Concept explainers
Interpretation:
The value of
Concept introduction:
In
Answer to Problem 11.19E
The value of
Explanation of Solution
The general wavefunction of harmonic oscillator is expressed as,
Substitute the value of
Substitute the value of
The value of
Substitute the value of
From Table 11.1, substitute the value of
The above equation show that
The integration of odd function going from
Therefore, equation (3) becomes,
Substitute the value of
From Table 11.1, substitute the value of
The above equation shows that
The integration of odd function going from
Therefore, equation (3) becomes,
Hence, the value of
The average momentum of harmonic oscillator is zero because the movement of mass takes place back and forth in both directions and momentum is a vector quantity.
The value of
Want to see more full solutions like this?
Chapter 11 Solutions
Physical Chemistry
- 4. Given these operators A=d/dx and B=x², can you measure the expectation values of the corresponding observables to infinite precision simultaneously?arrow_forwardConsider a 1D particle in a box confined between a = 0 and x = 3. The Hamiltonian for the particle inside the box is simply given by Ĥ . Consider the following normalized wavefunction 2m dz² ¥(2) = 35 (x³ – 9x). Find the expectation value for the energy of the particle inside the box. Give your 5832 final answer for the expectation value in units of (NOTE: h, not hbar!). In your work, compare the expectation value to the lowest energy state of the 1D particle in a box and comment on how the expectation value you calculated for the wavefunction ¥(x) is an example of the variational principle.arrow_forwardWithout evaluating any integrals, state the value of the expectation value of x for a particle in a box of length L for the case where the wavefunction has n = 2. Explain how you arrived at your answer.arrow_forward
- Determine the normalisation constant A for the ground-state wave function of an electron in a hydrogen atom at distance x from the nucleus ψ(x)=Axe-bx.arrow_forwardThe wavefunction for the motion of a particle on a ring is of the form ψ = Neimlϕ. Evaluate the normalization constant, N.arrow_forwardYou have a harmonic oscillator of mass 1.73 x 10-27 kg. There is an energy difference in adjacent energy levels that is measured to be 2.55 x 10-20 J. Calculate the force constant of the oscillator.arrow_forward
- Imagine 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_forwardCalculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 2.33 × 10−26 kg and force constant 155 N m−1.arrow_forward6. Calculate the expectation value of the radius (r) at which you would find the electron if the H atom wavefunction| is P1,0,0(r).arrow_forward
- Part A In normalizing wave functions, the integration is over all space in which the wave function is defined. Normalize the wave function x(a − x)y(b − y) over the range 0 ≤ x ≤ a, 0 ≤ y ≤ b. The element of area in two-dimensional Cartesian coordinates is dx dy; a and b are constants. Match the items in the left column to the appropriate blanks in the equations on the right. Make certain each equation is complete before submitting your answer. -19 6 30√√ 6 a5f5 [y(b - y)] [x(a − x)] [x(a − x)]² a 30 a³f³ 30 [y(by)]² N² 0 b 0 N || a dx dx || dy = 0 b dy = 1 Reset Helparrow_forwardFor an electron having a one-dimensional wavefunction Y = √2π sin x in the range x = 0 to 1, what is the probability that the electron is in the range x = 0.35 to 0.75?arrow_forward2. Calculate the amount of energy required to go from the ground state to the first excited state of an electron moving in a circular orbit with a radius of 100 pm. 3. The wavefunction of a particle on a ring can also be written in terms of sines and cosines as Y (4) = eimiø = cos(m¡4) + i sin(m¡4). How many nodes exist in the real part of the wavefunction for m=2 and m=4? At what values of p are they located?arrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,Chemistry & Chemical ReactivityChemistryISBN:9781337399074Author:John C. Kotz, Paul M. Treichel, John Townsend, David TreichelPublisher:Cengage LearningChemistry & Chemical ReactivityChemistryISBN:9781133949640Author:John C. Kotz, Paul M. Treichel, John Townsend, David TreichelPublisher:Cengage Learning