1. Getting used to quantum weirdness

2. Going from 1D to 2D to 3D

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

10. Compare

11. Circular 2D box

12. What about a circular 2D box?
Angular orthogonality m
Radial orthogonality
This has nodes along r and q
Angular nodes .. Orbitals l

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 Fnm = Jm(rxnm/a)eimj 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 !!