# Lect18 - Lecture 18 p 1 Lecture 18 3D Review MRI Examples A...

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Unformatted text preview: Lecture 18, p 1 Lecture 18: 3D Review, MRI, Examples A real (2D) “quantum dot” http://pages.unibas.ch/phys- meso/Pictures/pictures.html Lecture 18, p 2 Lect. 16: Particle in a 3D Box (3) The energy eigenstates and energy values in a 3D cubical box are: where n x ,n y , and n z can each have values 1,2,3,…. This problem illustrates two important points: • Three quantum numbers (n x ,n y ,n z ) are needed to identify the state of this three-dimensional system. That is true for every 3D system. • More than one state can have the same energy: “Degeneracy”. Degeneracy reflects an underlying symmetry in the problem. 3 equivalent directions, because it’s a cube, not a rectangle. y z x L L L ( 29 2 2 2 2 2 sin sin sin 8 x y z y x z n n n x y z n n n N x y z L L L h E n n n mL π π π ψ = = + + Lecture 18, p 3 Consider a particle in a 2D well, with L x = L y = L . 1. Compare the energies of the (2,2), (1,3), and (3,1) states? a. E (2,2) > E (1,3) = E (3,1) b. E (2,2) = E (1,3) = E (3,1) c. E (2,2) < E (1,3) = E (3,1) 2. If we squeeze the box in the x-direction ( i.e. , L x < L y ) compare E (1,3) with E (3,1) . a. E (1,3) < E (3,1) b. E (1,3) = E (3,1) c. E (1,3) > E (3,1) Act 1 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 Lecture 18, p 6 Radial Eigenstates of Hydrogen Here are graphs of the s-state wave functions, R no (r) , for the electron in the Coulomb potential of the proton. The zeros in the subscripts are a reminder that these are states with l = 0 ( zero angular momentum! ). 2 / 3 3,0 2 ( ) 3 2 3 r a r r R r e a a- ∝- + / 2 2,0 ( ) 1 2 r a r R r e a- ∝- / 1,0 ( ) r a R r e- ∝ r R 30 r 15a r R 20 10a R 10 4a-13.6 eV-3.4 eV-1.5 eV E The “Bohr radius” of the H atom....
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Lect18 - Lecture 18 p 1 Lecture 18 3D Review MRI Examples A...

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