Derive an expression in terms of l and ml for the half-angle of the apex of the cone

Discuss the physical origin of quantization energy for a particle confined to moving inside a one-dimensional box or on a ring.

Discuss the correspondence principle and provide two examples.

Define, justify, and provide examples of zero-point energy.

Discuss the physical origins of quantum mechanical tunnelling. Why is tunnelling more likely to contribute to the mechanisms of electron transfer and proton transfer processes than to mechanisms of group transfer reactions, such as AB + CA + BC (where A, B, and C are large molecular groups)?

Distinguish between a fermion and a boson. Provide examples of each type of particle.

Describe the features that stem from nanometre-scale dimensions that are not found in macroscopic objects.

Calculate the energy separations in joules, kilojoules per mole, electronvolts, and reciprocal centimetres between the levels (a) n = 2 and n = 1, (b) n = 6 and n = 5 of an electron in a box of length 1.0 nm.

Calculate the energy separations in joules, kilojoules per mole, electronvolts, and reciprocal centimetres between the levels (a) n = 3 and n = 1, (b) n = 7 and n = 6 of an electron in a box of length 1.50 nm.

Calculate the probability that a particle will be found between 0.49L and 0.51L in a box of length L when it has (a) n = 1, (b) n = 2. Take the wavefunction to be a constant in this range.

Calculate the probability that a particle will be found between 0.65L and 0.67L in a box of length L when it has (a) n = 1, (b) n = 2. Take the wavefunction to be a constant in this range.

Calculate the expectation values of p and p2 for a particle in the state n = 1 in a square-well potential.

Calculate the expectation values of p and p2 for a particle in the state n = 2 in a square-well potential.

An electron is confined to a a square well of length L. What would be the length of the box such that the zero-point energy of the electron is equal to its rest mass energy, mec 2? Express your answer in terms of the parameter C = h/mec, the Compton wavelength of the electron.

Repeat Exercise 9.4a for a general particle of mass m in a cubic box.

What are the most likely locations of a particle in a box of length L in the state n = 3?

What are the most likely locations of a particle in a box of length L in the state n = 5?

Consider a particle in a cubic box. What is the degeneracy of the level that has an energy three times that of the lowest level?

Consider a particle in a cubic box. What is the degeneracy of the level that has an energy 143 times that of the lowest level?

Calculate the percentage change in a given energy level of a particle in a cubic box when the length of the edge of the cube is decreased by 10 per cent in each direction.

A nitrogen molecule is confined in a cubic box of volume 1.00 m3. Assuming that the molecule has an energy equal to 32 kT at T = 300 K, what is the value of n = (nx 2 + ny 2 + nz 2 )1/2 for this molecule? What is the energy separation between the levels n and n + 1? What is its de Broglie wavelength? Would it be appropriate to describe this particle as behaving classically?

Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 2.33 1026 kg and force constant 155 N m1.

Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 5.16 1026 kg and force constant 285 N m1.

For a harmonic oscillator of effective mass 1.33 1025 kg, the difference in adjacent energy levels is 4.82 zJ. Calculate the force constant of the oscillator.

For a harmonic oscillator of effective mass 2.88 1025 kg, the difference in adjacent energy levels is 3.17 zJ. Calculate the force constant of the oscillator.

Calculate the wavelength of a photon needed to excite a transition between neighbouring energy levels of a harmonic oscillator of effective mass equal to that of a proton (1.0078 u) and force constant 855 N m1.

Calculate the wavelength of a photon needed to excite a transition between neighbouring energy levels of a harmonic oscillator of effective mass equal to that of an oxygen atom (15.9949 u) and force constant 544 N m1.

Refer to Exercise 9.10a and calculate the wavelength that would result from doubling the effective mass of the oscillator.

Refer to Exercise 9.10b and calculate the wavelength that would result from doubling the effective mass of the oscillator.

Calculate the minimum excitation energies of (a) a pendulum of length 1.0 m on the surface of the Earth, (b) the balance-wheel of a clockwork watch ( = 5 Hz).

Calculate the minimum excitation energies of (a) the 33 kHz quartz crystal of a watch, (b) the bond between two O atoms in O2, for which k = 1177 N m1.

Confirm that the wavefunction for the ground state of a onedimensional linear harmonic oscillator given in Table 9.1 is a solution of the Schrdinger equation for the oscillator and that its energy is 12 $.

Confirm that the wavefunction for the first excited state of a onedimensional linear harmonic oscillator given in Table 9.1 is a solution of the Schrdinger equation for the oscillator and that its energy is 32 $.

