Solved: Estimate the lowest possible energy of a neutron

Problem 1Q Compare a matter wave ? to (a) a wave on a string, (b) an EM wave. Discuss similarities and differences.

Problem 2Q Explain why Bohr's theory of the atom is not compatible with quantum mechanics, particularly the uncertainty principle.

Explain why it is that the more massive an object is, the easier it becomes to predict its future position.

In view of the uncertainty principle, why does a baseball seem to have a well-defined position and speed whereas an electron does not?

(I) An electron remains in an excited state of an atom for Typically \(10^{-8} \ s\). What is the minimum uncertainty in the energy of the state (in eV)?

Would it ever be possible to balance a very sharp needle precisely on its point? Explain.

(I) The \(\mathrm {Z^0}\) boson, discovered in 1985, is the mediator of the weak nuclear force, and it typically decays very quickly. Its average rest energy is 91.19 GeV, but its short lifetime shows up as an intrinsic width of 2.5 GeV. What is the lifetime of this particle? ________________ Equation Transcription: Text Transcription: Z^0

Problem 6Q A cold thermometer is placed in a hot bowl of soup. Will the temperature reading of the thermometer be the same as the temperature of the hot soup before the measurement was made? Explain.

Problem 7P (II) What is the uncertainty in the mass of a muon (m =105.7 Me V/c2), Specified eV/c2 in given its lifetime of 2.20 ps?

Problem 7Q Does the uncertainty principle set a limit to how well you can make any single measurement of position? Explain.

Problem 8Q If you Knew the position of an object precisely, with no uncertainty, how well would you Know its momentum?

Problem 9Q When you check the pressure in a tire, doesn't some air inevitably escape? Is it possible to avoid this escape of air altogether?What is the relation to the uncertainty principle?

Problem 10Q It has been said that the ground-state energy in the hydrogen atom can be precisely Known but the excited states have some uncertainty in their values (an "energy width"). Is this consistent with the uncertainty principle in its energy form? Explain.

Problem 11P (Ill) Use the uncertainty principle to show that if an electron were present in the nucleus (r =10-15 m) its kinetic energy (use relativity) would be hundreds of MeV. (Since such electron energies are not observed, we conclude that electrons are not present in the nucleus.) [Hint Assume a particle can have an energy as large as its uncertainty ]

Which model of the hydrogen atom, the Bohr model or the quantum-mechanical model, predicts that the electron spends more time near the nucleus?

The size of atoms varies by only a factor of three or so. from largest to smallest, yet the number of electrons varies from one to over 100. Explain.

Problem 13P (I) For n=6 what values can I have?

Excited hydrogen and excited helium atoms both radiate light as they jump down to the \(n=1\), \(l=0\), \(m_l=0\) state. Yet the two elements have very different emission spectra. Why? ________________ Equation Transcription: Text Transcription: n=1 l=0 m_l=0

Problem 14Q How would the Periodic Table look if there were no electron spin but otherwise quantum mechanics were valid? Consider the first 20 elements or so.

Which of the following electron configurations are not allowed: \((a)\ 1s^22s^22p^43s^24p^2\); \((b)\ 1s^22s^22p^83s^1\); \((c)\ 1s^22s^22p^63s^23p^54s^24d^54f^1\)? If not allowed, explain why. ________________ Equation Transcription: Text Transcription: (a) 1s^22s^22p^43s^24p^2 (b) 1s^22s^22p^83s^1 (c) 1s^22s^22p^63s^23p^54s^24d^54f^1

(I) How many different states are possible for an electron whose principal quantum number is n =4? Write down the quantum numbers for each state.

Problem 18Q Why do chlorine and iodine exhibit similar properties?

Explain why potassium and sodium exhibit similar properties.

The ionization energy for neon (Z=10) is \(21.6 \mathrm{eV}\), and that for sodium (Z=11) is \(5.1 \mathrm{eV}\). Explain the large difference.

Problem 21P (II) If a hydrogen atom has ml =-3, what are the possible values of n, I and ms?

