1. Multistate Problems in Quantum & Statistical Mechanics
– Analytically Solvable Models
CTS Seminer - IIT Kharagpur,
February 11, 2016.
Aniruddha Chakraborty
Associate Professor, School of Basic Sciences.
Indian Institute of Technology Mandi, Himachal Pradesh, India.
Web:

2. Our Research Group Ph.D. Students:
Ms. Paromita Dutta – Inverse Problems in Scattering & Diffusion.
Ms. Moumita Ganguly – Reaction-Diffusion Systems in Biology.
M.Sc. Students:
 Ms. Shivani Verma – Electron correlation in chemical bonding.
Mr. Deepak Kumar – Nano Devices: Vibrational Dynamics.
Alumni:
 Dr. Diwaker Kumar – Assistant Professor, Central University of Himachal Pradesh.
Thesis Title: Exact solution of few multi-state problems in quantum & statistical mechanics.

3. Our Research Nano Devices. Multistate Problems in Quantum Mechanics.
Barrierless Reactions in Solution.
Heat Diffusion in Realistic Solid Material.
Inverse Eigenvalue Problem.
Electron Correlation.

4. Proton/electron Transfer
𝑯 +
𝑯 +
A B
A B
Assumption: RHA & RHB is fixed.
Potential Energy (Schematic)
Nuclear coordinate RAB
Li Cl 𝐿𝑖 + 𝐶𝑙 −

5. Atom-exchange Reaction
H D
H H D
𝐴𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛: <𝐻𝐻𝐷=180°
Potential Energy (Schematic)
H-D : Excited electronic State
H-D : Ground electronic State
Reaction coordinate (approx. Anti-sym. Stretch)

6. Long Range Proton/electron Transport
Potential Energy (Schematic)
Nuclear coordinate (x) H H H H H

7. Multi state Problems V1 V3 V2 V1 V2 V3 Nuclear coordinate RAB 𝑯 + H H
Potential Energy (Schematic) V2
Nuclear coordinate RAB
𝑯 + H H V1 V2 V3
A B
𝑨 B
A 𝑩 +

8. Multi - State Problem (Non-stationary state)
IF Ψ(𝒙,𝟎) 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛,
what is Ψ(𝒙,t) ? Or
at least, what is Ψ (𝑥,𝒔) ?
t = 0
Potential Energy (Schematic)
Nuclear coordinate RAB
Ψ 1 (𝒙,𝟎) ≠ 0
Ψ 2 (𝒙,𝟎) = 0
Ψ 1 (𝒙,t) = ?
Ψ 2 (𝒙,t) = ?

9. Multi - State Problem (stationary state)2 1
Potential Energy (Schematic)
𝑇 12 𝐸 = ?
Nuclear coordinate RAB

10. Plan: An analytical method …….
Multi-state Problem: Time independent Arbitrary Coupling.
Multi-state Problem: Time dependent Arbitrary Coupling
Arbitrary Coupling
Collection of Dirac Delta Function Couplings
Dirac Delta Function Coupling
Dirac Delta Function Potential

11. Dirac Delta Potential Barrier
V x =k0 δ(𝑥−𝑥𝑐)
𝐻(𝑥) ϕ (x)= E ϕ(𝑥)
Where,
𝐻 =𝐾. 𝐸. +𝑃.𝐸.
ϕ(𝑥)= 𝑒 𝑖𝑘𝑥 + R 𝑒 −𝑖𝑘𝑥
ϕ(𝑥)= T 𝑒 𝑖𝑘𝑥
𝑃.𝐸.=𝑘0 δ (𝑥−𝑥𝑐) x
Two Boundary Conditions
𝑘 2 = 2 𝑚 𝐸 ℏ 2
Where,
1. ϕ(𝑥𝑐+ε)= ϕ(𝑥𝑐−ε)
2. 𝑑ϕ(𝑥) 𝑑𝑥 𝑥𝑐−ε 𝑥𝑐+ε − 2𝑚k0 ℏ 2 ϕ(𝑥𝑐)=0
Analytical Solution possible!
Two Unknowns: R & T
Quantum Physics: S. Gasiorowicz, John Wiley & Sons. , USA, 1995.

