Physics for Scientists and Engineers with Modern Physics
10th Edition
ISBN: 9781337553292
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Question
Chapter 40, Problem 31AP
(a)
To determine
Prove that, first term of Schrodinger equation reduces to kinetic energy of quantum particle multiplied by wave function for a freely moving particle.
(b)
To determine
Prove that, first term of Schrodinger equation reduces to kinetic energy of quantum particle multiplied by wave function for particle in a box.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
A proton has the wave function (Capital_Psi)(x) = Ae^(-x/L) with total energy equal to one-seventh its potential energy. What is the potential energy (in eV) of the particle if the wave function satisfies Schrodinger's equation? Here L = 3.80 nm.
For a quantum particle of mass m in the ground state of a square well with length L and infinitely high walls, the uncertainty in position is Δx ≈ L. (a) Use the uncertainty principle to estimate the uncertainty in its momentum.(b) Because the particle stays inside the box, its average momentum must be zero. Its average squared momentum is then ⟨p2⟩ ≈ (Δp)2. Estimate the energy of the particle. (c) State how the result of part (b) compares with the actual ground-state energy.
An electron is moving past the square well shown in Fig. . The electron has energy E = 3U0 . What is the ratio of the de Broglie wavelength of the electron in the region x 7 L to the wavelength for 0 6 x 6 L?
Chapter 40 Solutions
Physics for Scientists and Engineers with Modern Physics
Ch. 40.1 - Prob. 40.1QQCh. 40.2 - Prob. 40.2QQCh. 40.2 - Prob. 40.3QQCh. 40.5 - Prob. 40.4QQCh. 40 - Prob. 1PCh. 40 - Prob. 2PCh. 40 - Prob. 3PCh. 40 - Prob. 4PCh. 40 - Prob. 5PCh. 40 - Prob. 6P
Ch. 40 - Prob. 7PCh. 40 - Prob. 9PCh. 40 - Prob. 10PCh. 40 - Prob. 11PCh. 40 - Prob. 12PCh. 40 - Prob. 13PCh. 40 - Prob. 14PCh. 40 - Prob. 15PCh. 40 - Prob. 16PCh. 40 - Prob. 17PCh. 40 - Prob. 18PCh. 40 - Prob. 19PCh. 40 - Prob. 20PCh. 40 - Prob. 21PCh. 40 - Prob. 23PCh. 40 - Prob. 24PCh. 40 - Prob. 25PCh. 40 - Prob. 26PCh. 40 - Prob. 27PCh. 40 - Prob. 28PCh. 40 - Prob. 29PCh. 40 - Two particles with masses m1 and m2 are joined by...Ch. 40 - Prob. 31APCh. 40 - Prob. 32APCh. 40 - Prob. 33APCh. 40 - Prob. 34APCh. 40 - Prob. 36APCh. 40 - Prob. 37APCh. 40 - Prob. 38APCh. 40 - Prob. 39APCh. 40 - Prob. 40APCh. 40 - Prob. 41APCh. 40 - Prob. 42APCh. 40 - Prob. 44CPCh. 40 - Prob. 46CPCh. 40 - Prob. 47CP
Knowledge Booster
Similar questions
- The wave function of a quantum particle of mass m is ψ(x) = A cos (kx) + B sin (kx)where A, B, and k are constants. (a) Assuming the particle is free (U = 0), show that ψ(x) is a solution of the Schrödinger equation (as shown). (b) Find the corresponding energy E of the particle.arrow_forwardWhat is the value N so that ψ(x) = N/(a2 + x2) can give rise to a valid probability density?arrow_forwardShow that normalizing the particle-in-a-box wave function ψ_n (x)=A sin(nπx/L) gives A=√(2/L).arrow_forward
- A one-dimensional harmonic oscillator wave function is ψ = Axe-bx2(a) Show that ψ satisfies as shown. (b) Find b and the total energy E. (c) Is this wave function for the ground state or for the first excited state?arrow_forwardQUESTION 9 Consider the case of a 3-dimensional particle-in-a-box. Given: Y = V8 sin(3nx) sin(2ny) sin(4rz) What is the energy of the system? O 24h2/8m O sh?/8m O none are correct O 29h2/8m QUESTION 10 3 πχ sin sin () sin is a normalized wavefunction for the 3D particle-in-a-box. 1.5 O True O Falsearrow_forwardConsider an electron in a one-dimensional box of length L= 6 Å. The wavefunction for the particle is given as follows: Pn(x) = where n is the quantum number. Sketch the 2 and |Þ2|². Calculate the probability of finding electron in the first half of the box at n=2 level.arrow_forward
- A quantum particle in an infinitely deep square well has a wave function given by ψ2(x) = √2/L sin (2πx/L)for 0 ≤ x ≤ L and zero otherwise. (a) Determine the expectation value of x. (b) Determine the probability of finding the particle near 1/2 L by calculating the probability that the particle lies in the range 0.490L ≤ x ≤ 0.510L. (c) What If? Determine the probability of finding the particle near 1/4L bycalculating the probability that the particle lies in the range 0.240L ≤ x ≤ 0.260L. (d) Argue that the result of part (a)does not contradict the results of parts (b) and (c).arrow_forwardAngular momentum in quantum mechanics is given by L = Lxi+Lyj+Lzk with components Lx = ypz- zpy, Ly = zpx - xpz, Lz = xpy - ypx. a) Use the known commutation rules for x, y, z, px, py and pz to show that [Ly, Lz] = ihLx. b) Consider the spherical harmonic Y1, -1([theta], [phi]) = (1/2)*sqrt(3/2pi)*sin[theta]*e-i[phi], where [theta] and [phi] are the polar and azimuthal angles, respectively. -> i) Express Y1, -1 in terms of cartesian coordinates. -> ii) Show that Y1, -1 is an eigenfunction of Lz. ci) Express the wavefunction [psi]210 for the 2pz orbital of the hydrogen atom (derived in the lectures and given in the notes) in cartesian coordinates. [Note: This involves a different spherical harmonic than in (b).] ii) Based on this expression, show that this wavefunction satisfies the three-dimensional stationary Schrodinger equation of the hydrogen atom, and determine the corresponding energy. I have attached the question better formatted, as well as the information from…arrow_forwardAn electron confined to a one-dimensional box of length 0.70 nm jumps from the n = 2 level to the ground state. What is the wavelength (in nm) of the emitted photon?arrow_forward
- An electron having total energy E = 4.50 eV approaches a rectangular energy barrier with U = 5.00 eV and L = 950 pm as shown in Figure P40.21. Classically, the electron cannot pass through the barrier because E < U. Quantum-mechanically, however, the probability of tunneling is not zero.(b) To what value would the width L of the potential barrier have to be increased for the chance of an incident 4.50-eV electron tunneling through the barrierto be one in one million?arrow_forwardD4arrow_forwardA particle of mass m is moving in an infinite 1D quantum well of width L. y,(x) = J? sinx. sin nAx L (a) How much energy must be given to the particle so it can transition from the ground state to the second excited state? (b) If the particle is in the first excited state, what is the probability of finding the particle between x = and x = ;? 2.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Physics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning
Physics for Scientists and Engineers with Modern ...
Physics
ISBN:9781337553292
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning