# Lect18 - Le cture18 3D Re w MRI Exam s vie ple A real(2D...

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Lecture 18, p 1 Lecture18: 3D Review, MRI, Examples A real (2D) “quantum dot” http://pages.unibas.ch/phys- meso/Pictures/pictures.html

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Lecture 18, p 2 D=1 6E o (2,1,1) (1,2,1) (1,1,2) D=3 (1,1,1) 3E o (n x ,n y n z ) Cubical Box Exercise Consider a 3D cubic box: Show energies and label (n x ,n y ,n z ) for the first 11 states of the particle in the 3D box, and write the degeneracy, D, for each allowed energy. Define E o = h 2 /8mL 2 . E y z x L L L Degeneracy
Lecture 18, p 3 For a cubical box, we just saw that the 5 th energy level is at 12 E 0 , with a degeneracy of 1 and quantum numbers (2,2,2). 1. What is the energy of the next energy level? a. 13E 0 b. 14E 0 c. 15E 0 2. What is the degeneracy of this energy level? a. 2 b. 4 c. 6 Act 1

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Lecture 18, p 4
Lecture 18, p 5 E 3E o 6E o 9E o 11E o Consider a non-cubic box: The box is stretched along the y-direction. What will happen to the energy levels? Define E o = h 2 /8mL 1 2 z x y L 1 L 2 > L 1 L 1 Non-cubic Box

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Lecture 18, p 6 What is the normalization constant for the hydrogen atom ground state? / 100 10 ( , , ) ( ) o r a r NR r Ne ψ θ φ - = = 5 r 0 4a 0 R 10 Wave Function Normalization
Lecture 18, p 7 5 0 0 ψ (r) = R 10 (r) / ( ) o r a r Ne - ψ = 4a 0 r s Estimate the probability of finding the electron within a small sphere of radius r s = 0.2 a o at the origin.

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