12. N-Dirac Delta Potential Barrier
Four Boundary Conditions
k1δ(𝑥−𝑥1)
k2δ(𝑥−𝑥2)
ϕ(𝑥)= 𝑒 𝑖𝑘𝑥 + R 𝑒 −𝑖𝑘𝑥
ϕ(𝑥)= 𝐴1 𝑒 𝑖𝑘𝑥 + B1 𝑒 −𝑖𝑘𝑥
ϕ(𝑥)= T 𝑒 𝑖𝑘𝑥 x
k1δ(𝑥−𝑥1)
k2δ(𝑥−𝑥2)
k3δ(𝑥−𝑥3)
Six Boundary Conditions
ϕ(𝑥)= 𝑒 𝑖𝑘𝑥 + R 𝑒 −𝑖𝑘𝑥
A1 , B1
A2 , B2
ϕ(𝑥)= T 𝑒 𝑖𝑘𝑥 x

13. Arbitrary Potential V(x) = N-Dirac Delta Potentials
V(x) = 𝑎 𝑏 𝑉 𝑥𝑖 δ(𝑥−𝑥𝑖)dxi (1)
V(x) = 𝑖=1 𝑁 𝑉 𝑥𝑖 δ(𝑥−𝑥𝑖) δx (2)
for
δx→0
V(x) = 𝑖=1 𝑁 𝑘𝑖 δ 𝑥−𝑥𝑖 (3)
where
Ki = 𝑉 𝑥𝑖 δx
δ(𝑥−𝑥𝑛)
δ(𝑥−𝑥1)
δ(𝑥−𝑥𝑁)
𝑁= 𝑏−𝑎 δx
V(x)
Potential Energy
x = a x δx
x = b

14. Two State Problem: Dirac Delta Coupling
𝐻1(𝑥) V12(x) V21(x) 𝐻2(𝑥) ϕ1(𝑥) ϕ2(𝑥) = E ϕ1(𝑥) ϕ2(𝑥)
ϕ2(𝑥)= R 2𝑒 −𝑖𝑘1𝑥
ϕ2(𝑥)= T 2𝑒 𝑖𝑘1𝑥
V2(x)
𝑉12=𝑉21=𝑘0 δ (𝑥−𝑥𝑐)
Four Boundary Conditions
ϕ1(𝑥)= 𝑒 𝑖𝑘𝑥 + R 1𝑒 −𝑖𝑘𝑥
ϕ1(𝑥)= T 1𝑒 𝑖𝑘𝑥
1. ϕ1(𝑥𝑐+ε)= ϕ1(𝑥𝑐−ε)
2. ϕ2(𝑥𝑐+ε)= ϕ2(𝑥𝑐−ε)
V1(x)
3. 𝑑ϕ1(𝑥) 𝑑𝑥 𝑥𝑐−ε 𝑥𝑐+ε − 2𝑚𝑘0 ℏ 2 ϕ2 𝑥𝑐 =0
4. 𝑑ϕ2(𝑥) 𝑑𝑥 𝑥𝑐−ε 𝑥𝑐+ε − 2𝑚𝑘0 ℏ 2 ϕ1 𝑥𝑐 =0
𝑇12=2 𝑘1 𝑘 m k0 k ℏ 2 k k1 ℏ 4 + 𝑚 2 𝑘
Four Unknowns: R1, R2, T1 & T2.

