Section 2-9 Summary Notes

Home Explore Exams

Daniel

@danisgreat

view

Rating

0.0(0)

Explore Top Notes

Thinking Scientifically - Skills Of Scientific Investigation (copy)

Chapter 14 Organic chemistry

Note

Studied by 28 people

5.0 Stars(1)

Chapter 20 - Collapse at the Center

Note

Studied by 80 people

5.0 Stars(1)

Chapter 7 - Atomic Structure and Periodicity

Section 2-9 Summary

Section 2-9 Summary Figure 2-18

(a) A rectangular-hill potential of height U and width 2a. (b) Fraction of beam

transmitted as a function of E/U , where U = h2/2π2ma2.

in the transmission function for E > U , with 100% transmission occurring at intervals in E instead of only in the inﬁnite limit. These come about because of interference between waves reﬂecting off the front and back edges of the barrier. This is most easily understood by recognizing that 100% transmission corresponds to no reﬂection, so then B = 0. This occurs when the wave reﬂecting back from x = −a is of opposite phase to that reﬂecting back from x = +a, and this happens whenever there is an integral number of de Broglie half-wavelengths between x = −a and a. At energies where this happens, the beam behaves as though the potential barrier is not there.

The variation of reﬂection from thin ﬁlms (e.g., soap bubbles) of light of different

wavelengths results in the perception of colors and is a familiar example of scattering interference. Less familiar is the variation in reﬂection of a particle beam, outlined above. However, once we recognize the wave nature of matter, we must expect particles to manifest the same sort of wave properties we associate with light. 2-9 Summary

In this chapter we have discussed the following points: 1. A particle constrained in the classical sense (i.e., lacking the energy to overcome

barriers preventing its motion over the entire coordinate range) will have quantized energy levels and a ﬁnite zero-point energy. In the mathematical analysis, this arises from requirements on ψ at boundaries.

Chapter 2 Quantum Mechanics of Some Simple Systems 2. ψ can be nonsmooth, or cusped, where V is inﬁnite at a point. If V is inﬁnite over

a ﬁnite range, ψ must be zero there. 3. Nondegenerate eigenfunctions of H must be symmetric or antisymmetric for any

operation that leaves H unchanged. 4. |ψ|2 may be regarded as a statistical measure—a summary of many measurements

of position on independent, but identically prepared, systems. 5. Quantum-mechanical predictions approach classical predictions in the limits of

large E, or large mass, or very high quantum number values. 6. Integrals with antisymmetric integrands must vanish. 7. |ψ|2 does not vanish in regions where V > E if V is ﬁnite. This is called “barrier

penetration.” 8. One-dimensional motion of a free particle has a continuum of energy levels. Except

for E = 0, the states are doubly degenerate. Therefore, any mixture of such a pair of states is still an eigenfunction of H . But only two eigenfunctions (for a given E = 0) are also eigenfunctions for the momentum operator. These are the exponential functions. Since they correspond to different momenta, mixing them produces functions that are not eigenfunctions for the momentum operator. 9. Motion of a particle on a ring has quantum-mechanical solutions very similar to

those for free-particle motion in one dimension. In both cases, there is no zero-point energy. Both are doubly degenerate for E > 0 because two directional possibilities are present. Both have a set of exponential solutions that are eigenfunctions for momentum. The main difference is that the particle-in-a-ring energies are quantized, due to head-to-tail “joining conditions” on ψ. 10. Increasing the dimensionality of a particle’s range of motion increases the number

of quantum numbers needed to deﬁne the wavefunctions. In cases where the hamiltonian operator can be written as a sum of operators for different coordinates (i.e., is “separable”), the problem greatly simpliﬁes; the wavefunctions become products, and the energies become sums. 11. Scattering problems are treated by selecting an energy of interest from the con

tinuum of possibilities, removing functions that describe nonphysical processes such as backscatter from the trap, and matching wave values and slopes at region boundaries. Resulting wavefunctions show wave interference effects similar to those observed for light. 2-9.A Problems 2-1. Ascertain that the expression (2-12) for energy has the proper dimensions. 2-2. Solve Eq. (2-9) for A.

