Top Banner
MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen... 1 of 20 23/10/2007 9:23 PM [ Print View ] PHYS 1003 Physics 1 (Technological) 2007 Quantum Physics - PHYS1003 Assignment 6 & Tutorial (Chap 39-40) Due at 5:00pm on Friday, October 26, 2007 View Grading Details Quantum Physics - Assignment 6 and Tutorial Questions (Chapter 39-40) There are two sections to this MasteringPhysics exercise. The first consists of 8 compulsory assignment questions, each worth 5 marks and counting towards your final marks. Assignment questions must be completed by the deadline given above. The second section consists of a number of tutorial questions selected by your lecturers as being valuable practice for the course. You are strongly encouraged to complete at least some of these tutorial questions. They are not counted towards your final marks and they can be completed at any time, even after the assignment deadline. Assignments are due on Fridays at 5pm local time. Hints for Using MasteringPhysics Eight attempts are allowed per part and you may request the solution. We recommend that you work through the questions off-line. Don't just use your eight attempts as chances to guess. Pay special attention to the multiple choice questions as the maximum mark declines rapidly with incorrect responses. If necessary, use the Hints to get the correct answer. The cost in marks is small, as discussed in the FAQ page . Make sure MasteringPhysics interprets what you type the way you mean it - e.g. slide the cursor across symbols to check their syntax; explicitly include * for multiplication; use brackets to control the order of operations in an expression; carefully read feedback provided to wrong answers. If using MasteringPhysics to calculate values employing angles, remember that angles are in radians. Remember that, for example, atan means inverse tan when using the math palette. Values of constants can be found using the 'constants' button near the top of the page. For other constants, use values from your textbook. See the Help linked from "?" at the right end of relevant boxes for more help with formatting. If several attemps at an answer are wrong, use "my answers" to carefully review your attempts rather than guessing. Trouble with these questions or other course work? Remember the resources available to help you: the textbook - your #1 reference from the unit WebCT page: Unit and Module outlines Lecture notes Assignment solutions Physics resources for each module Multiple Choice Questions Sample and Past Examination papers and more. ask your lecturer, tutor or fellow students on the WebCT discussion forum consult the Duty Tutor (12 to 2, four days per week, Room 201) Physics Student Office, Room 202 University Student Services Assignment Questions The de Broglie Relation
20

berg Uncertainty Principle

Oct 27, 2014

Download

Documents

Siddhi Mahajan
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

1 of 20 23/10/2007 9:23 PM

[ Print View ]

PHYS 1003 Physics 1 (Technological) 2007

Quantum Physics - PHYS1003 Assignment 6 & Tutorial (Chap 39-40)

Due at 5:00pm on Friday, October 26, 2007

View Grading Details

Quantum Physics - Assignment 6 and Tutorial Questions(Chapter 39-40) There are two sections to this MasteringPhysics exercise. The firstconsists of 8 compulsory assignment questions, each worth 5 marks and counting towards your final marks. Assignment questions must be completed by the deadline given above.The second section consists of a number of tutorial questions selected by your lecturers as being valuable practice for the course. You arestrongly encouraged to complete at least some of these tutorial questions. They are not counted towards your final marks and they canbe completed at any time, even after the assignment deadline.

Assignments are due on Fridays at 5pm local time.

Hints for Using MasteringPhysics

Eight attempts are allowed per part and you may request the solution. We recommend that you work through the questions off-line.Don't just use your eight attempts as chances to guess. Payspecial attention to the multiple choice questions as the maximum mark declines rapidly with incorrect responses. If necessary, use the Hints to get the correct answer. The costin marks is small, as discussed in the FAQ page.Make sure MasteringPhysics interprets what you type the way you mean it - e.g. slide the cursor across symbols to check their syntax; explicitly include * for multiplication; use brackets to control the order of operations in an expression; carefully read feedback provided to wrong answers.If using MasteringPhysics to calculate values employing angles, remember that angles are in radians.Remember that, for example, atan means inverse tan when using the math palette.Values of constants can be found using the 'constants' button near the top of the page. For other constants, use values from your textbook.See the Help linked from "?" at the right end of relevant boxes for more help with formatting.If several attemps at an answer are wrong, use "my answers" to carefully review your attempts rather than guessing.

