Concept explainers
(a)
Interpretation:
Whether the given integrals involving harmonic oscillator wavefunctions is identically zero, not identically zero or indeterminate is to be determined.
Concept introduction:
In
Answer to Problem 11.22E
The given integral is identically zero.
Explanation of Solution
The wavefunction
The wavefunction
Substitute the value of
The integration of harmonic oscillator is solved using Hermite polynomials. The integral of Hermite polynomials is solved by the formula given in Table 11.2.
In the given integral, the wavefunctions are
The given integral is identically zero.
(b)
Interpretation:
Whether the given integrals involving harmonic oscillator wavefunctions is identically zero, not identically zero or indeterminate is to be determined.
Concept introduction:
In quantum mechanics, the wavefunction is given by
Answer to Problem 11.22E
The given integral is identically zero.
Explanation of Solution
The wavefunction
Substitute the value of
In the above equation,
The given integral is identically zero.
(c)
Interpretation:
Whether the given integrals involving harmonic oscillator wavefunctions is identically zero, not identically zero or indeterminate is to be determined.
Concept introduction:
In quantum mechanics, the wavefunction is given by
Answer to Problem 11.22E
The given integral is not identically zero.
Explanation of Solution
The wavefunction
Substitute the value of
In the above equation,
The given integral is not identically zero.
(d)
Interpretation:
Whether the given integrals involving harmonic oscillator wavefunctions is identically zero, not identically zero or indeterminate is to be determined.
Concept introduction:
In quantum mechanics, the wavefunction is given by
Answer to Problem 11.22E
The given integral is identically zero.
Explanation of Solution
The wavefunction
The wavefunction
Substitute the value of
The integration of harmonic oscillator is solved using Hermite polynomials. The integral of Hermite polynomials is solved by the formula given in Table 11.2.
In the given integral, the wavefunctions are
The given integral is identically zero.
(e)
Interpretation:
Whether the given integrals involving harmonic oscillator wavefunctions is identically zero, not identically zero or indeterminate is to be determined.
Concept introduction:
In quantum mechanics, the wavefunction is given by
Answer to Problem 11.22E
The given integral is not identically zero.
Explanation of Solution
The wavefunction
Substitute the value of
The integration of harmonic oscillator is solved using Hermite polynomials. The integral of Hermite polynomials is solved by the formula given in Table 11.2.
In the given integral, the wavefunctions are
The given integral is not identically zero.
(f)
Interpretation:
Whether the given integrals involving harmonic oscillator wavefunctions is identically zero, not identically zero or indeterminate is to be determined.
Concept introduction:
In quantum mechanics, the wavefunction is given by
Answer to Problem 11.22E
The given integral is indeterminate.
Explanation of Solution
The wavefunction
Substitute the value of
In the above equation,
The given integral is indeterminate.
Want to see more full solutions like this?
Chapter 11 Solutions
Physical Chemistry
- What is the physical explanation of the difference between a particle having the 3-D rotational wavefunction 3,2 and an identical particle having the wavefunction 3,2?arrow_forwardImagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In each case, give your reasons for accepting or rejecting each function. (1) Þ(x) = x²; (iv) y(x) = x 5. (ii) ¥(x) = ; (v) (x) = e-* ; (iii) µ(x) = e-x²; (vi) p(x) = sinxarrow_forwardThe ground-state wavefunction for a particle confined to a one dimensional box of length L is Ψ =(2/L)½ sin (πx/L) Suppose the box 10.0 nm long. Calculate the probability that the particle is: (a) between x = 4.95 nm and 5.05 nm (b) between 1.95 nm and 2.05 nm, (c) between x = 9.90 and 10.00 nm, (d) in the right half of the box and (e) in the central third of the box.arrow_forward
- Consider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction ψn. (a) Without evaluating any integrals, explain why ⟨x⟩ = L/2. (b) Without evaluating any integrals, explain why ⟨px⟩ = 0. (c) Derive an expression for ⟨x2⟩ (the necessary integrals will be found in the Resource section). (d) For a particle in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why <p2x> = n2h2/4L2.arrow_forward[p* L₂ da 0 HO H + →arrow_forwardDetermine the number and location of the maxima in the plot r2R*(r) R(r) for the 2s wavefunction.arrow_forward
- By considering the integral ∫02π ψ*ml ψml dϕ, where ml≠m'l, confirm that wavefunctions for a particle in a ring with different values of the quantum number ml are mutually orthogonal.arrow_forward4. Given these operators A=d/dx and B=x², can you measure the expectation values of the corresponding observables to infinite precision simultaneously?arrow_forwardP7B.1 Imagine a particle confined to move on the circumference of a circle ('a particle on a ring'), such that its position can be described by an angle & in the range 0 to 2π. Find the normalizing factor for the wavefunctions: (a) e" and (b) eim, where m, is an integer.arrow_forward
- Consider 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_forwardFunctions of the form sin(nπx/L), where n = 1, 2, 3 …, are wavefunctions in a region of length L (between x = 0 and x = L). Show that the wavefunctions with n = 1 and 2 are orthogonal; you will find the necessary integrals in the Resource section.arrow_forwardFor the system described in Exercise E7C.8(a), evaluate the expectation value of the angular momentum represented by the operator(ħ/i)d/dϕ for the case ml = +1, and then for the general case of integer ml.arrow_forward
- Physical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,Chemistry: Principles and PracticeChemistryISBN:9780534420123Author:Daniel L. Reger, Scott R. Goode, David W. Ball, Edward MercerPublisher:Cengage LearningIntroductory Chemistry: A FoundationChemistryISBN:9781337399425Author:Steven S. Zumdahl, Donald J. DeCostePublisher:Cengage Learning