√ 2-3. There is a simple way to show that A in Eq. (2-9) must equal

2/L. It involves

sketching ψ2, recognizing that sin2 x + cos2 x = 1, and asking what A must equal

in order to make the area under ψ2 equal 1. Show this for n = 1, and argue why it must give the same result for all n. 2-4. Evaluate the probability for ﬁnding a particle in the middle third of a one

dimensional box in states with n = 1, 2, 3, 104. Compare your answers with the sketches in Fig. 2-5 to see if they are reasonable. 2-5. a) Estimate the probability for ﬁnding a particle in the n = 1 state in the line

element x centered at the midpoint of a one-dimensional box if x = 0.01L. How does this compare to the classical probability?

b) Repeat the problem, but with x centered one third of the way from a box

edge. 2-6. a) Use common sense to evaluate the following integral for the particle in a

one-dimensional box, assuming that ψ is normalized.

L/5

ψ2

5 dx

0

b) How does this value compare to that for the integral over the same range, but

using ψ1 instead of ψ5? (Larger, smaller, or equal?) Use a sketch to defend your answer. 2-7. Let S and A be respectively symmetric and antisymmetric functions for the

operator R. Evaluate the following, where R operates on every function to its right: (a) RS (b) RA (c) RSS (d) RAA (e) RAS (f) RAASASSA (g) RAASASAA. Can you think of a simple general rule for telling when a product of symmetric and antisymmetric functions will be antisymmetric? 2-8. Using the concept of odd and even functions, ascertain by inspection of sketches

whether the following need be identically zero:

π sin θ cos θ dθ

0

b)

π

− sin θ cos θ dθ

π

c)

1

− x cos x dx

1

− cos y sin2 y dy

a

π sin3 θ cos2 θ dθ

0

π sin2 θ cos3 θ dθ

0

g)

1

−

x2y dx dy

1 −1

h)

π

− x sin x cos x dx

π

π sin x d sin2 x dx

0

dx

j)

π

− sin2 x d sin x dx

π

dx 2-9. Verify Eq. (2-23) for the general case n = m by explicit integration. 2-10. For the potential of Fig. 2-8, when E < U the energies are discrete, and when

E > U , they are continuous. Is there a solution with E = U ? What special requirements are there, if any, for such a solution to exist? <img src="./tmp/raw_deb0d1ce85a5c5c4d1ce09953a8c9ef8-85_1.png"/>

Chapter 2 Quantum Mechanics of Some Simple Systems Figure P2-12 2-11. Prove the following statement: any linear combination of degenerate eigenfunc

tions of H is also an eigenfunction of H . 2-12. In a few words, indicate what is wrong with the wavefunctions sketched in the

potentials shown in Fig. P2-12. If the solution appears to be acceptable, indicate this fact. 2-13. A double-well potential ranges from x = 0 to x = 2L and has a thin (width =

0.01L) rectangular barrier of ﬁnite height centered at L.

a) Sketch the wavefunction that goes with the fourth energy level in this system,

assuming that its energy is less than the height of the barrier.

b) Estimate the energy of this level for a particle of mass m. 2-14. Use the simple approach presented in Problem 2-3 to demonstrate that A =

√

1/ π for the trigonometric particle-in-a-ring eigenfunctions and 1/ 2π for the exponential eigenfunctions.

√ 2-15. Explain why (2π )−1/2 exp(i 2φ) is unacceptable as a wavefunction for the

particle in a ring.

√ 2-16. For a particle in a ring, an eigenfunction is ψ = (1/ π) cos(3φ).

a) Write down H . b) Evaluate H ψ and identify the energy. c) Is this a state for which angular momentum is a constant of motion? Demon

strate that your answer is correct.

