Essential University Physics
4th Edition
ISBN: 9780134988566
Author: Wolfson, Richard
Publisher: Pearson Education,
expand_more
expand_more
format_list_bulleted
Question
Chapter 35, Problem 59P
(a)
To determine
Schrödinger equation for the harmonic oscillator if the potential energy is
U = m ω 2 x 2 / 2
(b)
To determine
To show: The substitution
(c)
To determine
The normalization constant for given ground state wave function.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Start by defining
1(1) = N1 sin(7r/a)
(1)
b2(x) = N2 sin(2ñr/a)
(2)
for the infinite square well. Fix N1 and N2 so that
%3D
2)
You should find that p(r) is periodic in time. That is p(x, t + T) = p(x,t). Find
that T, and draw p(x) for at t = 0, t = T/4, t = T/2, and T = 3T/4.
Let's consider a harmonic oscillator. The total energy of
this oscillator is given by E=(p²/2m) +(½)kx?.
A) For constant energy E, graph the energies in the
range E to E + dE, the allowed region in the classical
phase space (p-x plane) of the oscillator.
B) For k = 6.0 N / m, m = 3.0 kg and the maximum
amplitude of the oscillator xmax =2.3 m For the
region with energies equal to or less than E, the
oscillator number of states that can be entered D(E).
Q.3 a) Which type of molecules exhibit infra-red spectra? Calculate the vibrational
energy and force constant of 'H"F if its vibrational frequency is 3.0 x 1014 Hz.
b) Prove that the eigenvalues of a Hermitian operator are real.
Chapter 35 Solutions
Essential University Physics
Ch. 35.1 - Prob. 35.1GICh. 35.2 - Prob. 35.2GICh. 35.3 - Prob. 35.3GICh. 35.3 - Prob. 35.4GICh. 35.3 - Prob. 35.5GICh. 35.4 - Prob. 35.6GICh. 35 - Prob. 1FTDCh. 35 - Prob. 2FTDCh. 35 - Prob. 3FTDCh. 35 - Prob. 4FTD
Ch. 35 - Prob. 5FTDCh. 35 - Prob. 6FTDCh. 35 - Prob. 7FTDCh. 35 - What did Einstein mean by his re maxi, loosely...Ch. 35 - Prob. 9FTDCh. 35 - Prob. 11ECh. 35 - Prob. 12ECh. 35 - Prob. 13ECh. 35 - Prob. 14ECh. 35 - Prob. 15ECh. 35 - Prob. 16ECh. 35 - Prob. 17ECh. 35 - Prob. 18ECh. 35 - Prob. 19ECh. 35 - Prob. 20ECh. 35 - Prob. 21ECh. 35 - Prob. 22ECh. 35 - Prob. 23ECh. 35 - Prob. 24ECh. 35 - Prob. 28ECh. 35 - Prob. 29ECh. 35 - Prob. 30ECh. 35 - Prob. 31ECh. 35 - Prob. 32ECh. 35 - Prob. 33ECh. 35 - Prob. 34ECh. 35 - Prob. 35ECh. 35 - Prob. 36PCh. 35 - Prob. 37PCh. 35 - Prob. 38PCh. 35 - Prob. 39PCh. 35 - Prob. 40PCh. 35 - Prob. 41PCh. 35 - Prob. 42PCh. 35 - Prob. 43PCh. 35 - Prob. 44PCh. 35 - Prob. 45PCh. 35 - Prob. 46PCh. 35 - Prob. 47PCh. 35 - Prob. 48PCh. 35 - Prob. 49PCh. 35 - Prob. 50PCh. 35 - Prob. 51PCh. 35 - Prob. 52PCh. 35 - Prob. 53PCh. 35 - Prob. 54PCh. 35 - Prob. 55PCh. 35 - Prob. 56PCh. 35 - Prob. 57PCh. 35 - Prob. 58PCh. 35 - Prob. 59PCh. 35 - Prob. 60PCh. 35 - Prob. 61PCh. 35 - Prob. 62PPCh. 35 - Prob. 63PPCh. 35 - Prob. 64PPCh. 35 - Prob. 65PP
Knowledge Booster
Similar questions
- Given that at time t = 0 a particle’s wave function is given by ψ(x, 0) =Ax/a, if 0 ≤ x ≤ a,A(b − x)/(b − a), if a ≤ x ≤ b, with A0, Otherwise.a and b as constants, answer the following questions; a) Find the normalization constant A in terms of the constants a and b. b) Sketch ψ(x, 0) as a function of x. c) Where is the particle most likely to be found at time t = 0? d) What is the probability of finding the particle to the left of a?arrow_forwardA qubit is in state |v) = Vo|0) + ₁|1) at time t = 0. It then evolves according to the Schrödinger equation with the Hamiltonian Ĥ defined by its action on the basis vectors: Ĥ|0) = 0|0) and H|1) = E|1), where E is a constant with units of energy. a) Solve for the state of the qubit at time t.arrow_forwardProblem 2. Consider the double delta-function potential V(x) = a[8(x + a) + 8(x − a)], where a and a are positive constants. (a) Sketch this potential. (b) How many bound states does it possess? Find the allowed energies, for a = ħ²/ma and for a = ħ²/4ma, and sketch the wave functions.arrow_forward
