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physics
physics scientists and engineers

Questions and Answers of Physics Scientists and Engineers

Use the LST element to solve Example 6.2. Compare the results. In example 6.2 For a thin plate subjected to the surface traction shown in Figure 6?16, determine the nodal displacements and the
How would you treat a linearly varying thickness for a three-noded triangle?
Compute the stiffness matrix of element 1 of the two-triangle element model of the rectangular plate in plane stress shown in the following figure. Then use it to compute the stiffness matrix of
Show that the normalization constant Anfor the wave functions of a particle in a rigid box has the value given in Equation 40.26. n= 1, 2, 3, ... A,= A,= VI
What is the probability that an electron will tunnel through a 0.45 nm gap from a metal to a STM probe if the work function is 4.0 eV?
Tennis balls traveling faster than 100 mph routinely bounce off tennis rackets. At some sufficiently high speed, however, the ball will break through the strings and keep going. The racket is a
a. What is the probability that an electron will tunnel through a 0.50 nm air gap from a metal to a STM probe if the work function is 4.0 eV?b. The probe passes over an atom that is 0.050 nm
In most metals, the atomic ions form a regular arrangement called a crystal lattice. The conduction electrons in the sea of electrons move through this lattice. FIGURE CP40.47 is a one-dimensional
In a nuclear physics experiment, a proton is fired toward a Z = 13 nucleus with the diameter and neutron energy levels shown inFigure 40.17. The nucleus, which was initially in its ground state,
A proton’s energy is 1.0 MeV below the top of a 10-fm-wide energy barrier. What is the probability that the proton will tunnel through the barrier?
Even the smoothest mirror finishes are “rough” when viewed at a scale of 100 nm. When two very smooth metals are placed in contact with each other, the actual distance between the surfaces varies
Figure 40.17showed that a typical nuclear radius is 4.0 fm. As you€™ll learn in Chapter 42, a typical energy of a neutron bound inside the nuclear potential well is En= -20 MeV. To find out
A particle of mass m has the wave function when it is in an allowed energy level with E = 0.a. Draw a graph of ψ(x) versus x.b. At what value or values of x is the particle most likely to be
a. Derive an expression for the classical probability density Pclass(y) for a ball that bounces between the ground and height h. The collisions with the ground are perfectly elastic.b. Graph your
a. Derive an expression for the classical probability density Pclass(x) for a simple harmonic oscillator with amplitude A.b. Graph your expression between x = -A and x = +A.c. Interpret your graph.
a. Determine the normalization constant A1 for the n = 1 ground-state wave function of the quantum harmonic oscillator. Your answer will be in terms of b.b. Write an expression for the probability
Show that the constant b used in the quantum-harmonic oscillator wave functions(a) Has units of length(b) Is the classical turning point of an oscillator in the n = 1 ground state.
A typical electron in a piece of metallic sodium has energy -E0 compared to a free electron, where E0 is the 2.7 eV work function of sodium.a. At what distance beyond the surface of the metal is the
For a particle in a finite potential well of width L and depth U0, what is the ratio of the probability Prob(in δx at x = L + η) to the probability Prob(in δx at x = L)?
For the quantum-well laser ofFigure 40.16, estimate the probability that an electron will be found within one of the GaAlAs layers rather than in the GaAs layer. Explain your reasoning. GAAIAS GạAs
A neutron is confined in a 10-fm-diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than 2.0 fm from the
Consider a particle in a rigid box of length L. For each of the states n = 1, n = 2, and n = 3:a. Sketch graphs of |ψ(x)|2. Label the points x = 0 and x = L.b. Where, in terms of L, are the
A particle confined in a rigid one-dimensional box of length 10 fm has an energy level En = 32.9 MeV and an adjacent energy level En+1 = 51.4 MeV.a. Determine the values of n and n + 1.b. Draw an
a. Derive an expression for λ2→1, the wavelength of light emitted by a particle in a rigid box during a quantum jump from n = 2 to n = 1.b. In what length rigid box will an electron undergoing a 2
Model an atom as an electron in a rigid box of length 0.100 nm, roughly twice the Bohr radius.a. What are the four lowest energy levels of the electron?b. Calculate all the wavelengths that would be
A 2.0-μm-diameter water droplet is moving with a speed of 1.0 mm/s in a 20@mm@long box.a. Estimate the particle’s quantum number.b. Use the correspondence principle to determine whether quantum
Suppose that ψ1(x) and ψ2(x) are both solutions to the Schrödinger equation for the same potential energy U(x). Prove that the superposition ψ(x) = Aψ1(x) + Bψ2(x) is also a solution to the
An electron approaches a 1.0-nm-wide potential-energy barrier of height 5.0 eV. What energy electron has a tunneling probability of(a) 10%,(b) 1.0%,(c) 0.10%?
