A two-element bar as shown in Figure PA-12 with element lengths L, cross-sectional area A, and Young’s modulus E can be shown to have a stiffness
A large plate of stainless steel with thickness of 5 cm and thermal conductivity of k = 15W/ m-oC is subjected to an internal uniform heat generation
An aircraft cabin window of circular cross section and simple supports all around as shown in the following is made of polycarbonate with E = 0:345
For the cylindrical vessel with hemispherical ends (heads) under uniform internal pressure of intensity p ¼ 500 psi shown in Figure P9–21,
Write a computer program to solve plane stress problems using the LST element.
Use the LST element to solve Example 6.2. Compare the results.In example 6.2For a thin plate subjected to the surface traction shown in Figure
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.
Show that the normalization constant Anfor the wave functions of a particle in a rigid box has the value given in Equation 40.26.
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
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
In most metals, the atomic ions form a regular arrangement called a crystal lattice. The conduction electrons in the sea of electrons move through
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
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
Figure 40.17showed that a typical nuclear radius is 4.0 fm. As youll learn in Chapter 42, a typical energy of a neutron bound inside the
A particle of mass m has the wave functionwhen it is in an allowed energy level with E = 0.a. Draw a graph of Ï(x) versus x.b. At what
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
a. Derive an expression for the classical probability density Pclass(x) for a simple harmonic oscillator with amplitude A.b. Graph your expression
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
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
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
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
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
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
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
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.
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
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
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
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)
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)
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
Use the data fromFigure 40.24 to calculate the first three vibrational energy levels of a C = O carbon-oxygen double bond.
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
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
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
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
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
The graph in FIGURE EX40.16 shows the potential-energy function U(x) of a particle. Solution of the SchrÃ¶dinger equation finds that the
Sketch the n = 1 and n = 7 wave functions for the potential energy shown in FIGURE EX40.15.
Sketch the n = 8 wave function for the potential energy shown in FIGURE EX40.14.
Sketch the n = 4 wave function for the potential energy shown in FIGURE EX40.13.
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
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
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
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
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
a. Sketch graphs of the probability density |Ï(x)|2 for the four states in the finite potential well of Figure 40.14a. Stack them
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
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
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
Rank in order, from largest to smallest, the penetration distances Î·ato Î·cof the wave functions corresponding to the
FIGURE EX40.5 is the probability density for an electron in a rigid box. What is the electrons energy, in eV?
What is the quantum number of the particle in FIGURE Q40.4? How can you tell?
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
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?
The correspondence principle says that the average behavior of a quantum system should begin to look like the Newtonian solution in the limit that
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.
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,
Consider the electron wave functionwhere x is in nm.a. Determine the normalization constant c.b. Draw a graph of Ï(x) over the interval -5
The wave function of a particle isand zero elsewhere.a. You will learn in Chapter 40 that the wave function must be a continuous function. Assuming
The wave function of a particle iswhere b is a positive constant. Find the probability that the particle is located in the interval -b ¤
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
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
A small speck of dust with mass 1.0 Ã 10-13g has fallen into the hole shown in FIGURE P39.46 and appears to be at rest. According to the
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
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
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
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
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
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
The probability density for finding a particle at position x isand zero elsewhere.a. You will learn in Chapter 40 that the wave function must be a
Consider the electron wave functiona. Determine the normalization constant c. Your answer will be in terms of L.b. Draw a graph of Ï(x)
Consider the electron wave functionwhere x is in cm.a. Determine the normalization constant c.b. Draw a graph of Ï(x) over the interval -2
An electron that is confined to x ¥ 0 nm has the normalized wave functionwhere x is in nm.What is the probability of finding the electron
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 P39.32 shows |Ï(x)|2for the electrons in an experiment.a. Is the electron wave function normalized? Explain.b. Draw a graph of
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.
An experiment finds electrons to be uniformly distributed over the interval 0 cm ≤ x ≤ 2 cm, with no electrons falling outside this interval.a.
Consider a single-slit diffraction experiment using electrons. (Single-slit diffraction was described in Section 33.4.) Using Figure 39.5 as a model,
FIGURE P39.28 shows a pulse train. The period of the pulse train is T = 2 Ît, where Ît is the duration of each pulse. What is
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
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
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
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
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
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
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
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
FIGURE EX39.17 shows the wave function of an electron.a. What is the value of c?b. Draw a graph of |Ï(x)|2.c. What is the probability that
FIGURE EX39.16 shows the wave function of a neutron.a. What is the value of c?b. Draw a graph of |Ï(x)|2.c. What is the probability that
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