Concept explainers
For an object having mass
Interpretation:
The Lagrangian function L for the given one dimensional motion is to be determined and Lagrangian equation of motion is to be stated.
Concept introduction:
The Lagrangian function is the formulation of the classical mechanics. According to mechanics, the Lagrangian function is the difference of kinetic energy and potential energy which is expressed as functions of position and velocity.
Answer to Problem 9.1E
The Lagrangian function L for the given one dimensional motion in the z direction is,
The Lagrange equation of motion for this system is,
Explanation of Solution
It is given that an object of mass
The Lagrangian function L for the given one dimensional motion in the z direction is,
Where,
•
•
•
Substitute the values of kinetic energy and potential energy in the equation (1) as shown below.
Newton’s second law of motion can be represented in the form of Lagrange’s equation of motion as shown below.
The Lagrange equation of motion for this system is,
The partial derivative of L with respect to first derivative of z and partial derivative of L with respect to z is shown below.
The Lagrange equation of motion can be rewritten using the partial derivative of L as,
This equation can be rearranged as shown below.
The Lagrangian function L for the given one dimensional motion in the z direction is,
The Lagrange equation of motion for this system is,
Want to see more full solutions like this?
Chapter 9 Solutions
Physical Chemistry
- Consider burning ethane gas, C2H6 in oxygen (combustion) forming CO2 and water. (a) How much energy (in J) is produced in the combustion of one molecule of ethane? (b) What is the energy of a photon of ultraviolet light with a wavelength of 12.6 nm? (c) Compare your answers for (a) and (b).arrow_forwardIndicate which of these expressions yield an eigenvalue equation, and if so indicate the eigenvalue. a ddxcos4xb d2dx2cos4x c px(sin2x3)d x(2asin2xa) e 3(4lnx2), where 3=3f ddsincos g d2d2sincosh ddtanarrow_forwardAre mathematical expressions for the following potential energies positive or negative? Explain why in each case. a The attraction between an electron and a helium nucleus b The repulsion between two protons in a nucleus c The attraction between a north and a south magnetic pole d The force of gravity between the Sun and Earth e A rock perched on the edge of a cliff with respect to the base of the cliffarrow_forward
- particle is confined to a one-dimensional box of length L. Deduce the location of the posit ions with in the box at which the particle is most likely to be found when the quantum number of the particle is (a) n = 1. (b) n = 2. and(c) n = 3.arrow_forwardImagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In eachcase, give your reasons for accepting or rejecting each function. (i) Ψ(x)=x2; (ii) Ψ(x)=1/x; (iii) Ψ(x)=e-x^2.arrow_forwardP7D.8* A particle is confined to move in a one-dimensional box of length L. If the particle is behaving classically, then it simply bounces back and forth in the box, moving with a constant speed. (a) Explain why the probability density, P(x), for the classical particle is 1/L. (Hint: What is the total probability of finding the particle in the box?) (b) Explain why the average value of x" is (x")= , P(x)x"dx . (c) By evaluating such an integral, find (x) and (x*). (d) For a quantum particle (x)=L/2 and (x*)=L (}-1/2n°n²). Compare these expressions with those you have obtained in (c), recalling that the correspondence principle states that, for very large values of the quantum numbers, the predictions of quantum mechanics approach those of classical mechanics.arrow_forward
- Vanadium tetrachloride (VCl4 ) is a bright red colored liquid with a vapor pressure of 5x10-2 torr at 298K. State the Jahn Teller Theorem and briefly discuss whether or not VCl4(g) is expected to manifest a Jahn Teller effect . What is the expected orbital angular momentum in the ground state of VCl4(g) ?arrow_forwardHydrogen atoms can move along metal surfaces to catalyze chemical reactions. Determine whether quantum mechanics is likely to be essential in describing the motions of 'Hatoms over distances of about 1.0 um (10 m), assuming a kinetic energy of about 4.0 x 1021Joule. out ofarrow_forwardConsider a fictitious one-dimensional system with one electron.The wave function for the electron, drawn below, isψ (x)= sin x from x = 0 to x = 2π. (a) Sketch the probabilitydensity, ψ2(x), from x = 0 to x = 2π. (b) At what value orvalues of x will there be the greatest probability of finding theelectron? (c) What is the probability that the electron willbe found at x = π? What is such a point in a wave functioncalled?arrow_forward
- Consider an electron in the N shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of h and in SI units. (c) What is the largest orbital momentum this electron could have in any chosen direction? Express your answers in terms of h and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of h and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the z-direction to its orbital angular momentum in the z direction? E and d no answerarrow_forward(a) For a particle in the stationary state n of a one dimensional box of length a, find the probability that the particle is in the region 0xa/4.(b) Calculate this probability for n=1,2, and 3.arrow_forward(c) The kinetic energy, Kr, of electrons emitted from a metal surface after irradiation with UV light of wavelength λ is given by: hc K₁=-=- where h is Planck's constant (6.626 x 10-4 Js), c is the speed of light in a vacuum (2.99 x 108 m s¹), and is the work function of the metal surface. In a specific experiment, light with a wavelength of 266 nm was used to irradiate a cadmium (Cd) metal surface. (i) Calculate the photon energy of the light used in the experiment, in Joules. (II) The work function for cadmium is 4.08 eV. Calculate the kinetic energy of the emitted electrons. [Note: 1 eV = 1.60 x 10-19 J.)arrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,Chemistry: The Molecular ScienceChemistryISBN:9781285199047Author:John W. Moore, Conrad L. StanitskiPublisher:Cengage LearningPrinciples of Modern ChemistryChemistryISBN:9781305079113Author:David W. Oxtoby, H. Pat Gillis, Laurie J. ButlerPublisher:Cengage Learning
- Chemistry: Principles and ReactionsChemistryISBN:9781305079373Author:William L. Masterton, Cecile N. HurleyPublisher:Cengage LearningIntroductory Chemistry: A FoundationChemistryISBN:9781337399425Author:Steven S. Zumdahl, Donald J. DeCostePublisher:Cengage Learning