- Consider the Schrodinger equation for a one-dimensional linear harmonic oscillator: -(hbar2/2m) * d2ψ/dx2 + (kx2/2)*ψ(x) = Eψ(x) Substitute the wavefunction ψ(x) = e-(x^2)/(ξ^2) and find ξ and E required to satisfy the Schrodinger equation. [Hint: First calculate the second derivative of ψ(x), then substitute ψ(x) and ψ′′(x). After this substitution, there will be an overall factor of e-(x^2)/(ξ^2) on both sides of the equation which canbe an canceled out. Then, gather all terms which depend on x into one coefficient multiplying x2. This coefficient must be zero because the equation must be satisfied for any x, and equating it with zero yields the expression for ξ. Finally, the remaining x-independent part of the equation determines the eigenvalue for energy E associated with this solution.]arrow_forwardA rigid rotor is in an eigenstate Y (0,0) = 15 8K sin cos 0 ei. (a) Determine the eigenvalue of 2. (b) Determine the expectation value for (L₂). (c) What is the angle between the angular momentum vector L and the z-axis for this rigid rotor? (d) Sketch this wave function in the yz plane. Be sure to label the axes correctly.arrow_forwardA function of the form e^−gx2 is a solution of the Schrodinger equation for the harmonic oscillator, provided that g is chosen correctly. In this problem you will find the correct form of g. (a) Start by substituting Ψ = e^−gx2 into the left-hand side of the Schrodinger equation for the harmonic oscillator and evaluating the second derivative. (b) You will find that in general the resulting expression is not of the form constant × Ψ, implying that Ψ is not a solution to the equation. However, by choosing the value of g such that the terms in x^2 cancel one another, a solution is obtained. Find the required form of g and hence the corresponding energy. (c) Confirm that the function so obtained is indeed the ground state of the harmonic oscillator and has the correct energy.arrow_forward
- the wave functions px and dxz are linear combinations of the spherical harmonic functions, which are eigenfunctions of the operators H, 12, Iz for rotation in three dimensions. The combinations have been chosen to yield real functions. Are these functions still eigenfunctions of lz? answer this question by applying the operator to the functionsarrow_forwardSolve the Schrödinger equation for the potential V(x) = |x| and find the eigen values.arrow_forwardConsider a classical model of an electron as a uniform solid sphere with a radius of 8.17 x 10-17 m. Imagine that this electron possesses spin angular momentum L; = Iw = h/2 because of classical rotation about the z axis. Determine the speed of a point on the equator of the electron. State how this speed compares with the speed of light. What does your result suggest about the validity of this model?arrow_forward
- For a one dimensional harmonic oscillator, a) obtain y, (x) and y, (x) wave functions b) Using dipole moment operator, d, =e.â where e is electron charge, determine whether the transition from w, (x) to w, (x) is an allowed transition or not. Hint: For an allowed transition, transition electric dipole moment integral ((w, d, w,)) must have a nonzero value.arrow_forwardA conduction electron is confined to a metal wire of length (1.46x10^1) cm. By treating the conduction electron as a particle confined to a one-dimensional box of the same length, find the energy spacing between the ground state and the first excited state. Give your answer in eV. Note: Your answer is assumed to be reduced to the highest power possible. Your Answer: x10 Answerarrow_forwardVerify that for a harmonic oscillator, average potential energy <V> < E - for the quantum number (a) v=0 and (b) v=1. Here use ħ=1, m=1, k=1. Before averaging, do not forget to normalize the wave function, i.e. to specify the normalization constant. Also obtain <T> and the <T>/<V> ratio for each case.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
University Physics Volume 3PhysicsISBN:9781938168185Author:William Moebs, Jeff SannyPublisher:OpenStax
Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Physics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
University Physics Volume 3
Physics
ISBN:9781938168185
Author:William Moebs, Jeff Sanny
Publisher:OpenStax
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning
Physics for Scientists and Engineers with Modern ...
Physics
ISBN:9781337553292
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning