Decreases (3) Remains the same(b) If the beam of electrons is replaced with a beam of protons moving with speed v, what happens to the width of the central maximum compared with that of electrons moving with the same speed?(1) Increases (2) Decreases (3) Remains the same | ||

Suppose an electron is moving at speed v. In terms of v, what would be the speed of abaseball with the same deBroglie wavelength as the electron? (1) v(2) 10^10v(3) 10^−10v (4) 10^−20v(5) 10^−30v | ||

A beam of monoenergetic electrons is incident on a mask that contains a single narrow slit. A pattern of diffraction maxima and minima appears on the screen.(a) If the slit width is halved, the diffraction minima on the screen would then be:(1) closer together (2) farther apart (3) unchanged(b) If the kinetic energy of the electrons in the original experiment is halved, the diffraction minima on the screen would be:(1) closer together (2) farther apart (3) unchanged(c) Suppose the beam of electrons were replaced with a beam of particles of greater mass, such that the resultant diffraction pattern was exactly the same as that in the original experiment. To accomplish this, the kinetic energy of the new particles would be:(1) greater than that of the original electrons (2) less than that of the original electrons (3) equal to that of the original electrons | ||

(a) A packet of water waves of width Δ x contains a range of wavelengths Δλ about a central wavelength λ; that is, the range of wavelengths is from about λ – Δλ/2 to λ + Δλ/2. If the packet were made half as wide, what would be the new range of wavelengths?(1) 2Δλ (2) Δλ/2 (3) Δλ (4) None of these(b) A whistle blast lasts for a time interval Δt. It consists of a central frequency ν with a range Δν. If the blast were made twice as long, what would be the new range of frequencies? (1) 2Δν (2) Δν/2 (3) Δν (4) None of these(c) A beam of electrons of momentump x moving in thex direction passes through a slit of width Δ y = a. The beam diffracts through the slit so that the range in itsy momentum is Δ p y, that is, from -Δ p y/2 to + Δ p y/2. What is the new range in they momentum if the slit is made half as wide?(1) 2Δ p y (2) Δ p y/2 (3) Δ p y (4) None of these | ||

Consider the following three experiments:(a) Thex component of the position of an electron is measured to within ±Δ x, and simultaneously thex component of its momentum is measured to within ±Δ p x.(b) Thex component of the position of an electron is measured to within ±Δ x, and then later thex component of its momentum is measured to within ±Δ p x.(c) Thex component of the position of an electron is measured to within ±Δ x, and simultaneously they component of its momentum is measured to within ±Δ p y. In which of these cases does the uncertainty principle NOT impose a limitation on the outcome of the experiment?(1) a only (2) b only (3) c only (4) a and b only (5) a and c only (6) b and c only (7) a, b, and c | ||

A sodium atom, a neutron, a proton, and an electron all have the same nonrelativistic kinetic energy. Which has the smallest de Broglie wavelength? (a) sodium atom (b) neutron (c) proton (d) electron | ||

Which of the following does NOT provide evidence for the wave nature of matter? (a) the photoelectric effect (b) neutron diffraction (c) the Heisenberg relationships (d) electron diffraction | ||

The probability density for a particle in the ground state of a one-dimensional infinite potential energy well:(1) has a single maximum at the center of the well.(2) has a minimum at the center of the well and maxima at the sides of the well.(3) has several maxima and minima in the well.(4) is constant throughout the well. | ||

In the one-dimensional infinite well, how does the energy spacing between the excited states change as the energy of the states increases?(1) The spacing is constant.(2) The spacing decreases.(3) The spacing increases.(4) The spacing changes randomly. | ||

A beam of particles is incident from the negative x axis onto a positive potential energy step located at x = 0.The kinetic energy of the particles is less than the potential energy of the step.Which is the best description of the behavior of the particles?(1) All particles are reflected precisely at x = 0.(2) Some particles are reflected at the step and some are transmitted into the x0 region.(3) Some particles are reflected and some are absorbed.(4) All particles are reflected, but they can penetrate a short distance into the x0 region.(5) All particles are absorbed at the step. | ||

The probability to find a particle at any specific location in space:(1)is directly proportional to the amplitude of the wave function.(2)can never be zero.(3)depends on the squared amplitude of the wave function.(4)can sometimes be infinite. | ||