Verify that the n = 1 wave function ψ1(x) of the quantum harmonic oscillator really is a solution of the Schrödinger equation. That is, show that the right and left sides of the Schrödinger
Use the data fromFigure 40.24 to calculate the first three vibrational energy levels of a C = O carbon-oxygen double bond. Transmission (%) 100- 75- 1-2 transition of a C-CH, bond 50- 1-2
An electron is confined in a harmonic potential well that has a spring constant of 12.0 N/m. What is the longest wavelength of light that the electron can absorb?
An electron in a harmonic potential well absorbs a photon with a wavelength of 400 nm as it undergoes a 1 → 2 quantum jump. What wavelength is absorbed in a 1 → 3 quantum jump?
An electron confined in a harmonic potential well emits a 1200 nm photon as it undergoes a 3 → 2 quantum jump. What is the spring constant of the potential well?
Two adjacent energy levels of an electron in a harmonic potential well are known to be 2.0 eV and 2.8 eV. What is the spring constant of the potential well?
An electron is confined in a harmonic potential well that has a spring constant of 2.0 N/m.a. What are the first three energy levels of the electron?b. What wavelength photon is emitted if the
The graph in FIGURE EX40.16 shows the potential-energy function U(x) of a particle. Solution of the Schrödinger equation finds that the n = 3 level has E3= 0.5 eV and that the n = 6 level has E6=
Sketch the n = 1 and n = 7 wave functions for the potential energy shown in FIGURE EX40.15. U(x) ↑ E, E FIGURE EX40.15
Sketch the n = 8 wave function for the potential energy shown in FIGURE EX40.14. U(x) Eg L FIGURE EX40.14 8
Sketch the n = 4 wave function for the potential energy shown in FIGURE EX40.13. U(x) E4 FIGURE EX40.13 8 8.
A helium atom is in a finite potential well. The atom’s energy is 1.0 eV below U0. What is the atom’s penetration distance into the classically forbidden region?
An electron in a finite potential well has a 1.0 nm penetration distance into the classically forbidden region. How far below U0 is the electron’s energy?
The energy of an electron in a 2.00-eV-deep potential well is 1.50 eV. At what distance into the classically forbidden region has the amplitude of the wave function decreased to 25% of its value at
An electron has a 0.0100 probability (a 1.00% chance) of tunneling through a potential barrier. If the width of the barrier is doubled, will the tunneling probability be 0.0050, 0.0025, or 0.0001?
A finite potential well has depth U0 = 2.00 eV. What is the penetration distance for an electron with energy(a) 0.50 eV,(b) 1.00 eV(c) 1.50 eV?
Four quantum particles, each with energy E, approach the potential-energy barriers seen in FIGURE Q40.8 from the left. Rank in order, from largest to smallest, the tunneling probabilities
a. Sketch graphs of the probability density |Ïˆ(x)|2 for the four states in the finite potential well of Figure 40.14a. Stack them vertically, similar to theFigure 40.14agraphs of
FIGURE Q40.7 shows two possible wave functions for an electron in a linear triatomic molecule. Which of these is a bonding orbital and which is an antibonding orbital? Explain how you can distinguish
Show that the penetration distance h has units of m.