Trouble with these questions or other course work? Remember the resources available to help you:

the textbook - your #1 referencefrom the unit WebCT page:

Unit and Module outlinesLecture notesAssignment solutionsPhysics resources for each moduleMultiple Choice QuestionsSample and Past Examination papersand more.

ask your lecturer, tutor or fellow students on the WebCT discussion forumconsult the Duty Tutor (12 to 2, four days per week, Room 201)Physics Student Office, Room 202University Student Services

Assignment Questions

The de Broglie Relation

Page 2: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

2 of 20 23/10/2007 9:23 PM

Learning Goal: To understand de Broglie waves and the calculation of wave properties.

In 1924, Louis de Broglie postulated that particles such as electrons and protons might exhibit wavelike properties. His thinking was guided by the notion that light has both wave and particle characteristics, so he postulated that particles such as electrons and protons would obey the same wavelength-momentum relation as that obeyed by light: , where

is the wavelength, the momentum, and Planck's constant.

Part A

Find the de Broglie wavelength for an electron moving at a speed of . (Note that this speed is low

enough that the classical momentum formula is still valid.) Recall that the mass of an electron is , and Planck's constant is .

Express your answer in meters to three significant figures.

ANSWER: = 7.270×10−10

Part B

Find the de Broglie wavelength of a baseball pitched at a speed of 44.0 . Assume that the mass of the baseball is

.

Express your answer in meters to three significant figures

ANSWER: = 1.05×10−34

As a comparison, an atomic nucleus has a diameter of around . Clearly, the wavelength of a moving

baseball is too small for you to hope to see diffraction or interference effects during a baseball game.

Part C

Consider a beam of electrons in a vacuum, passing through a very narrow slit of width . The electrons then

head toward an array of detectors a distance 0.9000 away. These detectors indicate a diffraction pattern, with a broad maximum of electron intensity (i.e., the number of electrons received in a certain area over a certain period of time) with minima of electron intensity on either side, spaced 0.529 from the center of the pattern. What is the wavelength of one of the electrons in this beam? Recall that the location of the first intensity minima in a single slit

diffraction pattern for light is , where is the distance to the screen (detector) and is the width of the slit.

The derivation of this formula was based entirely upon the wave nature of light, so by de Broglie's hypothesis it will also apply to the case of electron waves.

Express your answer in meters to three significant figures.

ANSWER: = 1.18×10−8

Part D

What is the momentum of one of these electrons?

Express your answer in kilogram-meters per second to three significant figures.

ANSWER: = 5.64×10−26

Page 3: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

3 of 20 23/10/2007 9:23 PM

This is much smaller than the usual momentum of electrons used for standard diffraction experiments or electron microscopy. Correspondingly, the wavelength that you found in Part C is much larger than that of these electrons. In order to observe the wave nature of the electron, you need to work at scales similar to or smaller than the diameter of an atom. The momentum that you found in Part C could be given to an electron by accelerating it through a potential difference of around . Electron microscopes frequently use accelerating voltages on the

order of tens of kilovolts, yielding wavelengths roughly one thousand times smaller.

Heisenberg's Uncertainty Principle

Learning Goal: To understand, qualitatively and quantitatively, the uncertainty principle.

Understanding Heisenberg's uncertainty principle is one of the keys to understanding quantum mechanics. The principle states that you can never simultaneously know the exact location and momentum of a particle. Further, it states that the more you know about the position of the particle, the less you know about its momentum, and vice versa. The uncertainty principle is more than just a statement about the difficulty of measuring such things experimentally. Rather, it states that momentum and position are not simultaneously well defined for quantum particles. In fact, Heisenberg did not call his idea the uncertainty principle; he called it the indeterminacy principle, because position and momentum are fundamentally indeterminate, not just unknown, for the waves described by quantum mechanics.

