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Exploring Quantum Mechanics: Homework and Concepts on the Hydrogen Atom

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In this homework assignment for Quantum Mechanics I, we focus on key concepts regarding the hydrogen atom, emphasizing the probability density of the electron, energy quantization as introduced by Bohr, and the significance of principal quantum numbers. Students will solve problems involving the degeneracy of energy levels, energy transitions, and characteristics of radial wave functions using associated Laguerre polynomials. Additionally, quizzes challenge understanding of spectral lines and wavelengths emitted during electron transitions, deepening comprehension of atomic structure and quantum theory fundamentals.

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Exploring Quantum Mechanics: Homework and Concepts on the Hydrogen Atom

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  1. Physics 451 Quantum mechanics I Fall 2012 Nov 7, 2012 Karine Chesnel

  2. Quantum mechanics • Homework this week: • HW #18 Friday Nov 9 by 7pm • Pb 4.10, 4.11, 4.12, 4.13

  3. Quantum mechanics The hydrogen atom What is the density of probability of the electron?

  4. Quantization of the energy Bohr 1913 Ground state: “binding energy” Quantum mechanics The hydrogen atom Principal quantum number

  5. Bohr radius Quantum mechanics The hydrogen atom

  6. Stationary states l: azimuthal quantum number m: magnetic quantum number Degeneracy of nth energy level: Quantum mechanics The hydrogen atom Energies levels n: principal quantum number

  7. Quantum mechanics Quiz 24a What is the degeneracy of the 5th energy band of the hydrogen atom? A. 5 B. 9 C. 11 D. 25 E. 50

  8. E 0 E4 E3 Energy transition Paschen E2 Balmer E1 Rydberg constant Lyman Quantum mechanics The hydrogen atom Spectroscopy Energies levels Pb 4.16 Pb 4.17

  9. Quantum mechanics Quiz 24b What is the wavelength of the electromagnetic radiation emitted by electrons transiting from the 7th to the 5th band in the hydrogen atom? A. 465 nm B. 87.5 x 10-8 m C. 4.65 mm D. 87.5 x10-7 m E. 4.65 x 10-8 m

  10. Solution to the radial equation with Coulomb’s law: Quantum mechanics The hydrogen atom Pb 4.10 4.11

  11. Equivalent to associated Laguerre polynomials Quantum mechanics The hydrogen atom Pb 4.12

  12. Radial wave functions (table 4.7) Spherical harmonics (table 4.3) Legendre polynomials OR Power series expansion with recursion formula Quantum mechanics The hydrogen atom Laguerre polynomials

  13. Edmond Laguerre • 1834 – 1886 • Adrien-Marie Legendre • 1752 – 1833 Quantum mechanics French mathematicians

  14. Step1: determine the principal quantum number n Step 5: Multiply by the spherical harmonics (tables) and obtain 2l +1 functions Ynlm for given (n,l) Quantum mechanics The hydrogen atom How to find the stationary states? Step 2: set the azimuthal quantum number l (0, 1, …n-1) Step 3: Calculate the coefficients cj in terms of c0 (from the recursion formula, at a given l and n) Step 4: Build the radial function Rnl(r) and normalize it (value of c0) (Step 6): Eventually, include the time factor:

  15. Quantum mechanics The hydrogen atom Representation of

  16. Quantum mechanics The hydrogen atom Representation of Bohr radius

  17. Pb 4.13 Most probable values Pb 4.14 Quantum mechanics The hydrogen atom Expectation values

  18. Pb 4.15 Quantum mechanics The hydrogen atom Expectation values for potential

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