The Schrödinger equation is(a) a second-order differential equation.(b) an equation based on conservation of energy. (c) an equation whose solution gives the wave function that describes a particle.How many of the above statements are true?(1) Zero (2) One (3) Two (4) All three | ||

Which of the following is NOT a characteristic of the Bohr model of the structure of the atom?(1) Electrons move in circular orbits about the nucleus.(2) Photons are emitted when an electron jumps from one circular orbit to alower-energy orbit.(3) The circular motion of the electron is consistent with the uncertainty principle.(4) The angular momentum of each orbit can take only values that are integermultiples of the smallest value. | ||

In the Bohr model:(1) Electrons move in circular orbits of definite radius.(2) Electrons move in elliptical orbits.(3) Electrons moving in the same orbit can have different energies.(4) Electrons can never jump from one orbit to another. | ||

Which of the following is NOT used in the Bohr model of the atom?(1) Quantization of energy.(2) Relativistic energy and momentum.(3) Coulomb’s law for electrostatic forces.(4) Quantization of angular momentum. | ||

In a Rutherford scattering experiment:(1) most of the particles are not scattered at all or scattered only at small angles,but a few are scattered at large angles. (2) the experimental results verify that the positive charge of the atom is spread throughout the volume of the atom. (3) most particles are scattered only once, but the ones that are scattered at large angles are scattered many times. (4) scattering by the negatively charged electrons can cancel scattering by the positively charged nucleus. |

## Chapter 23 Material

Chapter 23: MULTIPLE OPTIONS FOR MATERIAL The following technical terminology cannot both describe an electron and describe a photon at the same time: 1. the wavelength * the massc. energyd. momentum of the wave The reason why your sports vehicle does not deflect off the road when you drive through a tunnel is thata. wave characteristics can only be observed in the case of atomic-scale particles. b. the waves of the particles in the automobile interact with one another in a damaging manner. • c.

- It is very, very little, and the breadth of the diffraction pattern is very, very small.
- a.
- the British Broadcasting Corporation They both operate on the same frequency.
- There is no specification for the wavelength.
- just like bullets; orb.
- When they are identified, they appear to be bullets, but the pattern is wave-like.
- 5.

a.

two humps showing no interference * c.

the sum of the patterns formed when each slit is opened by itself6.

in the centerb.

any placed.

the sum of the patterns formed when each slit is open 7.

A.

energy C.

wavelength 8.

act like waves and do not have well-defined orbits, the quantum-mechanical model of the atom does not have the problem of accelerating charges generating electromagnetic radiation.

are not subject to a fee.

have a fixed position.

have pathways where there is no acceleration Nine.

a.

2 * c.

410; a.

2 * c.

410; a.

2 * c.

410; a.

2 * c.

410; Certain families of elements in the periodic table exhibit characteristics that are strikingly similar to one another.

A.

B.

C.

D.

11.

2, 7, 8, 2 * b.

2, 6, 9d.

7, 8, 2 * With n = 4, how many different quantum states are there?

16b.

24 * d.

a.

18c.

1.

momentum in the same direction 3.

d.

Because it was stated by a well-known physicist, scientists believe in the complementarity principle.

As a result, western scientific thought is heading in the direction of eastern religious philosophy.

* It is necessary to understand the behavior of electrons and photons to have both particle and wave descriptions.

What is the de Broglie wavelength of a sprinter (mass = 60 kg) who runs at a speed of 6.63 meters per second?

When an electron (m = 9.11 x 10-31 kg) travels at the speed of light, it has a wavelength equal to the separation between crystals (10-10 meter).

For n=3, how many different electron states are there?

States with l = 1, m = -1,0,+1, and x 2=6 are possible.

l = 2, m = -2,-1,0,+1,+2 x 2 =10 states with l = 2 and m = -2,-1,0,+1,+2 x 2 =10 states with l = 2 and m = -2,-1,0,+1,+2 x 2 =10 states with l = 2 and m = -2,-1,0,+1,+2 x 2 =10 states with l = 2 and There are a total of 18 states. R.S. Panvini (February 23, 2000)

## Physics and Human Affairs Test 3 Flashcards

What is the Quantum Theory, and how does it work? the concept that, rather of energy growing evenly, it grows in little increments throughout time What was the subject of the Photoelectric Effect study, and who was the author? Einstein’s study on the photoelectric effect was written in 1905, and it was primarily concerned with how solar cells functioned. The concept that light is composed of little energy lumps, which we now refer to as photons, was originally proposed. In 1900, Planck discovered a formula.