15. 𝐻1(𝑥) ϕ1 𝑥 + 𝑘0 2 δ (𝑥−𝑥𝑐) 𝐺 2 0 xc, 𝐸 xc ϕ 1(𝑥) = E ϕ1(𝑥)
Two State Problem: Dirac Delta Coupling
V2(x)
𝐻1(𝑥) V12(x) V21(x) 𝐻2(𝑥) ϕ1(𝑥) ϕ2(𝑥) = E ϕ1(𝑥) ϕ2(𝑥)
𝑉12=𝑉21=𝑘0 δ (𝑥−𝑥𝑐)
V1(x)
𝑉12=𝑉21=𝑘0 δ (𝑥−𝑥𝑐)
𝐻1(𝑥) ϕ1 𝑥 + 𝑘0 2 δ (𝑥−𝑥𝑐) 𝐺 2 0 xc, 𝐸 xc ϕ 1(𝑥) = E ϕ1(𝑥)
Where, 𝐺 2 0 xc, 𝐸 xc = xc| (𝐸−𝐻2) −1 |𝑥𝑐
1. ϕ1(𝑥𝑐+ε)= ϕ1(𝑥𝑐−ε)
2. 𝑑ϕ1(𝑥) 𝑑𝑥 𝑥𝑐−ε 𝑥𝑐+ε − 2𝑚 𝑘0 2 ℏ 𝐺 2 0 xc, ω xc ϕ1 𝑥𝑐 =0 R T
R+T≠1
𝑘 𝐺 2 0 xc, 𝐸 xc δ(𝑥−𝑥𝑐)
P12 = 1 – (R+T)

16. N- State Problem: Dirac Delta Couplings
V3(x)
𝑉13=𝑉31=𝑘13 δ (𝑥−𝑥𝑑)
V2(x)
𝑉12=𝑉21=𝑘12 δ (𝑥−𝑥𝑐)
V1(x)
𝐻1(𝑥) ϕ1 𝑥 + 𝑘12 2 δ (𝑥−𝑥𝑐) 𝐺 2 0 xc, 𝐸 xc ϕ 1(𝑥) + 𝑘13 2 δ (𝑥−𝑥𝑑) 𝐺 3 0 xd, 𝐸 x𝑑 ϕ 1(𝑥) = E ϕ1(𝑥)
𝑘 𝐺 2 0 xc, 𝐸 xc δ(𝑥−𝑥𝑐)
𝑘 𝐺 3 0 xd, 𝐸 x𝑑 δ(𝑥−𝑥𝑑)
R+T≠1 R T
P12 = 1 – (R+T)

17. 𝐻1(𝑥) ϕ1 𝑥 + 𝑖=1 𝑁 𝑘𝑖 2 δ (𝑥−𝑥𝑖) 𝐺 2 0 xi, 𝐸 x𝑖 ϕ 1(𝑥) = E ϕ1(𝑥)
Two State Problem: Arbitrary Coupling V2
Potential Energy V1 x
k1 δ (𝑥−𝑥1)
k2 δ (𝑥−𝑥2)
k3 δ(𝑥−𝑥3)
k4 δ (𝑥−𝑥4)
k5 δ(𝑥−𝑥5)
𝑉12 𝑥 =𝑉21 𝑥 = 𝑖=1 𝑁 𝑘𝑖 δ (𝑥−𝑥𝑖)
𝐻1(𝑥) ϕ1 𝑥 𝑖=1 𝑁 𝑘𝑖 2 δ (𝑥−𝑥𝑖) 𝐺 2 0 xi, 𝐸 x𝑖 ϕ 1(𝑥) = E ϕ1(𝑥)

18. Multi State Problem: Arbitrary CouplingV3
Potential Energy V2 V1 x
k1 δ (𝑥−𝑥1)
k2 δ (𝑥−𝑥2)
k3 δ(𝑥−𝑥3)
k4 δ (𝑥−𝑥4)
k5 δ(𝑥−𝑥5)
𝑉12 𝑥 =𝑉21 𝑥 = 𝑖=1 𝑁 𝑘𝑖 δ (𝑥−𝑥𝑖)
𝑉13 𝑥 =𝑉31 𝑥 = 𝑗=1 𝑁 𝑘𝑗 δ (𝑥−𝑥𝑗)
𝐻1(𝑥) ϕ1 𝑥 𝑖=1 𝑁 𝑘𝑖 2 δ (𝑥−𝑥𝑖) 𝐺 2 0 xi, 𝐸 x𝑖 ϕ 1(𝑥) + 𝑗=1 𝑁 𝑘𝑗 2 δ (𝑥−𝑥𝑗) 𝐺 3 0 xj, 𝐸 x𝑗 ϕ 1(𝑥) = E ϕ1(𝑥)