Consider a quantum harmonic oscillator.a. What happens to the spacing between the nodes of the wave function as |x| increases? Why?b. What happens to the heights of the antinodes of the wave function
A 16-nm-long box has a thin partition that divides the box into a 4-nm-long section and a 12-nm-long section. An electron confined in the shorter section is in the n = 2 state. The partition is
Rank in order, from largest to smallest, the penetration distances ηato ηcof the wave functions corresponding to the three energy levels in FIGURE Q40.5. LELELE 16 eV -10 eV 10 eV 10 eV 5 eV 5 eV
FIGURE EX40.5 is the probability density for an electron in a rigid box. What is the electron??s energy, in eV? O nm 0.45 nm FIGURE EX40.5
What is the quantum number of the particle in FIGURE Q40.4? How can you tell? E FIGURE Q40.4
FIGURE EX40.4 shows the wave function of an electron in a rigid box. The electron energy is 12.0 eV. What is the energy, in eV, of the next higher state? (x) FIGURE EX40.4
A particle in a potential well is in the n = 5 quantum state. How many peaks are in the probability density P(x) = |ψ(x)|2?
FIGURE EX40.3 shows the wave function of an electron in a rigid box. The electron energy is 25 eV. How long is the box? (x) FIGURE EX40.3
The correspondence principle says that the average behavior of a quantum system should begin to look like the Newtonian solution in the limit that the quantum number becomes very large. What is meant
An electron in a rigid box absorbs light. The longest wavelength in the absorption spectrum is 600 nm. How long is the box?
FIGURE Q40.1 shows the de Broglie waves of three equal- mass particles. Which one is moving most slowly? Explain. a b. FIGURE Q40.1
The electrons in a rigid box emit photons of wavelength 1484 nm during the 3 → 2 transition.a. What kind of photons are they—infrared, visible, or ultraviolet?b. How long is the box in which the
Consider the electron wave function where x is in nm.a. Determine the normalization constant c.b. Draw a graph of ψ(x) over the interval -5 nm ?? x ?? 5 nm. Provide numerical scales on both axes.c.
The wave function of a particle is and zero elsewhere.a. You will learn in Chapter 40 that the wave function must be a continuous function. Assuming that to be the case, what can you conclude about
The wave function of a particle is where b is a positive constant. Find the probability that the particle is located in the interval -b ?? x ?? b. b ¥(x)= ㅠ(x2 + b?) IT
The probability density of finding a particle somewhere along the x-axis is 0 for x < 1 mm. At x = 1 mm, the probability density is c. For x ≥ 1 mm, the probability density decreases by a factor
a. Starting with the expression Δf Δt ≈ 1 for a wave packet, find an expression for the product ΔE Δt for a photon.b. Interpret your expression. What does it tell you?c. The Bohr model of
A small speck of dust with mass 1.0 × 10-13g has fallen into the hole shown in FIGURE P39.46 and appears to be at rest. According to the uncertainty principle, could this particle have enough energy
Soot particles, from incomplete combustion in diesel engines, are typically 15 nm in diameter and have a density of 1200 kg/m3. FIGURE P39.45 shows soot particles released from rest, in vacuum, just
Physicists use laser beams to create an atom trap in which atoms are confined within a spherical region of space with a diameter of about 1 mm. The scientists have been able to cool the atoms in an
Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha
You learned in Chapter 37 that, except for hydrogen, the mass of a nucleus with atomic number Z is larger than the mass of the Z protons. The additional mass was ultimately discovered to be due to
What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electron’s speed is no more than 10 m/s?
A pulse of light is created by the superposition of many waves that span the frequency range f0 - 1/2 Δf ≤ f ≤ f0 + 1/2 ≤f, where f0 = c/λ is called the center frequency of the pulse. Laser
The probability density for finding a particle at position x is and zero elsewhere.a. You will learn in Chapter 40 that the wave function must be a continuous function. Assuming that to be the case,
Consider the electron wave function a. Determine the normalization constant c. Your answer will be in terms of L.b. Draw a graph of ψ(x) over the interval -L ?? x ?? 2L.c. Draw a graph of |ψ(x)|2
Consider the electron wave functionwhere x is in cm.a. Determine the normalization constant c.b. Draw a graph of Ïˆ(x) over the interval -2 cm ‰¤ x ‰¤ 2 cm. Provide
An electron that is confined to x ?? 0 nm has the normalized wave function where x is in nm.What is the probability of finding the electron in a 0.010-nm-wide region at x = 1.0 nm? So (1.414 nm-V2
FIGURE P39.33 shows the probability density for finding a particle at position x. a. Determine the value of the constant a, as defined in the figure.b. At what value of x are you most likely to find
FIGURE P39.32 shows |ψ(x)|2for the electrons in an experiment. a. Is the electron wave function normalized? Explain.b. Draw a graph of ψ(x) over this same interval. Provide a numerical scale on
FIGURE P39.31 shows the wave function of a particle confined between x = 0 nm and x = 1.0 nm. The wave function is zero outside this region. a. Determine the value of the constant c, as defined in
An experiment finds electrons to be uniformly distributed over the interval 0 cm ≤ x ≤ 2 cm, with no electrons falling outside this interval.a. Draw a graph of |ψ(x)|2 for these electrons.b.