- The Energy Formula is as follows: Energy equals h multiplied by frequency What does the letter “h” stand for in Planck’s equation?
- Which of the following did Einstein have to say about photons of light?
- This then provided us with the characteristics of light, or “particle,” which were derived from the characteristics of light, or “wave.” Both in terms of a particle and in terms of a wave.
- The only time that light behaves like a particle is when it is not transmitting its energy to something (such as when it passes through glass, orifices, or holes).
- It “hits” as a particle, which means that the individual dots are made up of photons rather than waves.
- It has been suggested that, in the same way that radiation (light, etch.) acts at times like particles, matter should behave at times like waves.
- Wavelength may be calculated using the de Broglie formula, which is The formula for wavelength is wavelength=h/mass x speed.

P.

What characteristics does each and every material particle possess?

What does the de Brogile wavelength formula teach us about the universe?

Because electrons have such a small mass, it is much easier to observe this phenomenon.

Who was Erwin Schroedinger, and what was his significance?

The Schroedinger Equation is one of the most important in quantum physics, and it can be expressed in several ways.

In some ways, they behave like particles, but in others, they behave more like waves, and vice versa.

When it comes to electrons and light, what are the differences?

Radiation is what light is.

Assume that two electrons are traveling in opposite directions through a double-slit apparatus.

To say that we know the first electron came through slit A, and the second electron came through slit B, would be correct, but it is not.

What happens in the double slit experiment with electrons is predictable, but why?

We can predict the locations on the screen where no electrons are reflected.

Would you expect to see the pattern of light and dark lines on the screen if electrons only behaved like particles (with no wave aspect)?

The interference pattern is caused by the electron’s wave nature, which can be seen in the picture above.

If both a proton and a baseball are moving at the same speed, which has the longer wavelength?

The Proton has a smaller mass than the neutron.

A proton or an electron has a longer wavelength, so which has the longer wavelength?

A radio photon or an ultraviolet photon has more energy, so which is more powerful?

– The higher the frequency of the wave, the greater the energy of the photon.

What is meant by a red photon, and how does it work?

That is, a photon that corresponds to an electromagnetic wave with a frequency of approximately 4 x 10/14 Hz.

Is the color of the light changing as the frequency is increased?

Each individual color that we see has a unique frequency associated with it.

Yes, because energy is a function of its frequency.

Is there a difference in the speed of the photons as the frequency increases?

Photons always travel at the speed of light (c) regardless of their surroundings.

What is it about Einstein that is so important?

What exactly is the significance of Planck?

What exactly is the significance of de Broglie?

What exactly is the significance of Born?

What is it about Schroedinger that is so important?

when the colors blend seamlessly into one another with no breaks or dark lines in between) when it only emits thin lines of specific colors with dark areas in between) What is the number of odes each element has?

## Final Examination Solutions

Physicists 262-005-08 (December 2000) SOLUTIONS FOR THE FINAL EXAMINATION 1.Consider the hydrogen atom in its most fundamental condition. In this orbit, the de Broglie wavelength is equal toA)B)C)D)E)The ground state comprises ONE de Broglie wave in the form of a standing wave, as seen in A). Consequently, the wavelength is equal to one hundredth of the circumference of the orbit, or The correct response is option A. Secondly, the units of Planck’s constant are A) the same as the units of momentum B) the same as the word force C) The same as the unit of energy EJoules.

- In this case, the right response isD.
- The distribution pattern of hits that pass through the slits and into a detector behind the slits is equal to the distribution pattern of interference.
- Compare and contrast the outcomes of the two experiments.
- B) The first demonstrates interference, but the second demonstrates just diffraction.
- D) In both cases, there is no discernible pattern at all.
- The interference pattern produced by both slits is precisely the same as if we were dealing with light waves (or photons!).
- It makes no difference that the total amount of time that the slits remain open is the same for all of them.

