The wave functions of 58, as well as their derivatives, need to be continuous at x = L

Explain qualitatively why a particle confined to a finite region cannot have zero energy

Does quantum tunneling violate energy conservation? Explain.

Bohrs correspondence principle states that quantum and classical mechanics must agree in a certain limit. Give an example

The ground-state wave function for a quantum harmonic oscillator has a single central peak. Why is this at odds with classical physics?

Whats the essential difference between the energy-level structures of infinite and finite square wells?

In terms of de Broglies matter-wave hypothesis, how does making the sides of a box different lengths remove the degeneracy associated with a particle confined to that box?

A particle is confined to a two-dimensional box whose sides are in the ratio 1:2. Are any of its energy levels degenerate? If so, give an example. If not, why not?

What did Einstein mean by his remark, loosely paraphrased, that God does not play dice?

Some philosophers argue that the strict determinism of classical physics is inconsistent with human free will, but that the indeterminacy of quantum mechanics does leave room for free will. Others claim that physics has no bearing on the question of free will. What do you think?

What fundamental principle of physics, when combined with quantum mechanics, provides a theoretical basis for the existence of antimatter?

Figure 35.19 shows an infinite square well with a step-like potential at the bottom. Sketch qualitatively what you think a wave function might look like for a particle whose energy is (a) less than the step height and (b) greater than the step height. U x Figure 35.19 Question 11

What are the units of the wave function c1x2 in a one-dimensional situation?

A particles wave function is c = Ae-x2 /a2 , where A and a are constants. (a) Where is the particle most likely to be found? (b) Where is the probability per unit length half its maximum value?

The solution to the Schrdinger equation for a particular potential is c = 0 for  x  7 a and c = A sin1px/a2 for -a x a,

Whats the quantum number for a particle in an infinite square well if the particles energy is 25 times the ground-state energy?

A particle in an infinite square well makes a transition from a higher to a lower energy state; the corresponding energy decrease is 33 times the ground-state energy. Find the quantum numbers of the initial and final states.

Determine the ground-state energy for an electron in an infinite square well of width 10.0 nm.

Find the width of a square well in which a protons first excited state has energy 1.5 keV

A carbon nanotube traps an electron in a hollow cylindrical structure 0.48 nm in diameter. Approximating the nanotube as a onedimensional infinite square well, find the energies in eV of (a) the ground state and (b) the first excited state.

One reason we dont notice quantum effects in everyday life is that Plancks constant h is so small. Treating yourself as a particle (mass 60 kg) in a room-sized one-dimensional infinite square well (width 2.6 m), how big would h have to be if your minimum possible energy corresponded to a speed of 1.0 m/s?

A particle is confined to a 1.0-nm-wide infinite square well. If the energy difference between the ground state and the first excited state is 1.13 eV, is the particle an electron or a proton?

A 3-g snail crawls at 0.5 mm/s between two rocks 15 cm apart. Treating this system as an infinite square well, determine the approximate quantum number. Does the correspondence principle permit the use of the classical approximation in this case?

An alpha particle (mass 4 u) is trapped in a uranium nucleus with diameter 15 fm. Treating the system as a one-dimensional square well, what would be the minimum energy for the alpha particle?

A quantum harmonic oscillator has ground-state energy 0.14 eV. What would be the systems classical oscillation frequency f ?

Find the ground-state energy for a particle in a harmonic oscillator potential whose classical angular frequency v is 1.0*1017 s -1 .

A harmonic oscillator emits a 1.1-eV photon as it undergoes a transition between adjacent states. Find its classical oscillation frequency f.

The ground-state energy of a harmonic oscillator is 4.0 eV. Find the energy separation between adjacent quantum states.

Your roommate is taking Newtonian physics, while youve moved on to quantum mechanics. He claims that QM cant be right, because he didnt see any evidence of quantized energy levels in a massspring harmonic oscillator experiment. You reply by calculating the spacing between energy levels of this system, which consists of a 1-g mass on a spring with k = 80 N/m. What is that spacing, and how does this help your argument?