Consider a single-slit diffraction experiment using electrons. (Single-slit diffraction was described in Section 33.4.) Using Figure 39.5 as a model, drawa. A dot picture showing the arrival
FIGURE P39.28 shows a pulse train. The period of the pulse train is T = 2 Δt, where Δt is the duration of each pulse. What is the maximum pulse-transmission rate (pulses per second) through an
Ultrasound pulses with a frequency of 1.000 MHz are transmitted into water, where the speed of sound is 1500 m/s. The spatial length of each pulse is 12 mm.a. How many complete cycles are contained
A 1.0-mm-diameter sphere bounces back and forth between two walls at x = 0 mm and x = 100 mm. The collisions are perfectly elastic, and the sphere repeats this motion over and over with no loss of
What is the minimum uncertainty in position, in nm, of an electron whose velocity is known to be between 3.48 × 105 m/s and 3.58 ×105 m/s?
A proton is confined within an atomic nucleus of diameter 4.0 m. Use a one-dimensional model to estimate the smallest range of speeds you might find for a proton in the nucleus.
Andrea, whose mass is 50 kg, thinks she’s sitting at rest in her 5.0-m-long dorm room as she does her physics homework. Can Andrea be sure she’s at rest? If not, within what range is her velocity
A thin solid barrier in the xy-plane has a 10-μm-diameter circular hole. An electron traveling in the z-direction with vx = 0 m/s passes through the hole. Afterward, is it certain that vx is still
What minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a 1.00 MHz oscillation?
A radio-frequency amplifier is designed to amplify signals in the frequency range 80 MHz to 120 MHz. What is the shortest-duration radio-frequency pulse that can be amplified without distortion?
A 1.5-μm-wavelength laser pulse is transmitted through a 2.0-GHz-bandwidth optical fiber. How many oscillations are in the shortest-duration laser pulse that can travel through the fiber?
Sound waves of 498 Hz and 502 Hz are superimposed at a temperature where the speed of sound in air is 340 m/s. What is the length Δx of one wave packet?
FIGURE EX39.17 shows the wave function of an electron. a. What is the value of c?b. Draw a graph of |ψ(x)|2.c. What is the probability that the electron is located between x = -1.0 nm and x = 1.0
FIGURE EX39.16 shows the wave function of a neutron.a. What is the value of c?b. Draw a graph of |Ïˆ(x)|2.c. What is the probability that the neutron is located between x = -1.0 mm and x =
FIGURE EX39.15 is a graph of |ψ(x)|2for a neutron. a. What is the value of a?b. Draw a graph of the wave function ψ(x). (There is more than one acceptable answer.)c. What is the probability that
FIGURE EX39.14 is a graph of |ψ(x)|2for an electron. a. What is the value of a?b. Draw a graph of the wave function ψ(x). (There is more than one acceptable answer.)c. What is the probability that
FIGURE EX39.13 shows the probability density for an electron that has passed through an experimental apparatus. What is the probability that the electron will land in a 0.010-mm-wide strip at (a) X =
In an interference experiment with electrons, you find the most intense fringe is at x = 7.0 cm. There are slightly weaker fringes at x = 6.0 and 8.0 cm, still weaker fringes at x = 4.0 and 10.0 cm,
What are the units of ψ? Explain.
When 5 × 1012 photons pass through an experimental apparatus, 2.0 × 109 land in a 0.10-mm-wide strip. What is the probability density at this point?

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