This idea is difficult to reconcile with common experience. To understand it better, you must consider the properties of a wave. According to the de Broglie equation, the momentum of a wave is directly related to its wavelength. For the wave in the first figure, the wavelength is clearly well defined. However, the position is not well defined at all. The question, "Where is the wave?" does not have a well-defined answer, as we expect for a particle. This is the essence of the indeterminacy principle. We could just as easily draw a single sharp point at some particular x coordinate. This could be

considered a wave with a very well determined position. However, any notion of wavelength for such a wave seems strange.

A wave like the one shown in the second figure can be built up by adding together waves with different wavelengths. Recall that if two waves with similar frequencies, and , are added together, a wave with a beat frequency of

is produced . This gives a wave with somewhat well-defined position and wavelength. If you add contributions from all of the frequencies between and , then you get a wave

packet, which looks essentially like a single isolated beat cycle. In this problem, you will consider such a wave packet as

Page 4: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

4 of 20 23/10/2007 9:23 PM

simply being one beat cycle of this wave. While not exactly correct, this will give a useful approxmation.

Let the distance between the two nodes of the wave be the uncertainty in position . Since the beat frequency is given

by , and the wave travels at speed , the uncertainty in position is given by

.

Part A

The de Broglie relation can be rewritten in terms of the wave number as . Recall that wave number

is defined by . Using the fact that , find the wave numbers and corresponding to frequencies

and .

Express your answer as two expressions separated by a comma. Use , , , and .

ANSWER: , =

Part B

Find an expression for the uncertainty in the wave number. Use your results from Part A.

Express your answer in terms of quantities given in Part A.

ANSWER: =

Part C

What is the value of the product ? Use to find the uncertainty in the momentum of the particle.

Express your answer in terms of quantities given in Part A and fundamental constants.

ANSWER: =

This gives you the general idea of what the uncertainty principle states mathematically. The product of the uncertainties in the momentum and position of a particle is on the order of Planck's constant. By looking more rigorously at the definition of the uncertainty, the uncertainty principle is found to state that . The

greater-than-or-equal-to sign indicates that some less than ideal waveforms have greater uncertainty that the minimum value of .

Part D

In an atom, an electron is confined to a space of roughly meters. If we take this to be the uncertainty in the

electron's position, what is the minimum uncertainty in its momentum?

Page 5: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

5 of 20 23/10/2007 9:23 PM

Express your answer in kilogram meters per second to two significant figures.

ANSWER: = 1.05×10−24

Part E

What is the kinetic energy of an electron with momentum kilogram meters per second?

Express your answer in electron volts to two significant figures.

ANSWER: = 3.8

Notice that this energy is similar to the energy scale for electrons in an atom, which typically ranges from a bit less than an electron volt up to a few dozen electron volts. A good estimate for the energy scale of a particle can often be found by calculating the energy the particle would have if you set the momentum equal to the minimum uncertainty in momentum. The justification for this sort of estimation lies in the rigorous statistical definition of the uncertainty; it is sufficient now for you to know that this will give a reasonably good order-of-magnitude estimate of the energy for a variety of quantum systems.

Part F

Suppose that you know the position of a 100-gram pebble to within the width of an atomic nucleus (

meters). What is the minimum uncertainty in the momentum of the pebble?

Express your answer in kilogram meters per second to one significant figure.

ANSWER: = 1.0×10−19

For a 100-gram pebble, this corresponds to an uncertainty in the speed of about meters per second. Such

tiny values are the reason that you are unaware of the uncertainty principle in everyday situations. In practice, it would be impossible to measure the position of a pebble to such accuracy, much less its speed.

Uncertainty in the Atomic NucleusRutherford's scattering experiments gave the first indications that an atom consists of a small, dense, positively charged nucleus surrounded by negatively charged electrons. His experiments also allowed for a rough determination of the size of the nucleus. In this problem, you will use the uncertainty principle to get a rough idea of the kinetic energy of a particle inside the nucleus.