19. Time dependent Schr 𝒐 dinger Equation
Quantum (Schr 𝑜 dinger Eqn.)
𝑘0 δ (𝑥−𝑥𝑐)
𝑖 𝜕Ψ(𝑥,𝑡) 𝜕𝑡 =𝐻 Ψ(𝑥,𝑡)
Potential Energy
where, H = − ℏ 2 2𝑚 𝜕 2 𝜕 𝑥 2 +𝑘0 δ (𝑥−𝑥𝑐)
Ψ 𝒙,𝟎 x
IF Ψ 𝒙,𝟎 𝑖𝑠 𝑘𝑛𝑜𝑤𝑛, 𝑤ℎ𝑎𝑡 𝑖𝑠 Ψ 𝒙,𝒕 ?
Ψ 𝒙,𝒔 ?

20. Solution for 𝑘0 δ(𝑥) i 𝜕 Ψ(𝑥,𝑡) 𝜕𝑡 =− 𝜕 2 𝛹 𝑥,𝑡 𝜕 𝑥 2 +𝑘0δ 𝑥 Ψ 𝑥,𝑡 (1)
Ψ 𝒙,𝒔 = a(s) exp(i 𝑖𝑠 |x|) 𝑖𝑠 −∞ ∞ 𝑑𝑥0 exp(i 𝑖𝑠 |x-x0|) Ψ 𝑥0, (4)
𝜕 Ψ 𝒙,𝒔 𝜕𝑥 0−ε 0+ε −𝑘0 Ψ 𝟎,𝒔 = (5)
𝑎 𝑠 = 𝑘0 Ψ 𝟎,𝒔 2𝑖 𝑖𝑠 (6)

21. Solution for 𝑘0 δ(𝑥)
Ψ 𝒙,𝒔 = 𝑘0 Ψ 𝟎,𝒔 2𝑖 𝑖𝑠 exp(i 𝑖𝑠 |x|) 𝑖𝑠 −∞ ∞ 𝑑𝑥0 exp(i 𝑖𝑠 |x-x0|) Ψ 𝑥0, (7)
Put x = 0
Ψ 𝟎,𝒔 = 𝑘0 Ψ 𝟎,𝒔 2𝑖 𝑖𝑠 𝑖𝑠 −∞ ∞ 𝑑𝑥0 exp(i 𝑖𝑠 |x0|) Ψ 𝑥0, (8)
Ψ 𝟎,𝒔 = 𝑖 2𝑖 𝑖𝑠 −𝑘0 −∞ ∞ 𝑑𝑥0 exp(i 𝑖𝑠 |x0|) Ψ 𝑥0, (9)

22. Two State Problem: Dirac Delta Coupling
V2(x)
i 𝜕 𝜕𝑡 Ψ1(𝑥,𝑡) Ψ2(𝑥,𝑡) = 𝐻1(𝑥) V12(x) V21(x) 𝐻2(𝑥) Ψ1(𝑥,𝑡) Ψ2(𝑥,𝑡)
𝑉12=𝑉21=𝑘0 δ (𝑥−𝑥𝑐)
Ψ𝟏 𝒙,𝟎
V1(x)
ϕ2 (𝑥,0)=0
𝑉12=𝑉21=𝑘12 δ (𝑥−𝑥𝑐)
i s ϕ 𝟏 (𝒙,𝒔) =𝐻1(𝑥) ϕ 𝟏 (𝒙,𝒔) + 𝑘0 2 δ 𝑥−𝑥𝑐 𝐺 2 0 xc, 𝑖𝑠 x𝒄 ϕ 𝟏 𝒙,𝒔 +𝑖 ϕ1(𝑥,0)
𝑘0 2 𝐺 2 0 xc, 𝑖𝑠 x𝒄 δ (𝑥−𝑥𝑐)
Ψ𝟏 𝒙,𝟎
Where, 𝐺 2 0 xc, 𝑖𝑠 xc = xc| (𝑖𝑠−𝐻2) −1 |𝑥𝑐
V1(x)