The right response is B.4.An electron and a proton have the same amount of kinetic energy as each other.

A) They both have the same de Broglie wavelengths, which is correct.

C) The electron has a longer de Broglie wavelength than the positron.

We can use this to go from KE to momentum.

However, because high momentum implies a short wavelength, Consequently, it is true that the electron, because of its lower momentum, has a longer wavelength.

Is it correct to say which of the following statements?

B) The proton has a longer de Broglie wavelength than the neutron.

D) There is insufficient information to make a determination.

The right response is A.6.An electron and a proton both move at the same speed as a neutron.

A) They both have the same de Broglie wavelengths, which is correct.

C) The electron has a longer de Broglie wavelength than the positron.

When traveling at equal speeds, the proton has the greater momentum and, as a result, the shorter wavelength than the neutron.

Compton scattering occurs when two photons, one of x-ray wavelengthpm and one of visible light wavelengthnm, collide with an electron in a substance at a certain scattering angle.

As a result of the equal wavelength shift experienced by both photons, they both give the same amount of energy to the electron.

C) The two photons have the same wavelength shift, with the visible photon imparting more energy to the electron than the infrared photon.

E) There is insufficient information provided.

However, the energy and momentum of the x-ray photon are far more than those of the visible light photon, and as a result, the recoil momentum of the impacted electron will be bigger in the x-ray collision than it will be in the visible light collision.

Rank the following radiations in order of the photon energies associated with them, with the highest photon energy coming first.

(2) In a microwave oven, photons from microwaves are produced.

Fourteenth, radio waves from the KUNM station at 89.95 MHz A) I, II, III, IVB) IV, III, II, IC) III, II, I, IVD) III, IV, I, IIE) III, I, II, IVD) III, I, II, IVD) III, I, II, IVD) III, I, II, IVD) III, I, II, IVD) III, I, II, IVD) III, I, II, IVD) III, I, II, IVD) III, I This shouldn’t be too difficult.

- As a result, the gamma ray would come first, followed by yellow light, microwaves, and radio waves.
- Rank the states in order of the energy disparities that exist between them, with the most energy differences appearing first.
- As a result, for have the following sequence of integers in the order specified.
- a) 8b) 9c) 10d) 11e) 12) The number of nodes (zeroes) in its waves function is a) 8b) 9c) 10 a) 9c) 10d) 11e) 12 There are n+1 nodes (zeroes) in an infinite square well, which means there are 2 nodes for n=1, 3 nodes for n=2, etc., in an infinite square well.

The number of maximums in its probability distribution is A) 8B) 9C) 10D) 11E) 12 while the number of minimums is A) 8B) 9C) 10D) 11E) 12 We square the wave function in order to achieve maximum probability distributions for all exterema of the wavefunction (i.e., all maximum or minimum points).

- In this case, the right answer is C.12.A collection of quantum states involving the hydrogen-atom hasn=5.
- The highest permissible value of the angular momentumLaren-1 is a positive integer.
- The correct response is E.13.
- 7B) 6C) 5D) 4E) 3 is the number of possible directions in which the angular momentum vector might point.
- Because ZERO is an allowable value, we have a total of SEVEN possible possibilities.
- Forl=3 is the maximum number of allowed values (directions).
- It is A.14 that is the right answer.

According to this definition, which of the following claims regarding the ground state is correct?

B) There is just one node and one maximum in the ground state wave function.

D) Neither nodes nor maxima can be found in the ground state wave function.

In a finite well, the ground state wavefunction does not reach zero until we reach the end of the well.

There is, of course, a maximum, which is located in the middle of the well.

Thanks for your help.

According to this definition, which of the following claims regarding the ground state is correct?

B) The probability function is smaller as one moves away from the well.

Outside of the well, the wave function is devoid of nodes; in addition, as one moves away from the well, the probability function becomes increasingly constant.

E) More information is required in order to explain the story.

As we get further away from the well, the likelihood of finding it diminishes significantly.

There are two times as many photons in the universe as there are photons in the universe.