Consider a nucleus with a diameter of roughly meters.

Part A

Consider a particle inside the nucleus. The uncertainty in its position is equal to the diameter of the nucleus. What

is the uncertainty of its momentum? To find this, use .

Hint A.1 The uncertainty relation

Hint not displayed

Express your answer in kilogram-meters per second to two significant figures.

Page 6: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

6 of 20 23/10/2007 9:23 PM

ANSWER: = 2.10×10−20

Part B

The uncertainty sets a lower bound on the average momentum of a particle in the nucleus. If a particle's average

momentum were to fall below that point, then the uncertainty principle would be violated. Since the uncertainty principle is a fundamental law of physics, this cannot happen. Using kilogram-meters per second as

the minimum momentum of a particle in the nucleus, find the minimum kinetic energy of the particle. Use

kilograms as the mass of the particle. Note that since our calculations are so rough, this serves as the

mass of a neutron or a proton.

Hint B.1 Choosing the kinetic energy formula

Hint not displayed

Express your answer in millions of electron volts to two significant figures.

ANSWER: = 0.81

Compare this to the normal energy scale for electrons in an atom, which is on the order of single electron volts. The characteristic energy scale for the nucleus seems to be roughly one million times that for electrons in an atom. This difference can be seen in calculations of the energy output per unit mass from coal-burning power plants, which utilize chemical energy (energy associated with the electrons of an atom) compared to the energy output from nuclear reactors (power plants that harness the much higher energies of nuclei).

The Wavelength of an Electron

An electron has de Broglie wavelength 2.85×10−10 .

Part A

Determine the magnitude of the electron's momentum .

Hint A.1 The de Broglie wavelength

Hint not displayed

Express your answer in kilogram meters per second to three significant figures.

ANSWER: = 2.33×10−24

Part B

Determine the kinetic energy of the electron.

Part B.1 Find the kinetic energy as a function of momentum

Part not displayed

Express your answer in joules to three significant figures.

ANSWER: = 2.97×10−18

Page 7: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

7 of 20 23/10/2007 9:23 PM

Part C

Determine the electron's kinetic energy in electron volts.

Hint C.1 The relation between electron volts and joules

Hint not displayed

Express your answer in electron volts to three significant figures.

ANSWER: = 18.6

Width of a Wave Function

A particle is described by a wave function , where and are real, positive constants.

Part A

If the value of is increased, what effect does this have on the particle's uncertainty in position?

Hint A.1 How to approach the problem

Hint not displayed

Part A.2 Find the FWHM of

Part not displayed

Part A.3 Find how the FWHM varies with respect to

Part not displayed

ANSWER: The particle's uncertainty in position will decrease.The particle's uncertainty in position will increase.There is no effect on the particle's uncertainty in position.

Part B

If the value of is increased, what effect does this have on the particle's uncertainty in momentum?

Hint B.1 How to approach the problem

Hint not displayed

ANSWER: The particle's uncertainty in momentum will decrease.The particle's uncertainty in momentum will increase.There is no effect on the particle's uncertainty in momentum.

Schrödinger Equation and the Particle in a Box

Learning Goal: To become familiar with the Schrödinger equation and its solution for the simple case of the particle ina box.

Page 8: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

8 of 20 23/10/2007 9:23 PM

The most important equation in quantum mechanics is the Schrödinger equation,

,

where is Planck's constant divided by (i.e., ).

Given a potential energy function , solving the Schrödinger equation allows you to determine the particle wave

functions. Finding solutions to the Schrödinger equation, for most potentials, is beyond the scope of introductory physics.However, you are able to check a solution, once it is presented to you. You will do this for the simple case of the particle in a box.

The quantum mechanical particle in a box has a particularly simple potential energy function. Although it does have some real-world applications, the particle in a box is also important as an illustration of many key concepts from quantum mechanics.

Consider a particle in a potential well with infinitely high walls. The potential energy function is formally written as

where is the width of the box. It is claimed that each of the functions

,

for is a solution to the Schrödinger equation for the particle in a box. You will prove this and calculate the

proper value for .

