1 / 19

Getting used to quantum weirdness

This article discusses the concept of multiple dimensions in quantum mechanics and explores the calculation of eigenvalues in 2D and 3D systems. It covers the separation of variables, radial and angular node patterns, and the solutions for free electron and Coulomb electron systems.

hannaj
Download Presentation

Getting used to quantum weirdness

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Getting used to quantum weirdness http://math-fail.com/images-old/quantum1.jpg

  2. Going from 1D to 2D to 3D http://www.intechopen.com/source/html/42048/media/fig2.png

  3. Multiple Dimensions(Separation of Variables) (-ħ22/2m + U)f = Ef If U(x,y) = Ux(x) + Uy(y) Then f(x,y) = X(x).Y(y) and E = Ex + Ey [Solve two 1-D problems]

  4. Multiple Dimensions(Separation of Variables) Nx = 100, Ny=100, N = 10,000 10,000 eigenvalues But we can also solve two 1d problems involving two sets of 100 x 100 matrices But this only gives us 200 eigenvalues! Where are the rest?

  5. Multiple Dimensions(Separation of Variables) Can mix and match any 100 x eigenvalues with any 100 y eigenvalues 1 1 2 2 3 3 100 100

  6. 2-D Box fpq = 4/L2 sin(ppx/L).sin(qpy/L) Epq = ħ2p2(p2+q2)/2mL2 (p, q = 1, 2, 3, …)

  7. 2-D Box fpq = 4/L2 sin(ppx/L).sin(qpy/L) Epq = ħ2p2(p2+q2)/2mL2 (p, q = 1, 2, 3, …) f22 8E0 f21 f12 5E0 2E0 f11

  8. Square 2-D Box f22 8E0 f21 f12 5E0 This has nodes along x and y (or x+y, x-y etc) 2E0 f11

  9. Circular 2-D Box 8E0 5E0 2E0 http://ej.iop.org/images/0957-4484/24/5/055501/Full/nano447184f3_online.jpg

  10. Compare http://hyperphysics.phy-astr.gsu.edu/hbase/music/imgmus/mem8.gif

  11. Circular 2D box http://www.jeffreythompson.org/blog/wp-content/uploads/2012/01/KettleDrumTablaModeDiagram.jpg

  12. What about a circular 2D box? Angular orthogonality m Radial orthogonality This has nodes along r and q Angular nodes .. Orbitals l http://projects.kmi.open.ac.uk/role/moodle/pluginfile.php/1250/mod_page/content/1/ta212_2_020i.jpg http://www.rsc.org/ej/AN/2013/c3an36260d/c3an36260d-f8.gif

  13. Where do these nodes come from? 2F = (-2mE/ħ2)F 1/r∂/∂r(r∂F/∂r)+ 1/r2∂2F/∂j2 F = Rnm(r)Qm(j) ∂2Q/∂j2 = -m2Q, solution e±imj m integer

  14. Radial Solution for free electron 2F = (-2mE/ħ2)F r2∂2R/∂r2 + r∂R/∂r+ (r2/l2- m2 ) R = 0 • = ħ/√2mE is the de Broglie wavelength Oscillatory radial solutions Rnm(r) = Jm(r/ln) Bessel Functions

  15. Quantization Rnm(a) = Jm(a/ln) = 0 a/ln = xnm, nth root of Jm En = ħ2xnm2/2ma2 Fnm= Jm(rxnm/a)eimj

  16. Radial Solution for Coulomb electron 2F = (-2m(E-U)/ħ2)F r2∂2R/∂r2 + r∂R/∂r+ (r2/l2+ r/a0- m2) R = 0 Different boundary condition R(∞) = 0

  17. Radial Solution for Coulomb electron r2∂2R/∂r2 + r∂R/∂r+ (r2/l2+ r/a0- m2) R = 0 Near r  0, get R = rm Near r  ∞, get R = eir/l = e-r/|l| Set R = rme-r/|l| f(r) rf’’+ f’[(2m+1)-2r/l] + f’[1/a0-(2m+1)/l] = 0 Solutions are associated Laguerre polynomials This equation has an interpretation we’ll come back to later

  18. Radial Solution for Coulomb electron Try f(r) = Spcprp cp/cp-1 = [(2p+2m-1)/l-1/a0]/(p+2m) For series to be finite, need (2p+2m-1)/l-1/a0 = 0 E = -ħ2/2ml2 = -(ħ2/2ma02) x 1/(2p+2m-1)2 = 4E0/(2n-1)2 = E0/(n-1/2)2 Twice the energy of 3D Hydrogen !!

  19. http://wanna-joke.com/wp-content/uploads/2013/08/funny-pictures-schrodinger-cat.jpghttp://wanna-joke.com/wp-content/uploads/2013/08/funny-pictures-schrodinger-cat.jpg

More Related