1. 1 Practical Quantum Mechanics

2. 2 For all time-independent problems iħ∂   ∂t = (-ħ 2  2 /2m + U)  = Ĥ  Separation of variables for static potentials  (x,t) =  (x)e -iEt/ħ Ĥ  =  E , Ĥ = -ħ 2  2 /2m + U BCs : Ĥ  n = E n  n (n = 1,2,3...) E n : eigenvalues (usually fixed by BCs)  n (x): eigenvectors/stationary states Oscillating solution in time

3. 3 Finite Difference Method x n-1 xnxn x n+1  n-1 nn  n+1  = =   n-1 nn  n+1 U n-1 UnUn U n+1 =  n-1 nn  n+1 U n-1  n-1 UnnUnn U n+1  n+1 U =U = One particular mode = [U][  ]

4. 4 What about kinetic energy? x n-1 xnxn x n+1  n-1 nn  n+1  = =   n-1 nn  n+1 (d  /dx) n = (  n+1/2 –  n-1/2 )/a (d 2  /dx 2 ) n = (  n+1 +  n-1 -2  n )/a 2

5. 5 What about kinetic energy? x n-1 xnxn x n+1  n-1 nn  n+1  = =   n-1 nn  n+1 -ħ 2 /2m(d 2  /dx 2 ) n = t(2  n -  n+1 -  n-1 ) t = ħ 2 /2ma 2

6. 6 What about kinetic energy? x n-1 xnxn x n+1  n-1 nn  n+1  = =   n-1 nn  n+1 -t 2t -t  n-1 nn  n+1 T =T = -t 2t -t -ħ 2 /2m(d 2  /dx 2 ) n = t(2  n -  n+1 -  n-1 )

7. 7 What about kinetic energy? x n-1 xnxn x n+1  n-1 nn  n+1  = =   n-1 nn  n+1 [H] = [T + U]

8. 8 What next? x n-1 xnxn x n+1   n-1 nn  n+1 Now that we’ve got H matrix, we can calculate its eigenspectrum >> [V,D]=eig(H); % Find eigenspectrum >> [D,ind]=sort(real(diag(D))); % Replace eigenvalues D by sorting, with index ind >> V=V(:,ind); % Keep all rows (:) same, interchange columns acc. to sorting index (n th column of matrix V is the n th eigenvector along the x values)

9. 9 Particle in a Box Results agree with analytical results E ~ n 2 Finite wall heights, so waves seep out

10. 10 Add a field

11. 11 Or asymmetry

12. 12 Harmonic Oscillator Shapes change from box: sin(  x/L)  exp(-x 2 /2a 2 ) Need polynomial prefactor to incorporate nodes (Hermite) E~n 2 for box, but box width increases as we go higher up  Energies equispaced E = (n+1/2)ħ , n = 0, 1, 2...

13. 13 Add asymmetry

14. 14 Grid issues For Small energies, finite diff. matches exact result Deviation at large energy, where  varies rapidly Grid needs to be fine enough to sample variations

15. 15 Grid issues Exact result: E = ħ 2 k 2 /2m = tka 2 Finite diff result: E = 2t[1-coska] -t(  n+1 +  n-1 -2  n ) = E  n Setting  n =  0 e ikna, we get the above The two agree for ka << 1

16. 16 Ring boundary conditions instead of box bcs Twice as many allowed solutions  n = Asin(k n x), Bcos(k n x) or equivalently,  n = Ae ±ikx (clockwise and anticlockwise) But allowed k’s only half as many k n L = 2n  (n = 1, 2, 3, …) For box,  must go to zero and ends  L = n /2, ie, k n L=n  For ring,  must match up at ends  L = n, ie, k n L=2n  (Recall Bohr condition) e ik  e -ik 

17. 17 -t 2t -t Ĥ = -t 2t -t -t -t 2t 2t -t Ring boundary conditions instead of box bcs Hard wall (Box) Periodic (Ring)

18. 18 Box vs Ring boundary conditions Quantization condition different for both For periodic bcs, half as many allowed k points, but each twice degenerate (two solutions per point), sin and cos (or, e ikx and e -ikx )

19. 19 Hydrogen Atom (-ħ 2  2 /2m –Zq 2 /4  0 r)  = E  Coulomb potential V

20. 20 Multiple Dimensions (Separation of Variables) (-ħ 2  2 /2m + U)  = E  If U(x,y) = U x (x) + U y (y) Then  (x,y) = X(x).Y(y) and E = Ex + Ey [Solve two 1-D problems] Nx = 100, Ny=100, N = 10,000 But solve two sets of 100 x 100 matrices (200 eigenvalues) Can match any 100 x eigenvalues with any 100 y eigenvalues

21. 21 2-D Box  pq = 4/L 2 sin(p  x/L).sin(q  y/L) E pq = ħ 2  2 (p 2 +q 2 )/2mL 2 (p, q = 1, 2, 3, …)

22. 22  pq = 4/L 2 sin(p  x/L).sin(q  y/L) E pq = ħ 2  2 (p 2 +q 2 )/2mL 2 (p, q = 1, 2, 3, …) 2-D Box  11  12  21  22 2E 0 5E 0 8E 0

23. 23 Hydrogen Atom (Variable separation in radial coords) (-ħ 2  2 /2m –Zq 2 /4  0 r)  = E  -(ħ 2  2 /2m)  = -ħ 2 /2m[1/r.∂ 2 (r  )/∂r 2 ] + +1/2mr 2 [- ħ 2 {1/sin .∂/∂  (sin  ∂  /∂  ) + 1/sin 2  ∂ 2  /∂  2 } ] Coulomb potential V L2L2 ˆ L = r x p = -iħr x  ˆˆˆ L 2 = L. L ˆ ˆˆ

24. 24 z

25. 25 - ħ 2 {1/sin .∂/∂  (sin  ∂  /∂  ) + 1/sin 2  ∂ 2  /∂  2 } L2L2 ˆ L = r x p = -ihr x  ˆˆˆ L 2 = L. L ˆ ˆˆ z

26. 26 -(ħ 2  2 /2m)  = -ħ 2 /2m[1/r.∂ 2 (r  )/∂r 2 ] + L 2 ħ 2 /2mr 2 ˆ (AxB).(AxB) = A 2 B 2 sin 2  = A 2 B 2 (1-cos 2  ) = (A.A)(B.B) – (A.B)(B.A) L = r x p = -iħr x  ˆˆˆ L 2 = L. L = -ħ 2 (r x  ).(r x  ) ˆ ˆˆ (rx  ).(rx  ) = r 2  2 – (r.  )( .r)

27. 27 Hydrogen Atom (Variable separation in radial coords) -(ħ 2  2 /2m)  = -ħ 2 /2m[1/r.∂ 2 (r  )/∂r 2 ] + +1/2mr 2 [- ħ 2 {1/sin .∂/∂  (sin  ∂  /∂  ) + 1/sin 2  ∂ 2  /∂  2 } ] L2L2 ˆ

28. 28 Hydrogen Atom L2  L2L2  L2 ˆ Angular Mom Quantized l(l+1)ħ 2 l = 0, 1, 2,.. (n-1) Lz  LzLz  Lz ˆ z-component Quantized mħ m = -l, -(l-1),... (l-1), l Also, Lx, Ly, Lz are not simultaneously measurable