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physical chemistry

Physical Chemistry 3rd edition Thomas Engel, Philip Reid - Solutions

Have a closer look at Equation (17.6) and Figure 17.4. How would Figure 17.4 change if Δk decreases for constant m? How well is the momentum known if Δk → 0?
How did Stern and Gerlach conclude that the operator “measure the z component of the magnetic moment of an Ag atom” has only two eigenfunctions with eigenvalues that have the same magnitude and opposite sign?
An electron of energy 5.0 eV approaches a step potential of height 2.0 eV. Calculate the probabilities that the electron will be reflected and transmitted.
In this problem, you will solve for the total energy eigenfunctions and eigenvalues for an electron in a finite depth box. We first go through the calculation for the box parameters used in Figure 16.1. You will then carry out the calculation for a different set of parameters. We describe the
The maximum safe current in a copper wire with a diameter of 3.0 mm is about 20 amperes. In an STM, a current of 1.0 × 10−9 A passes from the tip to the surface in a filament of diameter ~1.0 nm. Compare the current density in the copper wire with that in the STM.
Calculate the energy levels of the π-network in octatetraene, C8H10, using the particle in the box model. To calculate the box length, assume that the molecule is linear and use the values 135 and 154 pm for C=C and C-C bonds. What is the wavelength of light required to induce a transition from
Calculate the energy levels of the π-network in hexatriene, C6H8, using the particle in the box model. To calculate the box length, assume that the molecule is linear and use the values 135 and 154 pm for C=C and C - C bonds. What is the wavelength of light required to induce a transition from the
For the Ï€-network of Î²-carotene modeled using the particle in the box, the position-dependent probability density of finding 1 of the 22 electrons is given byThe quantum number n in this equation is determined by the energy level of the electron under consideration. As we saw
Semiconductors can become conductive if their temperature is raised sufficiently to populate the (empty) conduction band from the highest filled levels in the valence band. The ratio of the populations in the highest level of the conduction band to that of the lowest level in the conduction band
In this problem, you will calculate the transmission probability through the barrier illustrated in Figure 16.10. We first go through the mathematics leading to the solution. You will then carry out further calculations.The domain in which the calculation is carried out is divided into three
Why were quantum dots emitting in the near infrared region used for the surgery experiment shown in Figure 16.25?Figure 16.25 Color video 5 min post-injection NIR fluorescence 5 min post-injection Pre-injection autofluorescence 1 cm Color video NIR fluorescence
Why is atomic level resolution obtained on pentacene in the AFM mode as shown in Figure 16.18, but not in the STM mode?Figure 16.18 1.3Å A OA +1Hz -2Hz D 20A 5A -7Hz. -5Hz.
Figure 16.17 shows that atomic level resolution is only attainable in the repulsive portion of the tip€“surface potential. What does this tell you about the range of the attractive and repulsive parts of the potential?Figure 16.17 Tip Sample
The reflection probability from a step potential was calculated for E > V0in Section 16.5. Is Equation (16.18) valid for E < V0? What information can you extract from Figure 16.1 that will allow you to state the value of R for a step potential if E < V0?Figure 16.1 Vo n=5 n=4 n=3 n=2 n=1
Explain why the speed of the particle needs to be taken into account in calculating the probability for transmission over a step potential.
Explain how a quantum dot can absorb light over a range of wavelengths and emit light over a much smaller range of wavelengths.
Why must the amplitudes of the energy eigenfunctions in the finite depth box and in the adjoining barrier regions have the same value at the boundary?
Why must the amplitudes of the first derivatives of the energy eigenfunctions in the finite depth box and in the adjoining barrier regions have the same value at the boundary?
Why is it necessary to functionalize CdSe quantum dots with groups such as organic acids to make them useful in bioanalytical applications?
For CdSe quantum dots, the emission wavelength increases from 450. nm to 650. nm as the dot diameter increases from 2 to 8 nm. Calculate the band gap energy for these two particle diameters.
An STM can also be operated in a mode in which electrons tunnel from the surface into the tip. Use Figure 16.12 to explain how you would change the experimental setup to reverse the tunneling current.Figure 16.12 (a) (b) ev
Explain how you can use size-quantized quantum dots to create a protein with a barcode that can be read using light.
The overlap between wave functions can either be constructive or destructive, just as for waves. Can you distinguish between constructive and destructive overlap for the various energy levels in Figure 16.3?Figure 16.3 15 10 0.2 0.4 0.6 0.8 1 хlа E in units of h/8ma?