The relationship between photon’s energy and momenta is represented by Consequently, the ratio of momenta equals the ratio of energies in a system.

There are four different velocities: A), B), C), D), and E).

All photons have the same velocity, which is c, and hence have the same energy.

It is one of the following factors that influences whether or not electrons are released.

When exposed to light for an extended period of time, D is also important.

None of the other numbers are significant.

An electron is enclosed within a potential well with an infinite square potential.

4 We’ve already established that As a result, doubling the diameter of the hole reduces the energy by a factor of FOUR.

When an electron in a gold atom is in the then=4 state, it is said to be in then=4 state.

A)-3,-1,0,+1,+4B)-3,0,+1,+2,+3C) -4,-2,+2,+3D) -1,0,+2,+3,+4E`)-3,-1,+1,+3,+5 We dismiss A), D), E), and likewise C) from consideration for then=4state since the biggest value oflis We are left with just option B. It’s B that’s the right answer.

## The Wave Nature of Matter

You will be able to do the following by the conclusion of this section:

- Demonstrate your understanding of the Davisson-Germer experiment and how it gives evidence for the wave nature of electrons.

## De Broglie Wavelength

Prince Louis-Victor de Broglie (1892–1987), a PhD student in physics at the University of Paris, proposed a bold suggestion in 1923, based on the assumption that nature is symmetric. If electromagnetic radiation possesses both particle and wave qualities, then nature would be symmetrical if matter has both particle and wave properties as well as electromagnetic radiation. It is possible that what we originally considered to be an unequivocal wave (EM radiation) is actually a particle, and that what we previously considered to be an unequivocal particle (matter) is actually a wave.

- When he submitted his thesis to Einstein, he received positive feedback, stating that it was not only likely right, but that it may be of fundamental importance.
- For his notion that all particles have a wavelength, de Broglie took into consideration both relativity and quantum mechanics.
- Notice that we already have this information for photons, according to the equationp=frac.) Interference is the distinguishing characteristic of a wave.
- Why isn’t this something that everyone does on a regular basis?
- Because it is so microscopic, it is likewise quite small, especially when compared to macroscopic items.
- When waves interact with objects that are many times greater in size than their wavelength, the interference effects are minimal and the waves flow in straight lines (such as light rays in geometric optics).
- As a result, electrons were the first to demonstrate this phenomenon.
- Davisson and Lester H.
- P.
- J.

These patterns are precisely compatible with the interference of electrons with the de Broglie wavelength and are equivalent to light interacting with a diffraction grating in a diffraction-limited environment. (See Illustration 1.)

### Making Connections: Waves

Wave characteristics may be found in all minuscule particles, whether they are massless, such as photons, or mass-containing, such as electrons. For all particles, the link between momentum and wavelength is important to their behavior. Figure 1 shows an example of a formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formalized formal It was possible to obtain this diffraction pattern by diffracting electrons through crystalline silicon.

- Contrary to destructive interference, bright regions are caused by constructive interference, whereas dark parts are caused by destructive interference.
- When the Austrian physicist Erwin Schrödinger (1887–1961) published four papers in 1926, he did it clearly with wave equations, demonstrating how the wave character of particles could be dealt directly.
- Among them was the German scientist Werner Heisenberg (1901–1976), who, among his many other contributions to quantum mechanics, devised a mathematical account of the wave character of matter that relied on matrices rather than wave equations to describe the wave nature of matter.
- As a result of his vision, de Broglie was given the Nobel Prize in 1929, while Davisson and G.
- Thomson were awarded the Nobel Prize in 1937 for their experimental proof of de Broglie’s theory.

### Example 1. Electron Wavelength versus Velocity and Energy

If you have an electron with a de Broglie wavelength of 0.167 nm (which is acceptable for interacting with crystal lattice structures that are around this size), you can do the following:

- Calculate the electron’s velocity on the assumption that it is nonrelativistic in nature. Calculate the kinetic energy of an electron in electron volts (eV).

#### Strategy

Because the de Broglie wavelength is known in Part 1, the electron’s velocity may be calculated from lambda=frac by using the nonrelativistic momentum formula, p=mv, to the equation of motion. With respect to Part 2, oncevis has been acquired (and it has been established thatvis is nonrelativistic), the classical potential energy is simply fracmv2.