By inspection, you should be able to see that is a mathematical solution to any Schrödinger equation, so the

functions are clearly valid solutions outside the interval .

Part A

Consider the interval . What is the second derivative, with respect to , of the wave function in this

interval?

Express your answer in terms of , , , and .

ANSWER: =

Part B

What is in the interval ?

Express your answer in terms of , , and .

ANSWER: = 0

Part C

Page 9: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

9 of 20 23/10/2007 9:23 PM

is an as yet undetermined constant: the energy of the particle. What is in the interval ?

Express your answer in terms of , , , and .

ANSWER: =

Part D

Combine your answers from Parts A and B. Find the expression for the left side of the Schrödinger equation valid onthe interval .

Express your answer in terms of , , , , , and .

ANSWER: =

This entire expression is just multipled by a positive constant. Since you have already found the right

side of the equation to be multiplied by a positive constant ( ), you have proven that, if the two

constants are equal, is a mathematical solution to the Schrödinger equation for the particle in a box. You

will soon determine if it is also a physical solution.

Part E

Combine your answers from Parts C and D to find the value of , the energy of a particle with wave function .

Express your answer in terms of , , , and .

ANSWER: =

Even in the simple case of the particle in a box, one of the main ideas of quantum theory--the quantization of energy--may be seen in the discrete allowed energy levels:

.

In this context, you can see how the quantization of energy is a natural consequence of applying the boundary conditions to solutions of the Schrödinger equation.

Part F

For a solution to be a physical solution, it must satisfy several criteria. First, it must be continuous everywhere. Second, it must have a continuous derivative everywhere, except for points where the potential energy becomes infinite (as it does at the walls of the box). Finally, it must be normalizable.

In this case, you can check the first criterion by noting that the two functions and 0 are continuous and

that they have the same value at and , where their domains meet.

To check the second criterion, simply take the first derivative of and 0. Both derivatives are continuous

functions in their domains. The points where the domains meet are exactly the points where the potential energy becomes infinite, so you don't have to check for continuity there.

Page 10: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

10 of 20 23/10/2007 9:23 PM

The third criterion requires that there exist some value of such that

.

Since is zero outside of the interval , this equation reduces to

Use this equation to find the unique positive value of .

Part F.1 Evaluate the integral

Part not displayed

Express your answer in terms of .

ANSWER: =

Vision and the Particle in a BoxThough the particle in a box (infinite potential well) seems like a very unrealistic potential, it can actually be used to explain a bit about how humans see. The important light-absorbing molecule in human eyes is called retinal. Retinal consists of a chain of carbon atoms, roughly long. An electron in this long chain molecule behaves very

much like a particle in a box.

Part A

Find the wavelength of the photon that must be absorbed by an electron to move it from the state of a box to the

state. Assume that the box has length and that the electron has mass .

Part A.1 Find the wavelength in terms of energy

Find the wavelength of a photon in terms of the photon's energy . Recall that photons are described by the

equations and , where is Planck's constant divided by , is the angular frequency, is the

frequency, and is the speed of light.

Hint A.1.a Relation between and

Hint not displayed

Express your answer in terms of , , and .

ANSWER: =

Part A.2 Find the energy of the photon

Find the difference between the energy of the state for the electron and the energy of the

state.

Hint A.2.a Energy of the state of a particle in a box

Page 11: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

11 of 20 23/10/2007 9:23 PM

Hint not displayed

Express your answer in terms of , , , and .

ANSWER: =

Express your answer in terms of , , Planck's constant (divided by ) , the speed of light , and .

ANSWER: =

Part B

The retinal molecule has 12 electrons that are free to move about the chain. For reasons that you may learn later, these 12 electons fill the first 6 states of the box (with 2 electrons in each state). Thus, the lowest energy photon that can be absorbed by this molecule would be the one that moves an electron from the 6th state to the 7th. Use the equation that you found in Part A to determine the wavelength of this photon. Use the length of

the retinal molecule given in the introduction as the length of the box and use for the mass of the electron.