23. Schr 𝒐 dinger Eqn. with time dependent Dirac Delta potential
Ψ 𝒙,𝒔 = a(s) exp(i 𝑖𝑠 |x|) 𝑖𝑠 −∞ ∞ 𝑑𝑥0 exp(i 𝑖𝑠 |x-x0|) Ψ 𝑥0, (4)
𝜕 Ψ 𝒙,𝒔 𝜕𝑥 0−ε 0+ε −𝑳 𝑘 𝑡 Ψ 0,𝑡 = (5)
𝑎 𝑠 = 𝑳 𝑘 𝑡 Ψ 0,𝑡 2𝑖 𝑖𝑠 (6)

24. Schr 𝒐 dinger Eqn. with time dependent Dirac Delta potential
𝑽 𝒙,𝒕 =𝒌(𝒕) 𝜹(𝒙−𝒙𝒄)
K(t) = a t
K(t) = a/t
K(t) = a 𝒆 −𝒃𝒕
K(t) = a Exp[i ω t ]

25. Solution for k(t) = k0t
Ψ 𝒙,𝒔 = a(s) exp(i 𝑖𝑠 |x|) 𝑖𝑠 −∞ ∞ 𝑑𝑥0 exp(i 𝑖𝑠 |x-x0|) Ψ 𝑥0, (4)
𝑎 𝑠 =− 𝑘0 2𝑖 𝑖𝑠 𝜕 𝑃 𝟎,𝒔 𝜕𝑠 (7)
Ψ 𝒙,𝒔 = - 𝑘0 2𝑖 𝑖 𝑠 exp(- 𝑠 |x|) 𝜕 Ψ 𝟎,𝒔 𝜕𝑠 𝑠 𝑑𝑥0 exp(- 𝑠 |x-x0|) Ψ 𝑥0, (8)
Put x = 0
Ψ 𝟎,𝒔 = - 𝑘0 2𝑖 𝑖 𝑠 𝜕 Ψ 𝟎,𝒔 𝜕𝑠 𝑠 𝑑𝑥0 exp(- 𝑠 |x0|) Ψ 𝑥0, (9)

26. Schr 𝒐 dinger Equation with moving Dirac Delta Potential
𝑖 𝜕 Ψ(𝑥,𝑡) 𝜕𝑡 =− ℏ2 2𝑚 𝜕 2 Ψ 𝑥,𝑡 𝜕 𝑥 2 +𝑘0 δ[𝑥 −𝑎 𝐿0+𝑣𝑡 ]Ψ 𝑥,𝑡
Phys. Rev. A. 65, (2002).
1. y = 𝑥 𝐿(𝑡) ; L(t)= L0 + v t
2. Ψ 𝑦,𝑡 = 1 𝐿(𝑡) exp⁡[𝑖 𝑚 2ℏ 𝐿 𝑡 𝑣 𝑦 2 ]Φ 𝑦,𝑇
3. T = 𝑡 𝐿0𝐿(𝑡)
𝑖 𝜕 Φ(𝑦,𝑇) 𝜕𝑇 =− ℏ 2 2𝑚 𝜕 2 Φ 𝑦,𝑇 𝜕 𝑦 2 +𝑘0 δ(y −𝑎)Φ 𝑦,𝑇