Why do we not expect perfect agreement between measured values of X-ray line wavelengths and those calculated using Bohr theory, as in Example 28-6?

Problem 22P (II) Show that there can be 18 electrons in a "g" subshell

Why do we not expect perfect agreement between measured values of characteristic X-ray line wavelengths and those calculated using the Bohr model, as in Example 28-7?

Problem 23Q How would you figure out which lines in an X-ray spectrum correspond to K?, K?, L, etc., transitions?

Why do we expect electron transitions deep within an atom to produce shorter wavelengths than transitions by outer electrons?

Problem 25Q Compare spontaneous emission to stimulated emission.

Problem 26P (II) Estimate the binding energy of the third electron in lithium using the Bohr model. [Hint. This electron has n =2 and “sees" a net charge of approximately +1e. ] The measured value is 5.36 eV

How does laser light differ from ordinary light? How is it the same?

Problem 27P (II) Show that the total angular momentum is zero for a filled subshell.

Problem 27Q Explain how a 0.0005-W laser beam, photographed at a distance, can seem much stronger than a 1000-W street lamp at the same distance.

Does the intensity of light from a laser fall off as the inverse square of the distance? Explain.

(II) An excited \(\mathrm{H}\) atom is in a 6d state. (a) Name all the states (n, l) to which the atom is "allowed" to jump with the emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

(I) Show that the cutoff wavelength \(\lambda_0\) is given by \(\lambda_0=\frac{1240 \mathrm{~nm}}{V}\), where is the X-ray tube voltage in volts. ________________ Equation Transcription: Text Transcription: lambda_0 lambda_0=frac{1240 nm}{V}

(II) Estimate the wavelength for an n=2 to n=1 transition in iron (Z=26).

Problem 35P (II) Use the Bohr model to estimate the wavelength for an N =3 to n =1 transition in molybdenum (Z =42) The measured value is 0.063 nm. Why do we not expect perfect agreement?

Problem 36P (II) A mixture of iron and an unknown material is bombarded with electrons. The wavelengths of the Ka lines are 194 pm for iron and 229 pm for the unknown. What is the unknown material?

(II) A laser used to weld detached retinas puts out 28-ms-long pulses of 640-nm light which average 0.68-W output during a pulse. How much energy can be deposited per pulse and how many photons does each pulse contain? [Hint:See Example 27-5 ]

(II) A low-power laser used in a physics lab might have a power of 0.50 mW and a beam diameter of 3.0 mm. Calculate (a) the average light intensity of the laser beam, and (b) compare it to the intensity of a lightbulb producing 100-W light viewed from 2.0 m.

Problem 39P (II) Estimate the angular spread of a laser beam due to diffraction if the beam emerges through a 3.0-mm-diameter mirror. Assume that A =694 nm. What would be the diameter of this beam if it struck (a) a satellite 340 km above the Earth, or (b) the Moon? [Hint See Sections 24-5 and 25-7.]

(II) What is the wavelength of the He–Ne laser?

Problem 42GP An electron in the n = 2 state of hydrogen remains thereon average about 10-8 s before jumping to the n = 1 state.(a) Estimate the uncertainty in the energy of the n = 2state. (b) What fraction of the transition energy is this?(c) What is the wavelength, and width (in nm), of this linein the spectrum of hydrogen?

An electron in the n = 2 state of hydrogen remains thereon average about \(10^{-8} \ s\) before jumping to the n = 1 state.(a) Estimate the uncertainty in the energy of the n = 2 state. (b) What fraction of the transition energy is this?(c) What is the wavelength, and width (in nm), of this line in the spectrum of hydrogen?

Problem 44GP Estimate (a) the quantum number for the orbital angular momentum of the Earth about the Sun, and (b) the number of possible orientations for the plane of Earth’s orbit.

Problem 46GP Using the Bohr formula for the radius of an electron orbit.estimate the average distance from the nucleus for anelectron in the innermost (n = 1) orbit of a uranium atom(Z = 92). Approximately how much energy would berequired to remove this innermost electron?