Express your answer in nanometers to two significant figures.

ANSWER: = 570

A photon with this wavelength lies in the green part of the spectrum.

Part C

In a human eye, there are three types of cones that allow us to see colors. The three different types are most sensitive to red, green, and blue light, respectively. All three contain retinal bonded to a large protein. The way that retinal bonds to the protein can change the length of the potential well within which the electrons are confined. How would the length have to change from that given in the introduction to make the molecule more sensitive to blue or red light?

Hint C.1 Comparing wavelengths

Hint not displayed

ANSWER: The molecule would have to be shorter to be more sensitive to both red and blue light.The molecule would have to be shorter to be more sensitive to red light and longer to be more

sensitive to blue light.The molecule would have to be longer to be more sensitive to red light and shorter to be more

sensitive to blue light.The molecule would have to be longer to be more sensitive to both red and blue light.

Page 12: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

12 of 20 23/10/2007 9:23 PM

Once retinal has absorbed a photon, it changes shape, initiating a cascade of effects that eventually creates a nerve impulse. This impulse gets fed to the brain, where it is processed along with the signals from all of the other light-sensing cells in the eye, allowing you to see.

The following questions are from the textbook and have no hints or other feedback.

Problem 40.34A particle is in the ground level of a box that extends from to .

Part A

What is the probability of finding the particle in the region between 0 and ? Calculate this by integrating

, where is normalized, from to .

ANSWER:

Part B

What is the probability of finding the particle in the region to ?

ANSWER:

Part C

Add the probabilities calculated in parts (a) and (b).

ANSWER: 0.500

Tutorial QuestionsThe following questions have been selected to allow you to use MasteringPhysics to build your Physics skills. They are for nocredit but you are strongly encouraged to complete as many as possible. They can be done at any time, even after theassignment deadline. Named questions have full MasteringPhysics hints and feedback. Questions marked as 'Problem n.nn' are from the textbook and have no hints and other feedback. Chapter 39

De Broglie Waves in the Bohr ModelThe hypothesis that was put forward by Louis de Broglie in 1924 was astonishing for a number of reasons. An obvious reason is that associating a wavelike nature with particles is far from intuitive, but another astonishing aspect was how well the hypothesis fit in with certain parts of existing physics. In this problem, we explore the correspondence between the de Broglie picture of the wave nature of electrons and the Bohr model of the hydrogen atom.

Part A

What is the de Broglie wavelength of the electron in the first Bohr energy level of the hydrogen atom?

Hint A.1 Speed in the Bohr model

Hint not displayed

Page 13: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

13 of 20 23/10/2007 9:23 PM

Hint A.2 De Broglie wavelength

Hint not displayed

Express your answer in terms of , , the mass of the electron , and the magnitude of the charge on the

electron .

ANSWER: = Answer not displayed

Part B

Part not displayed

Part C

What is the de Broglie wavelength of the electron in the third ( ) Bohr energy level of the hydrogen atom?

Hint C.1 Speed in the Bohr model

Hint not displayed

Hint C.2 De Broglie wavelength

Hint not displayed

Express your answer in terms of , , , and .

ANSWER: = Answer not displayed

Part D

Part not displayed

Part E

In the previous parts, you saw that there is not equality between the de Broglie wavelength of an electron in the hydrogen atom and the circumference of its orbit. However, there does exist a definite relationship. What is the relationship between the circumference of the orbit of the th energy level and the de Broglie wavelength ?

Part E.1 Find the Broglie wavelength for the th energy level.

Part not displayed

Part E.2 Find the orbital circumference at the th energy level in hydrogen

Part not displayed

Express your answer in terms of and .

ANSWER: = Answer not displayed

Simultaneous Measurements of Position and Velocity

Page 14: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

14 of 20 23/10/2007 9:23 PM

Part A

The x coordinate of an electron is measured with an uncertainty of . What is , the x component of the electron's

velocity, if the minimum percentage uncertainty in a simultaneous measurement of is ? Use the following

expression for the uncertainty principle:

,

where is the uncertainty in the x coordinate of a particle, is the particle's uncertainty in the x component of

momentum, and , where is Planck's constant.