What is the advantage of using quantum dots that fluoresce in the near infrared for surgical applications?
Explain, without using equations, why tunneling is more likely for the particle with E = 3/4V0than for E = 1/4V0in Figure 16.10.Figure 16.10 Vo х E= 1/4 Vo
Explain how it is possible to create a three-dimensional electron conductor that has a continuous energy spectrum in two dimensions and a discrete energy spectrum in the third dimension.
Redraw Figure 16.7 for an insulator. (a) (b) In (c)
Why is a tunneling current not observed in an STM when the tip and the surface are 1 mm apart?
The amplitude of the wave on the right side of the barrier in Figure 16.10 is much smaller than that of the wave incident on the barrier. What happened to the €œrest of the wave€?Figure 16.10 Vo E= 1/4 Vo E=3/4Vo
Why is it necessary to apply a bias voltage between the tip and surface in a scanning tunneling microscope?
Calculate the probability that a particle in a one-dimensional box of length a is found between 0.32a and 0.35a when it is described by the following wave functions:a. √2/a sin(πx/a)b. √2/a sin(3 πx/a)What would you expect for a classical particle? Compare your results for (a) and (b) with
Use your result from P15.17 and make an energy level diagram for the first five energy levels of a square two dimensional box of edge length b. Indicate which of the energy levels are degenerate and the degeneracy of these levels.
Consider a particle in a one-dimensional box defined by V (x) = 0, a > x > 0 and V (x) = ∞, x ≥ a, x ≤ 0. Explain why each of the following un-normalized functions is or is not an acceptable wave function based on criteria such as being consistent with the boundary conditions, and with
Use the eigenfunction ψ (x) = A′e+ikx + B′e−ikx rather than ψ (x) = A sin kx + B cos kx to apply the boundary conditions for the particle in the box.a. How do the boundary conditions restrict the acceptable choices for A′ and B′ and for k?b. Do these two functions give different
Are the total energy eigenfunctions for the free particle in one dimension, ψ+ (x) A+ e +i√(2mE/h2)x and ψ−(x) A–e –i√(2mE/h2)x eigenfunctions of the one-dimensional linear momentum operator? If so, what are the eigenvalues?
The smallest observed frequency for a transition between states of an electron in a one-dimensional box is 3.0 × 1013 s−1. What is the length of the box?
Is the superposition wave function ψ (x) = √2/a[sin(nπ x/a) + sin(mπ x/a)] an eigenfunction of the total energy operator for the particle in the box?
Two wave functions are distinguishable if they lead to a different probability density. Which of the following wave functions are distinguishable from sin kx?a. (eikx − e−Ikx )/2b. eiθ sin kx, θ a constantc. cos(kx − π /2)d. I cos (kx + π/2)(sinθ + I cosθ) (√2/2 + I √2/2), θ is
Show by examining the position of the nodes that Re[A+e i(kx−ω t) ] and Re[A–e i(−kx−ω t) ] represent plane waves moving in the positive and negative x directions, respectively. The notation Re [ ] refers to the real part of the function in the brackets.
In discussing the Boltzmann distribution in Chapter 13, we used the symbols gi and gj to indicate the degeneracies of the energy levels i and j. By degeneracy, we mean the number of distinct quantum states (different quantum numbers) all of which have the same energy.a. Using your answer to Problem
What is the zero point energy and what are the energies of the lowest seven energy levels in a three-dimensional box with a = b = c? What is the degeneracy of each level?
Are the eigenfunctions of Hˆ for the particle in the one-dimensional box also eigenfunctions of the position operator x̂ ? Calculate the average value of x for the case where n = 3. Explain your result by comparing it with what you would expect for a classical particle. Repeat your calculation
Generally, the quantization of translational motion is not significant for atoms because of their mass. However, this conclusion depends on the dimensions of the space to which they are confined. Zeolites are structures with small pores that we describe by a cube with edge length 1.00 nm. Calculate
Normalize the total energy eigenfunction for the rectangular two-dimensional box,in the interval 0 ‰¤ x ‰¤ a, 0 ‰¤ y ‰¤ b. Wn», (x, y) = Nsin n„AX sın
Calculate a. The zero point energy of a He atom in a one-dimensional box of length 1.00 cm b. The ratio of the zero point energy to kBT at 300.K.
Using your result from P 15.17, how many energy levels does a particle of mass m in a two-dimensional box of edge length a have with E ≤ 29h2/8ma2 ? What is the degeneracy of each level?
