1. Schrödinger and Maxwell's Equations: on their similarities
Friday Talk, 16th April 2010
Oka Kurniawan
Computational Electronics and Photonics
1

2. Electron and Photon
E2 E2
ħω ħω
E1 E1
2

3. The equations
2 2
 ℏ 2  E 2
iℏ =−  V  0 0 = E
t 2m t 2
i 2 m  2 B
−
2
=  0 0 2 = 2 B
ℏ t t
2
2 f 2
c 2
= f
t
3

4. Classical Case
E
EC2
EC1
z 0 1 T(E)
4

5. Comparison with Classical Case
E
EC2
EC1
z 0 1 T(E)
5

6. The case of a single barrier
EC2
EC1 ε1 ε2
z z
ik x x ik y y −iEt/ ħ x x y y −i  t
 r ,t =C  z  e e e E y r , t=C E y0  z  e e e
2
   
2
d  2m 2 2 d E y0 2
 n z 
2
2 2
2
 2  E −E C  z −k x −k y =0  − x − y E y0 =0
dz ħ dz 2 c2
6

7. The analogy: Quantum and Optics
■ Streams of electrons ■ Streams of photons
(EM wave)
■ At interface: ■ At interface:
 Reflection  Reflection
 Transmission  Transmission
■ Interface → energy ■ Interface → refractive
barrier index difference
2 2
R=

k 1−k 2
k 1k 2  R=

n1−n2
n1n2 
2 2
T=

k 2 2 k1
k 1 k 1k 2  T=

2 v 2 2 n1
1 v 1 n1n 2 
1 2
I =  v E0
2 7

8. Another simple case
V(x)
a x a
d 2  ħ2 d 2 E0 2
2
 E =0 k E 0 =0
dx 2m dx
2
= An sin k n x  E 0 = Aq sin  k q x 
=0 , for x=0 and x=a E 0 =0, for x =0 and x=a
n q
k n= k q=
a a
8

9. Simple example: discrete frequencies
V(x)
a x a
v = c/2a
E = π2Ћ2/2ma2
9

10. Quantum and Optical Confinement
EC
ħω
EV
Taken from wikipedia
10

11. Laser: Light Amplification by Stimulated Emission of Radiation
E2
ħω ħω
ħω Active medium
E1
Active medium
a
EC
ħω
EV
11

12. Nano Laser
[1] M.T. Hill,et al., “Lasing in metallic-coated nanocavities,”
Nat Photon, vol. 1, Oct. 2007, pp. 589-594.
12

13. References
■ Datta, “Quantum Phenomena,” Modular Series on Solid State Devices, Vol VIII, Addison-
Wesley (1989). page 12-28.
■ Joannopoulus, et al., “Photonic Crystals: Molding the flow of light,” 2nd Ed, Princeton
(2008). page 22 and Appendix A. (E-book download from TWiki)
13

14. Solid-State and Photonic Crystal
14

15. Summary
15