Hint A.1 How to approach the problem

Hint not displayed

Part A.2 Find the electron's minimum uncertainty in velocity

Part not displayed

Express your answer in meters per second to three significant figures.

ANSWER: = Answer not displayed

Part B

Part not displayed

The Uncertainty Principle: Virtual ParticlesThe uncertainty principle can be expressed as a relation between the uncertainty in the energy state of a system and

the time interval during which the system remains in that state. In symbols,

,

where , is Planck's constant.

The energy-time uncertainty principle says that the longer a system remains in the same energy state, the higher the accuracy (or the smaller the uncertainty) a measurement of that energy can be. Another implication is that physical processes can violate the law of energy conservation as long as the violation occurs for only a short time, determined by the uncertainty principle. This idea is at the base of the theory of virtual particles.

Part A

Consider two electrons that interact with each other. Classically, their interaction would be described in terms of the electrostatic force. In quantum mechanics, their interaction is interpreted in terms of emission and absorption of photons: One of the two electrons emits a photon with energy , which is then absorbed by the other electron after a

short period of time.

How long can the photon survive before it is absorbed without violating the uncertainty principle?

Hint A.1 How to approach the problem

Hint not displayed

Page 15: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

15 of 20 23/10/2007 9:23 PM

Express your answer in terms of , and or .

ANSWER: = Answer not displayed

Part B

Part not displayed

Problem 39.3

Part A

An electron moves with a speed of 5.60×106 . What is its de Broglie wavelength?

Use 6.63×10−34 for Planck's constant and 9.11×10−31 for the mass of an electron.

ANSWER: Answer not displayed m

Part B

A proton moves with the same speed. Determine its de Broglie wavelength.

Use 1.67×10−27 for the mass of a proton.

ANSWER: Answer not displayed m

Problem 39.17By extremely careful measurement, you determine the x-coordinate of a car's center of mass with an uncertainty of only . The car has a mass of .

Part A

What is the minimum uncertainty in the x-component of the velocity of the car's center of mass as prescribed by the Heisenberg uncertainty principle?

Use for Planck's constant.

ANSWER: Answer not displayed

Problem 39.19A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of 0.170 and its momentum

component along this axis with a standard deviation of 3.20×10−25 .

Part A

Page 16: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

16 of 20 23/10/2007 9:23 PM

Use the Heisenberg uncertainty principle to evaluate the validity of this claim.

ANSWER: Answer not displayed

Problem 39.27

Part A

What is the de Broglie wavelength of an electron that has been accelerated from rest through a potential increase of 900 ?

Use 6.63×10−34 for Planck's constant, 9.11×10−31 for the mass of an electron, and 1.60×10−19 for the

charge on an electron.

ANSWER: Answer not displayed m

Part B

What is the de Broglie wavelength of a proton accelerated from rest through a potential decrease of 900 ?

Use 1.67×10−27 for the mass of a proton.

ANSWER: Answer not displayed m

Chapter 40

The Finite Square-Well Potential: Bound States

Learning Goal: To understand the qualities of the finite square-well potential and how to connect solutions to theSchrödinger equation from different regions.

The case of a particle in an infinite potential well, also known as the particle in a box, is one of the simplest in quantum mechanics. The closely related finite potential well is substantially more complicated to solve, but it also shows more of the qualities that are characteristic of quantum systems. The potential energy function for a finite square-well potential is

where is a positive number that measures the depth of the potential well and is the width of the well. The figure is a

graph of potential energy versus position, which shows why this is called the square-well potential. Inside the well (i.e., for

) the solutions take the form

, where and are constants

and . Outside the well, the solutions take the

form , where and are constants

and . In this problem, you will

consider a particle in a state with energy . Such states

are called bound states, because classically the particle would be trapped in the potential well.