Section 2-9 Summary 2-17. Consider two related systems—a particle in a ring of constant potential and

another just like it except for a very thin, inﬁnitely high barrier inserted at φ = 0.

When this barrier is inserted,

a) are any energies added or lost?

b) do any degeneracies change?

c) are exponential and sine–cosine forms both still acceptable for eigenfunc

tions?

d) is angular momentum still a constant of motion? 2-18. Consider a particle of mass m in a two-dimensional box having side lengths Lx

and Ly with Lx = 2Ly and V = 0 in the box, ∞ outside.

a) Write an expression for the allowed energy levels of this system. b) What is the zero-point energy? c) Calculate the energies and degeneracies for the lowest eight energy levels. d) Sketch the wavefunction for the fourth level. e) Suppose V = 10 J in the box. What effect has this on (i) the eigenvalues?

(ii) the eigenfunctions? 2-19. Consider the particle in a three-dimensional rectangular box with Lx = Ly =

Lz/2. What would be the energy when nx = 1, ny = 2, nz = 2? For nx = 1, ny = 1, nz = 4? Can you guess the meaning of the term “accidental degeneracy?” 2-20. Consider a particle of mass m in a cubical box with V = 0 at 0 < x, y, z < L.

√

a) Is 1/ 2

− ψ

an eigenfunction for this system? Explain your

5,1,1

3,3,3

reasoning.

b) Estimate the probability for ﬁnding the particle in a volume element V =

0.001V at the box center when the system is in its lowest-energy state. What is the classical value? 2-21. Kuhn [1] has suggested that the mobile π electrons in polymethine dyes can be

modeled after the one-dimensional box. Consider the symmetric carbocyanine dyes (I) where the positive charge “resonates” between the two nitrogen atoms. The zigzag polymethine path along which the π electrons are relatively free to move extends along the conjugated system between the two nitrogens. Kuhn assumed a box length L equal to this path length plus one extra bond length on each end (so that the nitrogens would not be at the very edge of the box where they would be prevented from having any π -electron charge).

This

gives L = (2n + 10)l where l is 1.39 Å, the bond length of an intermediate (i.e., between single and double) C–C bond. The number of π electrons in the polymethine region is 2n + 10. Assume that each energy level in the box is capable of holding no more than two electrons and that the electronic transition responsible for the dye color corresponds to the promotion of an electron from the highest ﬁlled to the lowest empty level, the levels having initially been ﬁlled starting with the lowest, as shown in Fig. P2-13. Calculate E and λ for the cases n = 0, 1, 2, 3 and compare with the observed values of maximum absorption of about 5750, 7150, 8180, and 9250 Å, respectively. <img src="./tmp/raw_deb0d1ce85a5c5c4d1ce09953a8c9ef8-87_1.png"/>

Chapter 2 Quantum Mechanics of Some Simple Systems Figure P2-13 2-22. Show whether momentum in the x direction is a constant of motion for a free par

ticle of mass m in states described by the following functions. In cases where it is a constant of motion, give its value. In all cases, evaluate the kinetic energy of the particle.

a) ψ = sin 3x

b) ψ = exp(3ix)

c) ψ = cos 3x

d) ψ = exp(−3ix) 2-23. Show whether angular momentum perpendicular to the plane of rotation is a

constant of motion for a particle of mass m moving in a ring of constant potential in states described by the following functions. In cases where it is a constant of motion, give its value. In all cases, evaluate the kinetic energy of the particle.

√

a) ψ = (1/ π) cos 3φ

√

b) ψ = (1/ 2π) exp(−3iφ)

√

c) ψ = (1/ π) sin 3φ

√

d) ψ = (1/ 2π) exp(3iφ) 2-24. Demonstrate that the requirement that ψ = A exp(ikt) + B exp(−ikt) have real

value and slope at a point in x sufﬁces to make |A| = |B|. 2-25. Derive relations (2-73) and (2-74) by matching wave values and slopes at the step. </body></html>

Home