Problem 50GP (a) Show that the number of different states possible for agiven value of I is equal to 2(21 + 1). (b) What is thisnumber for I = 0, 1, 2, 3, 4, 5, and 6?

Show that the number of different electron states possible for a given value of is \(2n^2\). (See Problem 50 .) ________________ Equation Transcription: Text Transcription: 2n^2

A beam of electrons with kinetic energy 45 keV is shot through two narrow slits in a barrier. The slits are a distance \(2.0 \times 10^{-6} \ \mathrm {m}\) apart. If a screen is placed 45.0 cm behind the barrier, calculate the spacing between the ‘'bright fringes of the interference pattern produced on the screen.

Problem 53GP The angular momentum in the hydrogen atom is givenboth by the Bohr model and by quantum mechanics.Compare the results for n = 2.

An atomic spectrum contains a line with a wavelength centered at \(488 \mathrm{~nm}\). Careful measurements show the line is really spread out between 487 and \(489 \mathrm{~nm}\). Estimate the lifetime of the excited state that produced this line.

Problem 57GP An electron and a proton, each initially at rest, are accelerated across the same voltage. Assuming that the uncertaintyin their position is given by their de Broglie wavelength.find the ratio of the uncertainty in their momentum.

Problem 58GP If the principal quantum number n were limited to the rangefrom 1 to 6, how many elements would we find in nature?

Problem 60GP Suppose that the spectrum of an unknown element showsa series of lines with one out of every four matching a linefrom the Lyman series of hydrogen. Assuming that theunknown element is an ion with Z protons and oneelectron. determine Z and the element in question.

Problem 61GP Photons of wavelength 0.154 nm are emitted from thesurface of a certain metal when it is bombarded with highenergy radiation. If this photon wavelength corresponds tothe Ka line, what is the element?

Show that the diffractive spread of a laser beam, \(\approx \lambda/D\) as described in (Section 28–11), is precisely what you might expect from the uncertainty principle. [Hint: Since the beam’s width is constrained by the dimension of the aperture D, the component of the light’s momentum perpendicular to the laser axis is uncertain.]

(II) The neutrons in a parallel beam, each having kinetic energy \(\frac{1}{40}\mathrm{eV}\), are directed through two slits 0.50 mm apart. How far apart will the interference peaks be on a screen 1.0 m away? [Hint: first find the wavelength of the neutron.] ________________ Equation Transcription: Text Transcription: frac{1}{40}eV

Bullets of mass 3.0 g are fired in parallel paths with speeds of 220 m/s through a hole 3.0 mm in diameter. How far from the hole must you be to detect a 1.0-cm-diameter spread in the beam of bullets?

A proton is traveling with a speed of \((6.560 \pm 0.012) \times 10^5\) m/s, With what maximum accuracy can its position be ascertained? [Hint: \(\Delta p = m \Delta v\).]

Problem 4P If an electron’s position can be measured to an accuracy of 2.0 × 10?8 m, how accurately can its speed be known?

(II) A free neutron \(\left(m=1.67 \times 10^{-27} \mathrm{~kg}\right)\) has a mean life of \(900 \mathrm{~s}\). What is the uncertainty in its mass (in \(\mathrm{kg}\) )?

Problem 9P An electron and a 140-g baseball are each traveling 150 m/s measured to an accuracy of 0.055%. Calculate and compare the uncertainty in position of each.

Problem 10P Estimate the lowest possible energy of a neutron contained in a typical nucleus of radius 1.0 × 10?l5m. [Hint: a particle can have an energy at least as large as its uncertainty.]

Problem 12P How accurately can the position of a 3.00-keV electron be measured assuming its energy is known to 1.00%?

Problem 14P For n = 5, l = 3, what are the possible values of ml and ms?

(I) How many electrons can be in the n = 6, l = 3, subshell?

Give the complete electron configuration for a uranium atom (careful scrutiny across the periodic table on the inside back cover will provide useful hints).

