1 Practical Quantum Mechanics

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

10 Add a field

11 Or asymmetry

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 Add asymmetry

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

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 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  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 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 z

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 -(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 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 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