#### Solution for Part 1

The de Broglie wavelength is obtained by substituting the nonrelativistic formula for momentum (p=mv) into the de Broglie wavelength.

The result is lambda=frac=frac. When you solve forv, you get v=frac. When known values are substituted, the result is: displaystyle =frac cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text cdot text c

#### Solution for Part 2

This electron’s speed is not extremely relativistic when compared to that of an automobile, so we can confidently apply the classical formula to calculate the electron’s kinetic energy and convert it to electron volts (eV) when the question is asked. Text at the start of a sentence is fracmv. 2 text= frac left(9.11 times 10 text right) left(4.36 times 10 6 text right) 2 text= frac left(86.4 times 10 text right) left(frac frac text right) 2 text= frac left(86.4 times 10 text right) 2 text= frac left(86.4 times 10 text right) 2 text= frac left(86.4 times 10 text right) 2 text= frac left(86.4 times 10 “text= 54.0 “text “end” “text= 54.0” “text=”” “text=”” “text=”” “end” “text=”” “text=”” “text=”” “text=”” “end” “text=”” “text=”” “text=”” “text=”” “text=”” “text=”” “text=”” “text=”” “text=”” “text=”” “text=”” “text=””” “text=”” “text=”” “text=””

#### Discussion

Because of their low energy, these 0.167-nm electrons might be produced by accelerating them via a 54.0-V electrostatic potential, which would be a simple process to do. Additionally, the findings support the hypothesis that electrons are nonrelativistic, given that their velocity is slightly greater than one percent of the speed of light and their kinetic energy is around 0.01 percent of the rest energy of an electron (0.511 MeV). It is possible that the electrons were relativistic, in which case we would have had to perform more complicated computations including relativistic formulae.

## Electron Microscopes

The electron microscope is an example of a result or application of the wave aspect of matter in nature. The level of detail that can be viewed with any probe that has a wavelength has a limit, as we have shown. The amount of viewable detail, or resolution, is restricted to around one wavelength. Because electrons with sub-nanometer wavelengths may be produced at a potential of just 54 V, it is possible to get electrons with wavelengths that are far smaller than those of visible light (hundreds of nanometers).

- (See Figure 2 for an example.) There are two fundamental types of electron microscopes: scanning and scanning transmission.
- It is necessary to expand the beam before it can pass through the sample.
- The transmission electron microscope (TEM) is similar to the optical microscope in that it requires a thin sample to be studied in a vacuum.
- The transmission electron microscope (TEM) has allowed us to see individual atoms as well as the structure of cell nuclei.
- In addition, magnetic lenses are used to concentrate the beam onto the sample in the SEM.
- To analyze the data for each electron point, a CCD detector is employed, which results in pictures similar to the one shown at the beginning of this chapter.
- However, it has a resolution that is approximately 10 times lower than that of a TEM.
- (Image courtesy of Dallas Krentzel on Flickr) Electrons were the first particles with mass to have their wavelength directly confirmed by de Broglie, and they were the first to do so.
- Unlike photons, the de Broglie wavelength for massless particles was clearly established in the 1920s.
- Nature’s universal trait is that all particles have a wave nature, regardless of their size.

For example, the following section demonstrates that, no matter how hard we try, there are limits to the precision with which we can make predictions, despite our best efforts. There are even limits to the precision with which we can determine the position or energy of an item in our environment.

### Making Connections: A Submicroscopic Diffraction Grating

Because of the wave nature of matter, it may display all of the qualities of other, more recognizable, waves, such as sound. In the case of light, diffraction gratings, for example, create diffraction patterns that rely on the spacing between the gratings and the wavelength of the light. As is true of most wave phenomena, this impact is most noticeable when the wave interacts with things that have a size that is comparable to the wavelength of the wave. (In gratings, this is the distance between each of many slits.) As seen in the upper left of Figure 3, when electrons contact with a system with a spacing that is similar to the electron wavelength, they produce interference patterns that are comparable to those produced by light when it interacts with diffraction gratings.