If all sides of a cubical box are doubled, what happens to the ground-state energy of a particle in that box?

A very crude model for an atomic nucleus is a cubical box 1 fm on a side. What would be the energy of a gamma ray emitted if a proton in such a nucleus made a transition from its first excited state to the ground state?

An electron is confined to a cubical box. For what box width will a transition from the first excited state to the ground state result in emission of a 950-nm infrared photon?

Find an expression for the normalization constant A for the wave function given by c = 0 for  x  7 b and c = A(b2 - x 2 ) for -b x b.

Suppose c1 and c2 are solutions of the Schrdinger equation for the same energy E. Show that the linear combination ac1 + bc2 is also a solution, where a and b are arbitrary constants.

An electron is trapped in an infinite square well 25 nm wide. Find the wavelengths of the photons emitted in these transitions: (a) n = 2 to n = 1; (b) n = 20 to n = 19; (c) n = 100 to n = 1.

An electron drops from the n = 7 to the n = 6 level of an infinite square well 1.5 nm wide. Find (a) the energy and (b) the wavelength of the photon emitted.

Show explicitly that the difference between adjacent energy levels in an infinite square well becomes arbitrarily small compared with the energy of the upper level, in the limit of large quantum number n.

An electron is in a narrow molecule 4.4 nm long, a situation that approximates a one-dimensional infinite square well. If the electron is in its ground state, what is the maximum wavelength of electromagnetic radiation that can cause a transition to an excited state?

The ground-state energy for an electron in infinite square well A is equal to the energy of the first excited state for an electron in well B. How do the wells widths compare?

Electrons in an ensemble of 0.834-nm-wide square wells are all initially in the n = 4 state. (a) How many different wavelengths of spectral lines could be emitted as the electrons cascade to the ground state through all possible downward transitions? (b) Find those wavelengths. (c) What regions of the electromagnetic spectrum do these spectral lines encompass?

Sketch the probability density for the n = 2 state of an infinite square well extending from x = 0 to x = L, and determine where the particle is most likely to be found.

An infinite square well extends from -L/2 to L/2. (a) Find expressions for the normalized wave functions for a particle of mass m in this well, giving separate expressions for even and odd quantum numbers. (b) Find the corresponding energy levels

A particle is in the ground state of an infinite square well. Whats the probability of finding the particle in the left-hand third of the well?

A laser emits 1.96-eV photons. If this emission is due to electron transitions from the n = 2 to n = 1 states of an infinite square well, whats the well width?

Whats the probability of finding a particle in the central 80% of an infinite square well, assuming its in the ground state?

Is quantization significant for macromolecules confined to biological cells? To find out, consider a protein of mass 250,000 u confined to a 10@m-diameter cell. Treating this as a particle in a one-dimensional square well, find the energy difference between the ground state and the first excited state. Given that biochemical reactions typically involve energies on the order of 1 eV, what do you conclude about the role of quantization?

In your physical chemistry course, you model hydrogen chloride as a hydrogen atom on a spring; the other end of the spring is attached to a rigid wall (the massive chlorine atom). In order to determine the spring constant in your model, you measure the

A particle detector has a resolution 15% of the width of an infinite square well. Whats the probability that the detector will find a particle in the ground state of the square well if the detector is centered on (a) the midpoint of the well and (b) a point onefourth of the way across the well?

Find the probability that a particle in an infinite square well is located in the central one-fourth of the well for the quantum states n = (a) 1, (b) 2, (c) 5, and (d) 20. (e) Whats the classical probability in this situation?

A particle of mass m is in a region where its total energy E is less than its potential energy U. Show that the Schrdinger equation has nonzero solutions of the form Ae { 12m1U-E2x/U . Such solutions describe the wave function in quantum tunneling, beyond the turning points in a quantum harmonic oscillator, or beyond the well edges in a finite potential well.

(a) Use Equation 35.8 to draw an energy-level diagram for the first six energy levels of a particle in a cubical box, in terms of h2 /8mL2 , and (b) give the degeneracy of each.