Consider the contour plots of Problem P15.17.a. What are the most likely area or areas Î”x Î”y to find the particle for each of the eigenfunctions of HË† depicted in plots a€“f?b. For the one-dimensional box, the nodes are points. What form do the nodes
For a particle in a two-dimensional box, the total energy eigenfunctions are a. Obtain an expression for Enx, ny in terms of nx, ny, a, and b by substituting this wave function into the two dimensional analog of Equation (15.19).b. Contour plots of
A bowling ball has a weight of 12 lb and the length of the lane is approximately 60. feet. Treat the ball in the lane as a one-dimensional box. What quantum number corresponds to a velocity of 7.5 miles per hour?
Calculate the wavelength of the light emitted when an electron in a one-dimensional box of length 5.0 nm makes a transition from the n = 7 state to the n = 6 state.
a. Show by substitution into Equation (15.19) that the eigenfunctions of HË† for a box with lengths along the x, y, and z directions of a, b, and c, respectively, are b. Obtain an expression for Enx, ny, nz in terms of nx, ny, nz, and a, b, and c. п.лz И,лу sin sin = N sin nn,n
Show that the energy eigenvalues for the free particle, E = h2k2/2m, are consistent with the classical result E = (1/2)mv2.
It is useful to consider the result for the energy eigenvalues for the one-dimensional box En = h2n2/8ma2 n = 1, 2, 3 , . . . as a function of n, m, and a.a. By what factor do you need to change the box length to decrease the zero point energy by a factor of 50 fora fixed value of m?b. By what
Derive an equation for the probability that a particle characterized by the quantum number n is in the first 25% (0 ≤ x ≤ a/4) of an infinite depth box. Show that this probability approaches the classical limit as n → ∞.
What is the solution of the time-dependent Schrödinger equation ψ (x,t) for the total energy eigenfunction ψ (x) = √2/a sin (3πx/a) for an electron in a one-dimensional box of length 1.00 × 10−10 m? Write explicitly in terms of the parameters of the problem. Give numerical values for the
Is the function ψ (x) = A( y/b)[1 − ( y/b)] an acceptable wave function for the particle in the one-dimensional infinite depth box of length b? Calculate the normalization constant A and the expectation values 〈 y〉 and〈 y2〉.
Evaluate the normalization integral for the eigenfunctions of Hˆ for the particle in the box ψ n (x) = A sin(nπx/a) using the trigonometric identity sin2 y = (1 − cos2y)/2.
Are the eigenfunctions of Hˆ for the particle in the one-dimensional box also eigenfunctions of the momentum operator pˆx? Calculate the average value of px for the case n = 3. Repeat your calculation for n = 5 and, from these two results, suggest an expression valid for all values of n. How does
Consider a free particle moving in one dimension whose probability of moving in the positive x direction is four times that for moving in the negative x direction. Give as much information as you can about the wave function of the particle.
Suppose that the wave function for a system can be written asand that Ï†1(x), Ï†2(x), and Ï†3(x) are orthonormal eigenfunctions of the operator Ë†E kinetic with eigenvalues E1, 2E1, and 4E1 respectively.a. Verify that Ïˆ (x) is normalized.b. What
Is the superposition wave function for the free particle an eigenfunction of the momentum operator? Is it an eigenfunction of the total energy operator? Explain your result. 2mE -x- 2mE v (x) = Ae
Normalize the total energy eigenfunctions for the three-dimensional box in the interval 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c.
Calculate the expectation values < x > and (x2) for a particle in the state n = 5 moving in a one-dimensional box of length 2.50 × 10−10 m. Is (x2) = (x2)? Explain your answer.
This problem explores under what conditions the classical limit is reached for a macroscopic cubic box of edge length a. A nitrogen molecule of average translational energy 3/2kBT is confined in a cubic box of volume V = 1.250 m3 at 298 K. Use the result from Equation (15.25) for the dependence of
What are the possible results for the energy that would be obtained in a measurement on the particle in a one-dimensional box if the wave function is ψ n (x) = √2/a sin(7π x/a)?
Show that the correct energy eigenvalues for the particle in a one-dimensional box are obtained even if the total energy eigenfunctions are not normalized.
What is the relationship between the zero point energy for a H atom and a H2 molecule in a one-dimensional box?
What are the units of the probability density for the particle in a three-dimensional box?
What are the units of the probability density for the particle in a one-dimensional box?
Why is the zero point energy lower for a He atom in a box than for an electron?
Invoke wave-particle duality to address the following question: How does a particle get through a node in a wave function to get to the other side of the box?