Page 17: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

17 of 20 23/10/2007 9:23 PM

Part A

For a one-dimensional wave function to be normalizable, it must go to zero as goes to infinity or negative infinity. Consider the wave function in the region . As goes to infinity, this must become

zero. What does this imply about the constants and ?

ANSWER:

Notice that the wave function is nonzero in the entire domain , even though this region would be forbidden

by classical mechanics since . This "tunneling" into the classically forbidden region is a key difference

between classical and quantum mechanics. Also notice that, since anywhere in the region, there

is some small, but nonzero, probability of finding the particle hundreds of kilometers away from the potential well.

Part B

Now, consider the wave function in the region . As goes to negative infinity, this must

become zero. What does this imply about the constants and ? (Be careful of signs.)

ANSWER: Answer not displayed

Part C

Part not displayed

Part D

Part not displayed

Part E

Part not displayed

Part F

Part not displayed

Part G

Part not displayed

Classical and Quantum Harmonic OscillatorsConsider a harmonic oscillator with mass and . You may have worked similar problems

before, as a mass on a spring using classical mechanics, but this time you will use the solution to the Schrödinger equationfor the harmonic oscillator. Keep in mind that this system would be enormous by quantum standards, and in practice you would never expect to use quantum mechanics to describe a mass on a spring. Nonetheless, it is interesting to see what

Page 18: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

18 of 20 23/10/2007 9:23 PM

quantum mechanics predicts here.

Throughout this problem, use .

Part A

Let this oscillator have the same energy as a mass on a spring, with the same and , released from rest at a

displacement of from equilibrium. What is the quantum number of the state of the harmonic oscillator?

Hint A.1 Finding from and

Hint not displayed

Part A.2 Find

Part not displayed

Part A.3 Find the energy of the oscillator

Part not displayed

Express the quantum number to three significant figures.

ANSWER: = Answer not displayed

Part B

Part not displayed

Part C

Part not displayed

Problem 40.3It takes of energy to excite an electron in a box from the ground level to the first excited level.

Part A

What is the width L of the box?

Use for Planck's constant and for the mass of an electron.

ANSWER: Answer not displayed

Problem 40.5A particle with mass of 5.50 is in a box with a width L.

Part A

Classically, what is the kinetic energy of the particle if it has a speed of 1.70 ?

ANSWER: Answer not displayed J

Page 19: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

19 of 20 23/10/2007 9:23 PM

Part B

What is L if the ground state energy of the particle equals the kinetic energy calculated in part (A)?

Use 6.63×10−34 for Planck's constant.

ANSWER: Answer not displayed m

Problem 40.8

Part A

Find the excitation energy from the ground level to the third excited level for an electron confined to a box that has a width of .

Use for Planck's constant and for the mass of an electron.

ANSWER: Answer not displayed

Part B

The electron makes a transition from the n = 1 to n = 4 level by absorbing a photon. Calculate the wavelength of this photon.

Use for the speed of light in a vacuum.

ANSWER: Answer not displayed

Problem 40.39

Photon in a Dye Laser. An electron in a long, organic molecule used in a dye laser behaves approximately like a particle in a box with width 4.20 .

Part A

What is the wavelength of the photon emitted when the electron undergoes a transition from the first excited level to the ground level?

Use 3.00×108 for the speed of light in a vacuum, 6.63×10−34 for Planck's constant, and 9.11×10−31

for the mass of an electron.

ANSWER: Answer not displayed m

Part B

What is the wavelength of the photon emitted when the electron undergoes a transition from the second excited level to the first excited level?

ANSWER: Answer not displayed m

Further QuestionsThere are many more questions available at the end of every chapter of the textbook. Odd numbered problems have brief

Page 20: berg Uncertainty Principle

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmen...

20 of 20 23/10/2007 9:23 PM

answers at the back of the book. Answers to all 'Exercises' and 'Problems' can be requested during lunch time Duty Tutorsessions.

Summary 8 of 21 items complete (37.86% avg. score)39.76 of 40 points