Locate the nodes of the harmonic oscillator wavefunction with v = 4.

Locate the nodes of the harmonic oscillator wavefunction with v = 5.

Assuming that the vibrations of a 35Cl2 molecule are equivalent to those of a harmonic oscillator with a force constant k = 329 N m1, what is the zeropoint energy of vibration of this molecule? The mass of a 35Cl atom is 34.9688 u.

Assuming that the vibrations of a 14N2 molecule are equivalent to those of a harmonic oscillator with a force constant k = 2293.8 N m1, what is the zero-point energy of vibration of this molecule? The mass of a 14N atom is 14.0031 u.

The wavefunction, (), for the motion of a particle in a ring is of the form = Neiml. Determine the normalization constant, N.

Confirm that wavefunctions for a particle in a ring with different values of the quantum number ml are mutually orthogonal.

A point mass rotates in a circle with l = 1, Calculate the magnitude of its angular momentum and the possible projections of the angular momentum on an arbitrary axis.

A point mass rotates in a circle with l = 2, Calculate the magnitude of its angular momentum and the possible projections of the angular momentum on an arbitrary axis.

Draw scale vector diagrams to represent the states (a) s = 12 , ms = + 12 , (b) l = 1, ml = +1, (c) l = 2, ml = 0.

Draw the vector diagram for all the permitted states of a particle with l = 6.

Calculate the second-order correction to the energy for the system described in Problem 9.6 and calculate the ground-state wavefunction. Account for the shape of the distortion caused by the perturbation. Hint. The following integrals are useful _x sin ax sin bx dx = _cos ax sin bx dx _cos ax sin bx dx= +constant

Suppose that 1.0 mol perfect gas molecules all occupy the lowest energy level of a cubic box. How much work must be done to change the volume of the box by V? Would the work be different if the molecules all occupied a state n 1? What is the relevance of this discussion to the expression for the expansion work discussed in Chapter 2? Can you identify a distinction between adiabatic and isothermal expansion?

Derive eqn 9.20a, the expression for the transmission probability.

Consider the one-dimensional space in which a particle can experience one of three potentials depending upon its position. They are: V=0 for \(-\infty<x \leq 0,0, V=V_2\) for \(0 \leq x \leq L\), and \(V=V_3\) for \(L \leq x<\infty\). The particle wavefunction is to have both a component \(\mathrm{e}^{i k_1 x}\) that is incident upon the barrier \(V_2\) and a reflected component \(\mathrm{e}^{-i k_1 x}\) in region \(1(-\infty<x \leq 0)\). In region 3 the wavefunction has only a forward component, \(\mathrm{e}^{\mathrm{i} k_3 x}\), which represents a particle that has traversed the barrier. The energy of the particle, E, is somewhere in the range of the \(V_2>E>V_3\). The transmission probability, T, is the ratio of the square modulus of the region 3 amplitude to the square modulus of the incident amplitude. (a) Base your calculation on the continuity of the amplitudes and the slope of the wavefunction at the locations of the zone boundaries and derive a general equation for T. (b) Show that the general equation for T reduces to eqn 9.20b in the high, wide barrier limit when \(V_1=V_3=0\). (c) Draw a graph of the probability of proton tunnelling when \(V_3=0, L=50 \mathrm{pm}\), and \(E=10 \mathrm{~kJ} \mathrm{~mol}^{-1}\) in the barrier range \(E<V_2<2 E\).

The wavefunction inside a long barrier of height V is = Ne x. Calculate (a) the probability that the particle is inside the barrier and (b) the average penetration depth of the particle into the barrier.

Confirm that a function of the form egx2 is a solution of the Schrdinger equation for the ground state of a harmonic oscillator and find an expression for g in terms of the mass and force constant of the oscillator.

Calculate the mean kinetic energy of a harmonic oscillator by using the relations in Table 9.1.

Calculate the values of _x3_ and _x4_ for a harmonic oscillator by using the relations in Table 9.1.

Determine the values of x = (_x2_ _x_2)1/2 and p = (_p2_ _p_2)1/2 for (a) a particle in a box of length L and (b) a harmonic oscillator. Discuss these quantities with reference to the uncertainty principle.

We shall see in Chapter 13 that the intensities of spectroscopic transitions between the vibrational states of a molecule are proportional to the square of the integral vxvdx over all space. Use the relations between Hermite polynomials given in Table 9.1 to show that the only permitted transitions are those for which v = v } 1 and evaluate the integral in these cases.