The generalization of the Schrdinger equation to three dimensions is - U2 2m a 0 2 c 0 x 2 + 0 2 c 0 y2 + 0 2 c 0z2 b + U1x, y, z2c = Ec (a) For a particle confined to the cubical region 0 x L, 0 y L, 0 z L, show by direct substitution that the equation is satisfied by wave functions of the form c(x, y, z) = A sin(nxpx/L) sin(nypy/L) sin(nzpz/L), where the ns are integers and A is a constant. (b) In the process of working part (a), verify that the energies E are given by Equation 35.8.

A 9-W laser beam shines on an ensemble of 1024 electrons, each in the ground state of a one-dimensional infinite square well 0.72 nm wide. The photon energy is just high enough to raise an electron to its first excited state. How many electrons can be excited if the beam shines for 10 ms?

A large number of electrons are confined to infinite square wells 1.2 nm wide. Theyre undergoing transitions among all possible states. How many visible lines (400 nm to 700 nm) are in the spectrum emitted by this ensemble of square-well systems?

A particle is in the nth quantum state of an infinite square well. (a) Show that the probability of finding it in the left-hand quarter of the well is P = 1 4 - sin1np/22 2np (b) Show that for odd n, the probability approaches the classical value 1 4 as n S .

(a) Using the potential energy U = 1 2mv2 x 2 discussed on page 675, develop the Schrdinger equation for the harmonic oscillator. (b) Show by substitution that c0(x) = A0e-a2 x2 /2 satisfies your equation, where a2 = mv/U and the energy is given by Equation 35.7 with n = 0. (c) Find the normalization constant A0. You then have the ground-state wave function for the harmonic oscillator

Youre trying to convince a friend that nuclear energy represents a much more concentrated energy source than fossil fuels, whose combustion involves rearranging atomic electrons. For a rough comparison, you calculate the ground-state energy of a proton

The table below lists the wavelengths emitted as electrons in identical square-well potentials drop from various states n to the ground state. Determine a quantity that, when you plot l against it, should yield a straight line. Make your plot, establish a best-fit line, and use your line to determine the width of the square well. Initial state, n 4 5 7 8 10 Wavelength, l (nm) 1110 674 354 281 169

The next three problems solve the Schrdinger equation for finite square wells like that shown in Fig. 35.14. Its convenient to work in dimensionless forms of the particle energy E and well depth U0, given respectively by P = 2mL2 E/U2 and m = 2mL2 U0/U2 . Assuming that E 6 U0, or, equivalently, P 6 m, show by substitution that the following wave functions satisfy the Schrdinger equation in the regions indicated: c1 = A sin12Px/L2, 0 x L c2 = Be -2m- Px/L , x L where A and B are constants

The wave functions of Problem 58, as well as their derivatives, need to be continuous at x = L if these functions are to represent the quantum state of a particle in the finite square well. (a) Show that these conditions lead to two equations: A sin12P2 = Be -2- P 2PA cos12P2 = - 2 - PBe -2- P (b) then show that these lead to the single equation tan12P2 = - A P - P

Solve the final equation of Problem 59 to find all possible values of P for (a) m = 2, (b) m = 20, and (c) m = 50. Youll need to use a numerical root-finding routine on a calculator or computer. The number of solutions may vary with m, and its possible that there are no solutions for some values of m.

If a qdots size is decreased, what happens to the wavelength of the photon emitted in a transition from the dots first excited state to the ground state? a. The wavelength increases. b. The wavelength decreases. c. The wavelength is unchanged

If the dot behaves as a perfectly cubical 3-D square well, the first excited state is a. nondegenerate. b. twofold degenerate. c. threefold degenerate. d. You cant tell without knowing the energy.

If the dot behaves as a perfectly cubical 3-D square well, the ground state is a. nondegenerate. b. twofold degenerate. c. threefold degenerate. d. You cant tell without knowing the energy

If all three sides of a qdot are halved, its ground-state energy a. is halved. b. drops to one-fourth its original value. c. doubles. d. quadruples