Explanation Given: An hydrogen atom initially is an n = 3 , l = 1 state makes a transition to the n = 2 , l = 0 , j = 1 2 . One photon emitted in a transition starts in the n = 3 , l = 0 , j = 3 2 and other one in the state of n = 2 , l = 0 , j = 1 2 . Write the formula for energy level of the hydrogen atom. E n = − 13.60 eV n 2 (1) Here, n is the principal quantum number. When an hydrogen atom makes the transition from the state of n = 3 to n = 2 . Substitute 3 for n in equation (1) to find E 3 . E 3 = − 13.60 eV 3 2 = − 1.511 eV Substitute 2 for n in equation (1) to find E 2 . E 2 = − 13.60 eV 2 2 = − 3.4 eV The difference in the energies for the states of n = 3 , n = 2 is Δ E . Δ E = E 3 − E 2 Substitute − 1.511 eV for E 3 and − 3.4 eV for E 2 in above equation. Δ E = − 1.511 eV − ( − 3.4 eV ) = 1.889 eV Write the formula to find the wavelength of the photon. λ = h c E photon Here, h is the Plank’s constant, h = 4.1356 × 10 − 15 eV ⋅ s . c is the velocity of the light in vacuum, c = 3 × 10 8 m/s . E photon is the energy of the photon. Here, Δ E = E photon . Substitute 4.1356 × 10 − 15 eV ⋅ s for h , 3 × 10 8 m/s for c and 1.889 eV for E in above equation to find λ . λ = ( 4.1356 × 10 − 15 eV ⋅ s ) ( 3 × 10 8 m/s ) 1.889 eV = 656.79 × 10 − 9 m × 1 nm 10 − 9 m = 656.79 nm Write the formula for the energy of the photon including orbital and spin angular momenta . E n , j = − 13.60 eV n 2 [ 1 + α 2 n 2 ( n j + 1 2 − 3 4 ) ] (2) Here, α is the fine-structure constant, α = 7.2974 × 10 − 3 . j is the total angular momentum quantum number. When the electron is emitted from the state of n = 3 , l = 0 , j = 3 2 , Substitute 3 for n , 7

University Physics with Modern Physics (14th Edition)

14th Edition

ISBN: 9780321973610

Author: Hugh D. Young, Roger A. Freedman

Publisher: PEARSON

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Chapter 41, Problem 41.64P

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The difference in wavelength between two photons and which photon has the longer wavelength.

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Chapter 41, Problem 41.63P

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University Physics with Modern Physics (14th Edition)

Ch. 41.1 - Prob. 41.1TYU Ch. 41.2 - Prob. 41.2TYU Ch. 41.3 - Prob. 41.3TYU Ch. 41.4 - In this section we assumed that the magnetic field...Ch. 41.5 - In which of the following situations is the...Ch. 41.6 - Prob. 41.6TYU Ch. 41.7 - Prob. 41.7TYU Ch. 41.8 - Prob. 41.8TYU Ch. 41 - Prob. 41.1DQ Ch. 41 - Prob. 41.2DQ

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