Explanation Write the expression for wave function. ψ 200 = 1 32 π r 0 3 ( 2 − r r 0 ) e − r 2 r 0 Here, ψ 200 is the wave function, r is the radius of the sphere and r 0 is Bohr radius. Square both sides of the equation. | ψ 200 | 2 = 1 32 π r 0 3 ( 2 − r r 0 ) 2 e − r r 0 Write the expression for volume. d V = 4 π r 2 ( d r ) Here, d V is the volume, 4 π r 2 is the cross sectional area and d r is change in thickness. For ψ 200 to be normalized, integrate | ψ 200 | 2 d V over all the space. ∫ all space | ψ 200 | 2 d V = ∫ 0 ∞ 1 32 π r 0 3 ( 2 − r r 0 ) 2 e − r r 0 d V Here, ψ 200 is a wave function, r is the radius and r 0 is the Bohr radius. Substitute 4 π r 2 ( d r ) for d V in the above expression. ∫ all space | ψ 200 | 2 d V = ∫ 0 ∞ 1 32 π r 0 3 ( 2 − r r 0 ) 2 e − r r 0 ( 4 π r 2 d r ) (I) Let r r 0 be equal to x . r r 0 = x r = x r 0 The differential of r is, d r = r 0 d x Substitute x for r r 0 , x r 0 for r and r 0 d x for d r in equation (I). ∫ all space | ψ 200 | 2 d V = ∫ 0 ∞ 4 π 32 π r 0 3 ( 2 − x ) 2 e − x r 0 2 x 2 r 0 d x = 1 8 ∫ 0 ∞ ( 2 − x ) 2 e − x x 2 d x = 1 8 ∫ 0 ∞ ( 4 x 2 − 4 x 3 + x 4 ) e − x d x = 1 2 ∫ 0 ∞ x 2 e − x d x − 1 2 ∫ 0 ∞ x 3 e − x d x + 1 8 ∫ 0 ∞ x 4 e − x d x (II) Integrate the terms separately. Integration of first term is shown below. ∫ x 2 e − x d x = − x 2 e − x + ∫ 2 x e − x d x = − x 2 e − x + 2 ( − e − x ) ( x + 1 ) = − e − x ( x 2 + 2 x + 2 ) Integration of second term is shown below

Physics for Scientists & Engineers with Modern Physics [With Access Code]

4th Edition

ISBN: 9780321712592

Author: GIANCOLI

Publisher: Pearson College Div

See similar textbooks

Concept explainers

Quantum mechanics and hydrogen atom

Consider an electron of mass m moves with the velocity v in a hydrogen atom. If an electron is at a distance r from the proton, then the potential energy function of the electron can be written as follows:

Isotopes of Hydrogen Atoms

To understand isotopes, it's easiest to learn the simplest system. Hydrogen, the most basic nucleus, has received a great deal of attention. Several of the results seen in more complex nuclei can be seen in hydrogen isotopes. An isotope is a nucleus of t…

Mass of Hydrogen Atom

Hydrogen is one of the most fundamental elements on Earth which is colorless, odorless, and a flammable chemical substance. The representation of hydrogen in the periodic table is H. It is mostly found as a diatomic molecule as water H2O on earth. It is …

Question

Chapter 39, Problem 20P

To determine

The proof that ψ200 as given by equation 39-8 is normalized.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 39, Problem 19P

Chapter 39, Problem 21P

Students have asked these similar questions

Curve Fitter CURVE FITTER Open Update Fit Save New Exclusion Rules Select Validation Data Polynomial Exponential Logarithmic Auto Fourier Fit Fit Duplicate Data Manual FILE DATA FIT TYPE FIT Harmonic Motion X us 0.45 mi ce 0.4 0.35 0.3 0.25 0.2 Residuals Plot Contour Plot Plot Prediction Bounds None VISUALIZATION Colormap Export PREFERENCES EXPORT Fit Options COA Fourier Equation Fit Plot x vs. t -Harmonic Motion a0+ a1*cos(x*w) + b1*sin(x*w) Number of terms Center and scale 1 ▸ Advanced Options Read about fit options Results Value Lower Upper 0.15 a0 0.1586 0.1551 0.1620 a1 0.0163 0.0115 0.0211 0.1 b1 0.0011 -0.0093 0.0115 W 1.0473 0.9880 1.1066 2 8 10 t 12 14 16 18 20 Goodness of Fit Value Table of Fits SSE 0.2671 Fit State Fit name Data Harmonic Motion x vs. t Fit type fourier1 R-square 0.13345 SSE DFE 0.26712 296 Adj R-sq 0.12467 RMSE 0.030041 # Coeff Valic R-square 0.1335 4 DFE 296.0000 Adj R-sq 0.1247 RMSE 0.0300

What point on the spring or different masses should be the place to measure the displacement of the spring? For instance, should you measure to the bottom of the hanging masses?

