Lect18 - Lecture 18, p 1 Lecture 18: 3D Review, MRI,...

Info iconThis preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 its 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 = = + + 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....
View Full Document

Page1 / 22

Lect18 - Lecture 18, p 1 Lecture 18: 3D Review, MRI,...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online