- The gaps between these planes have the same effect as the apertures in a diffraction grating in terms of diffraction.
- Aside from these angles, the difference in route lengths does not correspond to an integral wavelength, and there is partial to entire destructive interference.
- It is known as the Bragg reflection, after the father-and-son pair who were the first to investigate and examine it in depth.
- 3rd illustration.
- In comparison to electrons dispersed from the top layer of atoms, those scattered from the second layer go far further.
- Let us put the distance between parallel planes of atoms in the crystal to rest for a while.
- Due to the fact that AB = BC =dsin, we get constructive interference whenn= 2 dsin.
- When it comes to matter, the wavelength is a submicroscopic feature that may be used to explain macroscopic phenomena such as Bragg reflection.

As with the wavelength of light, it is a submicroscopic feature that accounts for the macroscopic phenomena of diffraction patterns in the presence of a lens.

## Section Summary

- Additionally, matter has a wavelength, which is known as the de Broglie wavelength, which is determined by the equation lambda=frac, wherepis momentum. It has been discovered that matter exhibits the same interference characteristics as any other wave.

### Conceptual Questions

- What is the difference between the interference of water waves and the interference of electrons? What is the analogy between them
- Describe one sort of evidence for the fact that matter is made of waves. Define and describe one sort of evidence for the particle nature of electromagnetic radiation.

### ProblemsExercises

- A wavelength of 1.00 m is achieved by an electron traveling at a certain velocity. When an electron travels at 3.00 percent the speed of light, what is the wavelength of the electron? If a proton moves at the speed of light, it will have a wavelength of 6.00 fm (roughly the size of a nucleus). Assume that the proton has nonrelativistic behavior. 1 femtometer equals 10 15 meters
- Is it possible to calculate the velocity of a 0.400-kg pool ball if its wavelength is 7.50 cm (which is large enough for it to interfere with the movement of other pool balls)? To find out the wavelength of a proton travelling at one-hundredth the speed of light, do the following: Using ultracold neutrons with velocities as low as 1.00 m/s, experiments are carried out on the atomic scale. (a) What is the wavelength of a neutron of this type? (b) What is the kinetic energy of the object in eV? (a) Calculate the velocity of a neutron with a wavelength of 6.00 micrometers (about the size of a nucleus). Assume that the neutron is not a relativistic particle. How much energy does a neutron have in MeV? What is the wavelength of an electron accelerated via a 30.0-kV potential, such as that found in a television tube
- And What is the kinetic energy of an electron in a transmission electron microscope (TEM) with a wavelength of 0.0100 nm
- 1. Calculate the velocity of an electron with a wavelength of 1.00 m. 2. Calculate the velocity of an electron with a wavelength of 1.00 m. (b) At what voltage does the electron have to be accelerated in order to achieve this velocity? A proton exiting a Van de Graaff accelerator travels at a velocity equal to 25.0 percent of the speed of light, according to the International Atomic Energy Agency. (a) How long is the wavelength of a proton? (b) Given the assumption that it is nonrelativistic, what is its kinetic energy? (c) What was the equivalent voltage that was used to accelerate it
- And One hundred thousand electron volts (keV) is the kinetic energy of an electron that has been accelerated in an x-ray tube. What is the wavelength of the particle, assuming it is nonrelativistic? Incredibly Unreasonable Outcomes In the assumption that it is nonrelativistic, compute the velocity of an electron with a wavelength of 0.100 fm. (b) (small enough to detect details of a nucleus). (b) What is it about this outcome that is unreasonable? (c) Identify any assumptions that are irrational or inconsistent.

## Glossary

The De Broglie wavelength is the wavelength held by a particle of matter, and it is computed using the formula lambda=frac.

### Selected Solutions to ProblemsExercises

1. 7.28 104 m3 at a speed of 6.62 107 m/s 5. 1.32 x 10 x 13 m 5. 7: (a) 6.62 x 10 7 m/s; (b) 22.9 MeV9; 15.1 keV11; 7. (a) 5.29 fm; (b) 4.70 10 12J; (c) 29.4 MV13; (d) 5.29 fm; (e) 5.29 fm; (f) 5.29 fm; It moves at a rate of 7.28 10 12 meters per second; (b) this is millions of times the speed of light (an impossibility); (c) the assumption that the electron is non-relativistic is unworkable at this wavelength.