Can the particles in a one-dimensional box, a square two-dimensional box, and a cubic three-dimensional box all have degenerate energy levels?
Why are traveling-wave solutions for the particle in the box not compatible with the boundary conditions?
What is the difference between probability and probability density?
Why are standing-wave solutions for the free particle not compatible with the classical result x = x0 + v0t?
Show that for the particle in the box total energy eigenfunctions, ψn (x) = √2/a sin (nπ x/a), ψ (x) is a continuous function at the edges of the box. Is dψ /dx a continuous function of x at the edges of the box?
Can a guitar string be in a superposition of states or is such a superposition only possible for a quantum mechanical system?
Explain using words, rather than equations, why if V (x, y, z) ≠ Vx (x) + Vy ( y) + Vz (z), the total energy eigenfunctions cannot be written in the form ψ (x, y, z) = X (x)Y( y)Z(z).
The probability density for a particle in a box is an oscillatory function even for very large energies. Explain how the classical limit of a constant probability density that is independent of position is achieved for large quantum numbers.
Why is it not possible to normalize the free-particle wave functions over the whole range of motion of the particle?
Is the probability distribution for a free particle consistent with a purely particle picture, a purely wave picture, or both?
We set the potential energy in the particle in the box equal to zero and justified it by saying that there is no absolute scale for potential energy. Is this also true for kinetic energy?
How does an expectation value for an observable differ from an average of all possible eigenvalues?
Discuss why a quantum mechanical particle in a box has zero point energy in terms of its wavelength.
Show that the three wave functions in Equation (14.11) are normalized.
In classical mechanics, the angular momentum vector L is defined by L = r Ã— p. Determine the x component of L. Substitute quantum mechanical operators for the components of r and p to prove that -iħ Z- ay ze
Is the relation Aˆ[f (x) + g(x)] = Aˆ f (x) + Aˆg(x) always obeyed? If not, give an example to support your conclusion.
Is the relation (Aˆf (x))/f (x) = Aˆ always obeyed? If not, give an example to support your conclusion.
Show that if Ïˆ n (x) and Ïˆ m(x) are solutions of the time-independent SchrÃ¶dinger equation, Î¨ (x, t) =is a solution of the time-dependent SchrÃ¶dinger equation. iE +у (х)е (к)е А
For a Hermetian operator Aˆ, ∫ψ*(x)[Aˆψ (x)] dx = ∫ψ (x)[Aˆψ (x)]*dx. Assume that Aˆ f (x) = (a + ib) f (x), where a and b are constants. Show that if Aˆ is a Hermetian operator, b = 0 so that the eigenvalues of f (x) are real.
In combining operators sequentially, it is useful to insert an arbitrary function after the operator to avoid errors. For example, if the operators Aˆ and Bˆ are x and d /dx, then AˆBˆ f (x) = x df (x)/dx. Derive the operator for kinetic energy using the classical relation for the kinetic
Which of the following functions are acceptable wave functions over the indicated interval?a. e−x2/2 − ∞ < x < ∞ b. e−ix 0 < x < 2π c. x2e−2πix 0 < x < ∞d. xe−x 0 < x < ∞
Which of the following functions are acceptable wave functions over the indicated interval?a. e−x 0 < x < ∞b. e−2πix − 100 < x < 100c. e−x − ∞ < x < ∞d. 1/x 1 < < ∞
Consider the function f (x) = sinπ x/2, − 1 < x < 1 and f (x) = −1, x ≤ −1, f (x) = 1, x ≥ 1.Graph f(x) = df/dx, and d2f/dx2 over the interval –2 < x < 2.Which if any of these functions is continuous over the interval?
Is the function 1/(1 − x)2 continuous at x = 1? Answer this question using the criteria listed in Section 14.1.
Is the function (x2ˆ’ 1)/(x ˆ’ 1) continuous at x = 1? Answer this question by evaluating f (1) and lim f(x)
Graph f (x) = |x| and its first derivative over the interval −4 ≤ x ≤ 4. Are f (x) and df (x)/dx continuous functions of x?
Graphover the interval ˆ’4 ‰¤ x ‰¤ 4 . Is f (x) a continuous function of x? Sx+1 -15xs1 except x=0 F(x) = 10 x=0
What did Einstein mean in his famous remark “Do you really think that the moon is not there when we are not looking at it”?
Which of the following functions are single-valued functions of the variable x?a. Sin 2πx/ab. e3√xc. 1 – 3 sin2xd. e2πix

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