Problem 17P List the quantum numbers for each electron in the ground state of (a) carbon (Z = 6), (b) magnesium (Z = 12).

In what column of the periodic table would you expect to find the atom with each of the following configurations: (a) \(1 s^2 2 s^2 2 p^6 3 s^2\); (b) \(1 s^2 2 s^2 2 p^6 3 s^2 3 p^6\); (c) \(1 s^2 2 s^2 2 p^6 3 s^2 3 p^6 4 s^1 ;(d) 1 s^2 2 s^2 2 p^5 ?\)

(I) List the quantum numbers for each electron in the ground state of nitrogen (Z = 7).

Problem 19P Suppose a certain hydrogen atom has l = 4. What are the possible values for n, ml, and ms?

Problem 20P Calculate the magnitude of the angular momentum of an electron in the n = 4, l = 3 state of hydrogen.

Problem 23P What is the full electron configuration in the ground state for elements with Z equal to (a) 27, (b) 36, (c) 38? [Hint: see the periodic table inside the back cover.]

Problem 24P What is the full electron configuration for (a) selenium (Se), (b) gold (Au), (c) radium (Ra)? [Hint:see the periodic table inside the back cover.]

(II) A hydrogen atom is in the 6s state. Determine (a) the principal quantum number, (b) the energy of the state, (c) the orbital angular momentum and its quantum number l, and (d) the possible values for the magnetic quantum number.

(II) For each of the following atomic transitions, state whether the transition is allowed or forbidden, and if forbidden, what rule is being violated: (a) \(4 p \rightarrow 3 p\); (b) \(2 p \rightarrow 1 s\) ; (c) \(3 d \rightarrow 2 d\) ; (d) \(4 d \rightarrow 3 s\) ; (e) \(4 s \rightarrow 2 p\).

Problem 30P What are the shortest-wavelength X-rays emitted by electrons striking the face of a 33.5-kV TV picture tube? What are the longest wavelengths?

Problem 31P If the shortest-wavelength bremsstrahlung X-rays emitted from an X-ray tube have ? = 0.030 nm, what is the voltage across the tube?

(II) Use the result of Example 28-6 to estimate the X-ray wavelength emitted when a Co (Z=27) atom jumps from n=2 to n=1.

Problem 43GP What are the largest and smallest possible values for the angular momentum L of an electron in the n = 5 shell?

A 12-g bullet leaves a rifle at a speed of \(180 \mathrm{~m} / \mathrm{s}\). (a) What is the wavelength of this bullet? (b) If the position of the bullet is known to an accuracy of \(0.60 \mathrm{~cm}\) (radius of the barrel), what is the minimum uncertainty in its momentum?

An X-ray tube operates at 95 kV with a current of 25 mA and nearly all the electron energy goes into heat. If the specific heat of the 0.085-kg plate is \(0.11 \mathrm {kcal/kg}\cdot {C}^\circ\), what will be the temperature rise per minute if no cooling water is used?

Problem 48GP The ionization (binding) energy of the outermost electron in boron is 8.26 eV. (a) Use the Bohr model to estimate the “effective charge,” Zeff, seen by this electron. (b) Estimate the average orbital radius.

Use the Bohr theory (especially Eq. ) to show that the Moseley plot (Fig. ) can be written \(\sqrt{\frac{1}{\lambda}}=a(Z-b)\) where \(b\approx1\), and evaluate . ________________ Equation Transcription: Text Transcription: sqrt{frac{1}{lambda}}=a(Z-b) b{approx}1

Problem 54GP An 1100-kg car is traveling with a speed of (22 ± 0.22) m/s. With what maximum accuracy can its position be determined?

Problem 56GP Protons are accelerated from rest across 550 V. They are then directed at two slits 0.70 mm apart. How far apart will the interference peaks be on a screen 28 m away?

Problem 59GP If your de Broglie wavelength were 0.50 m, how fast would you be moving if your mass is 75.0 kg? Would you notice diffraction effects as you walk through a doorway? Approximately how long would it take you to walk through the doorway?