The potential energy of the rotation of one CH3 group relative to its neighbour in ethane can be expressed as V() = V0 cos 3. Show that for small displacements the motion of the group is harmonic and calculate the energy of excitation from v = 0 to v = 1. What do you expect to happen to the energy levels and wavefunctions as the excitation increases?

Show that, whatever superposition of harmonic oscillator states is used to construct a wavepacket, it is localized at the same place at the times 0, T, 2T, . . . , where T is the classical period of the oscillator.

Use the virial theorem to obtain an expression for the relation between the mean kinetic and potential energies of an electron in a hydrogen atom.

Evaluate the z-component of the angular momentum and the kinetic energy of a particle on a ring that is described by the (unnormalized) wavefunctions (a) ei, (b) e2i, (c) cos , and (d) (cos )ei + (sin )ei.

Is the Schrödinger equation for a particle on an elliptical ring of semimajor axes a and b separable? Hint. Although r varies with angle \(\varphi\), the two are related by \(r^2=a^2 \sin ^2 \phi+b^2 \cos ^2 \phi\).

Use mathematical software to construct a wavepacket of the form (,t) = ml= 0 ml,max cml ei(ml Eml t/$) Eml = ml 2 $2/2I with coefficients c of your choice (for example, all equal). Explore how the wavepacket migrates on the ring but spreads with time.

Confirm that the spherical harmonics (a) Y0,0, (b) Y2,1, and (c) Y3,+3 satisfy the Schrdinger equation for a particle free to rotate in three dimensions, and find its energy and angular momentum in each case.

Confirm that Y3,+3 is normalized to 1. (The integration required is over the surface of a sphere.)

Derive an expression in terms of l and ml for the half-angle of the apex of the cone used to represent an angular momentum according to the vector model. Evaluate the expression for an spin. Show that the minimum possible angle approaches 0 as l.

Show that the function f = cos ax cos cos cz is an eigenfunction of 2, and determine its eigenvalue.

Derive (in Cartesian coordinates) the quantum mechanical operators for the three components of angular momentum starting from the classical definition of angular momentum, l = r p. Show that any two of the components do not mutually commute, and find their commutator.

Starting from the operator lz = xpy ypx , prove that in spherical polar coordinates lz = i$/.

Show that the commutator [l2,lz] = 0, and then, without further calculation, justify the remark that [l2,lq] = 0 for all q = x, y, and z.

A particle is confined to move in a one-dimensional box of length L. (a) If the particle is classical, show that the average value of x is 12L and that the root-mean square value is L/31/2. (b) Show that for large values of n, a quantum particle approaches the classical values. This result is an example of the correspondence principle, which states that, for very large values of the quantum numbers, the predictions of quantum mechanics approach those of classical mechanics.

When-carotene is oxidized in vivo, it breaks in half and forms two molecules of retinal (vitamin A), which is a precursor to the pigment in the retina responsible for vision (Impact I14.1). The conjugated system of retinal consists of 11 C atoms and one O atom. In the ground state of retinal, each level up to n = 6 is occupied by two electrons. Assuming an average internuclear distance of 140 pm, calculate (a) the separation in energy between the ground state and the first excited state in which one electron occupies the state with n = 7, and (b) the frequency of the radiation required to produce a transition between these two states. (c) Using your results and Illustration 9.1, choose among the words in parentheses to generate a rule for the prediction of frequency shifts in the absorption spectra of linear polyenes: The absorption spectrum of a linear polyene shifts to (higher/lower) frequency as the number of conjugated atoms (increases/decreases).

Many biological electron transfer reactions, such as those associated with biological energy conversion, may be visualized as arising from electron tunnelling between protein-bound co-factors, such as cytochromes, quinones, flavins, and chlorophylls. This tunnelling occurs over distances that are often greater than 1.0 nm, with sections of protein separating electron donor from acceptor. For a specific combination of donor and acceptor, the rate of electron tunnelling is proportional to the transmission probability, with 7 nm1 (eqn 9.20). By what factor does the rate of electron tunnelling between two co-factors increase as the distance between them changes from 2.0 nm to 1.0 nm?