Let's assume that the brightness of a field-emission electron gun is given by β = 4iB π² d²α² a) Assuming a gun brightness of 5x108 A/(cm²sr), if we want to have an electron beam with a semi-convergence angle of 5 milliradian and a probe current of 1 nA, What will be the effective source size? (5 points) b) For the same electron gun, plot the dependence of the probe current on the parameter (dpa) for α = 2, 5, and 10 milliradian, respectively. Hint: use nm as the unit for the electron probe size and display the three plots on the same graph. (10 points)

Chapter 39 Solutions

Physics for Scientists & Engineers with Modern Physics [With Access Code]

Ch. 39.2 - Prob. 1AE Ch. 39.2 - Prob. 1BE Ch. 39.3 - Prob. 1CE Ch. 39.4 - Prob. 1DE Ch. 39.4 - Prob. 1EE Ch. 39.5 - Prob. 1FE Ch. 39.7 - Prob. 1GE Ch. 39 - Prob. 1Q Ch. 39 - Prob. 2Q Ch. 39 - Prob. 3Q

Ch. 39 - Prob. 4Q Ch. 39 - Prob. 5Q Ch. 39 - Prob. 6Q Ch. 39 - Prob. 7Q Ch. 39 - Prob. 8Q Ch. 39 - Prob. 9Q Ch. 39 - Prob. 10Q Ch. 39 - Prob. 11Q Ch. 39 - On what factors does the periodicity of the...Ch. 39 - Prob. 13Q Ch. 39 - Prob. 14Q Ch. 39 - Prob. 15Q Ch. 39 - Prob. 16Q Ch. 39 - Prob. 17Q Ch. 39 - Prob. 18Q Ch. 39 - Prob. 19Q Ch. 39 - Prob. 20Q Ch. 39 - Prob. 21Q Ch. 39 - Prob. 22Q Ch. 39 - Prob. 23Q Ch. 39 - Prob. 24Q Ch. 39 - Prob. 25Q Ch. 39 - Prob. 26Q Ch. 39 - Prob. 27Q Ch. 39 - Prob. 28Q Ch. 39 - Prob. 29Q Ch. 39 - Prob. 1P Ch. 39 - Prob. 2P Ch. 39 - Prob. 3P Ch. 39 - Prob. 4P Ch. 39 - Prob. 5P Ch. 39 - Prob. 6P Ch. 39 - Prob. 7P Ch. 39 - Prob. 8P Ch. 39 - Prob. 9P Ch. 39 - Prob. 10P Ch. 39 - Prob. 11P Ch. 39 - Prob. 12P Ch. 39 - Prob. 13P Ch. 39 - Prob. 14P Ch. 39 - Prob. 15P Ch. 39 - Prob. 16P Ch. 39 - Prob. 17P Ch. 39 - Prob. 18P Ch. 39 - Prob. 19P Ch. 39 - Prob. 20P Ch. 39 - Prob. 21P Ch. 39 - Prob. 22P Ch. 39 - Prob. 23P Ch. 39 - Prob. 24P Ch. 39 - Prob. 25P Ch. 39 - Prob. 26P Ch. 39 - Prob. 27P Ch. 39 - Prob. 28P Ch. 39 - Prob. 29P Ch. 39 - Prob. 30P Ch. 39 - Prob. 31P Ch. 39 - Prob. 32P Ch. 39 - Prob. 33P Ch. 39 - Prob. 34P Ch. 39 - Prob. 35P Ch. 39 - Prob. 36P Ch. 39 - Prob. 37P Ch. 39 - Prob. 38P Ch. 39 - Prob. 39P Ch. 39 - Prob. 40P Ch. 39 - Prob. 41P Ch. 39 - Prob. 42P Ch. 39 - Prob. 43P Ch. 39 - Prob. 44P Ch. 39 - Prob. 45P Ch. 39 - Prob. 46P Ch. 39 - Prob. 47P Ch. 39 - Prob. 48P Ch. 39 - Prob. 49P Ch. 39 - Prob. 50P Ch. 39 - Prob. 51P Ch. 39 - Prob. 52P Ch. 39 - Prob. 53P Ch. 39 - Prob. 54P Ch. 39 - Prob. 55P Ch. 39 - Prob. 56P Ch. 39 - Prob. 57P Ch. 39 - Prob. 58P Ch. 39 - Prob. 59P Ch. 39 - Prob. 60P Ch. 39 - Prob. 61GP Ch. 39 - Prob. 62GP Ch. 39 - Prob. 63GP Ch. 39 - Prob. 64GP Ch. 39 - Prob. 65GP Ch. 39 - Prob. 66GP Ch. 39 - Prob. 67GP Ch. 39 - Prob. 68GP Ch. 39 - Prob. 69GP Ch. 39 - Prob. 70GP Ch. 39 - Prob. 71GP Ch. 39 - Prob. 72GP Ch. 39 - Prob. 73GP Ch. 39 - Prob. 74GP Ch. 39 - Prob. 75GP Ch. 39 - Prob. 76GP Ch. 39 - Prob. 77GP

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.

The proof that ψ 200 as given by equation 39 - 8 is normalized.

Solution Summary: The author explains the proof that psi is normalized by equation 39-8. Write the expression for wave function.

Concept explainers

The proof that ψ200 as given by equation 39-8 is normalized.

Want to see the full answer?

Chapter 39 Solutions