Carbon monoxide binds strongly to the \(\mathrm{Fe}^{2+}\) ion of the haem group of the protein myoglobin. Estimate the vibrational frequency of \(\mathrm{CO}\) bound to myoglobin by using the data in Problem 9.2 and by making the following assumptions: the atom that binds to the haem group is immobilized, the protein is infinitely more massive than either the \(\mathrm{C}\) or \(\mathrm{O}\) atom, the \(\mathrm{C}\) atom binds to the \(\mathrm{Fe}^{2+}\) ion, and binding of \(\mathrm{CO}\) to the protein does not alter the force constant of the \(\mathrm{C} \equiv \mathrm{O}\) bond.

Of the four assumptions made in Problem 9.33, the last two are questionable. Suppose that the first two assumptions are still reasonable and that you have at your disposal a supply of myoglobin, a suitable buffer in which to suspend the protein, 12C16O, 13C16O, 12C18O, 13C18O, and an infrared spectrometer. Assuming that isotopic substitution does not affect the force constant of the C.O bond, describe a set of experiments that: (a) proves which atom, C or O, binds to the haem group of myoglobin, and (b) allows for the determination of the force constant of the C.O bond for myoglobinbound carbon monoxide.

The particle on a ring is a useful model for the motion of electrons around the porphine ring (2), the conjugated macrocycle that forms the structural basis of the haem group and the chlorophylls. We may treat the group as a circular ring of radius 440 pm, with 22 electrons in the conjugated system moving along the perimeter of the ring. As in Illustration 9.1, we assume that in the ground state of the molecule quantized each state is occupied by two electrons. (a) Calculate the energy and angular momentum of an electron in the highest occupied level. (b) Calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels.

When in Chapter 19 we come to study macromolecules, such as synthetic polymers, proteins, and nucleic acids, we shall see that one conformation is that of a random coil. For a one-dimensional random coil of N units, the restoring force at small displacements and at a temperature T is F = ln where l is the length of each monomer unit and nl is the distance between the ends of the chain (see Section 19.8). Show that for small extensions (n << N) the restoring force is proportional to n and therefore the coil undergoes harmonic oscillation with force constant kT/Nl2. Suppose that the mass to use for the vibrating chain is its total mass Nm, where m is the mass of one monomer unit, and deduce the root mean square separation of the ends of the chain due to quantum fluctuations in its vibrational ground state.

The forces measured by AFM arise primarily from interactions between electrons of the stylus and on the surface. To get an idea of the magnitudes of these forces, calculate the force acting between two electrons separated by 2.0 nm. Hints. The Coulombic potential energy of a charge q1 at a distance r from another charge q2 is V = where 0 = 8.854 1012 C2 J1 m1 is the vacuum permittivity. To calculate the force between the electrons, note that F = dV/dr. q1q2 40r DEF N + n N n ABC kT 2l

Here we explore further the idea introduced in Impact I9.2 that quantum mechanical effects need to be invoked in the description of the electronic properties of metallic nanocrystals, here modelled as three-dimensional boxes. (a) Set up the Schrdinger equation for a particle of mass m in a threedimensional rectangular box with sides L1, L2, and L3. Show that the Schrdinger equation is separable. (b) Show that the wavefunction and the energy are defined by three quantum numbers. (c) Specialize the result from part (b) to an electron moving in a cubic box of side L = 5 nm and draw an energy diagram resembling Fig. 9.2 and showing the first 15 energy levels. Note that each energy level may consist of degenerate energy states. (d) Compare the energy level diagram from part (c) with the energy level diagram for an electron in a one-dimensional box of length L = 5 nm. Are the energy levels become more or less sparsely distributed in the cubic box than in the one-dimensional box?

We remarked in Impact I9.2 that the particle in a sphere is a reasonable starting point for the discussion of the electronic properties of spherical metal nanoparticles. Here, we justify eqn 9.54, which shows that the energy of an electron in a sphere is quantized. (a) The Hamiltonian for a particle free to move inside a sphere of radius R is @ = 2 Show that the Schrdinger equation is separable into radial and angular components. That is, begin by writing (r,,) = X(r)Y(,), where X(r) depends only on the distance of the particle away from the centre of the sphere, and Y(,) is a spherical harmonic. Then show that the Schrdinger equation can be separated into two equations, one for X, the radial equation, and the other for Y, the angular equation: + + X(r) = EX(r) 2Y = l(l + 1)Y You may wish to consult Further information 10.1 for additional help. (c) Consider the case l = 0. Show by differentiation that the solution of the radial equation has the form X(r) = (2R)1/2 (e) Now go on to show that the allowed energies are given by: En = This result for the energy (which is eqn 9.54 after substituting me for m) also applies when l 0.