27. Two state problem with moving Dirac Delta Coupling
𝑖 𝜕 𝜕𝑡 Ψ1(𝑥,𝑡) Ψ2(𝑥,𝑡) = 𝐻1(𝑥) 𝑘0 δ (𝑥−𝑣𝑡) 𝑘0 δ (𝑥−𝑣𝑡) 𝐻2(𝑥) Ψ1(𝑥,𝑡) Ψ2(𝑥,𝑡)
i 𝜕 𝜕𝑇 Ψ1(𝑦,𝑇) Ψ2(𝑦,𝑇) = 𝐻1(𝑦) 𝑘0δ(𝑦−𝑎) 𝑘0δ(𝑦−𝑎) 𝐻2(𝑦) Ψ1(𝑦,𝑇) Ψ2(𝑦,𝑇)
Ψ1(𝑥,0)
𝑘0 δ (𝑥−𝑣𝑡)

28. Quantum Mechanics Statistical Mechanics
Quantum (Schr 𝑜 dinger Equation)
Diffusion (Smoluchowski Equation)
𝑖 𝜕Ψ(𝑥,𝑡) 𝜕𝑡 =𝐻 Ψ(𝑥,𝑡)
𝜕 𝑃(𝑥,𝑡) 𝜕𝑡 = L(x) 𝑃(𝑥,𝑡)
where, H = − ℏ 2 2𝑚 𝜕 2 𝜕 𝑥 2 +𝑉(𝑥)
where, L(x)=𝐴 𝜕 2 𝜕 𝑥 2 +𝐵 𝜕 𝜕𝑥 𝑑𝑉(𝑥) 𝑑𝑥
Ψ (x,t) P(x,t)
− ℏ 2 2𝑚 A
𝑉 𝑥 −−−−−− 𝐵 𝜕 𝜕𝑥 𝑑𝑉(𝑥) 𝑑𝑥 i

29. Two state Problems in Statistical Mechanics
𝜕 𝑃(𝑥,𝑡) 𝜕𝑡 = L(x) 𝑃(𝑥,𝑡)
𝑖 𝜕Ψ(𝑥,𝑡) 𝜕𝑡 =𝐻 Ψ(𝑥,𝑡)
where, L(x) = 𝐿1(𝑥) 𝑉12(𝑥) 𝑉21(𝑥) 𝐿2(𝑥)
where, H = 𝐻1(𝑥) 𝑉12(𝑥) 𝑉21(𝑥) 𝐻2(𝑥)
P(𝑥,𝑡) = 𝑃1(𝑥,𝑡) 𝑃2(𝑥,𝑡)
Ψ(𝑥,𝑡) = ϕ1(𝑥,𝑡) ϕ2(𝑥,𝑡)
where, Li(x)=𝐴 𝜕 2 𝜕 𝑥 2 +𝐵 𝜕 𝜕𝑥 𝑑𝑉𝑖(𝑥) 𝑑𝑥
Hi = − ℏ 2 2𝑚 𝜕 2 𝜕 𝑥 2 +𝑉𝑖(𝑥)

30. Current Research: Inverse Problem
𝑉 𝑥 = 𝑖=1 𝑁 𝑘𝑖 δ (𝑥−𝑥𝑖)
Arbitrary Potential
Collection of Dirac Delta Potentials x x
T(E) E

31. Current Research: Polymer cyclization in Solutionx
X : End to End Distance
Smoluchowski Equation for a long polymer with time dependent sink
k(t) δ (𝑥−𝑎)

32. References
Curve crossing effects on absorption & resonance Raman spectrum:A. Chakraborty, Mol. Phys. 107,165 (2009).
Multi-channel curve crossing problems: Analytically solvable model: A. Chakraborty, Mol. Phys. 107, 2459 (2009).
Curve crossing problem with arbitrary coupling: Analytically solvable model, Diwaker & A. Chakraborty, Mol. Phys., 110, 2197 ( 2012).
Multi-channel scattering problems: Analytically solvable model: Diwaker & A. Chakraborty, Mol. Phys., 110, 2257 (2012).
Nonadiabatic tunneling in an ideal one dimensional semi-infinite periodic potential systems: A.Chakraborty, Mol. Phys. 109, 429 (2011).
Barrierless electronic relaxation in solution: An analytically solvable model.: A. Chakraborty, J. Chem. Phys., 139, (2013).
Transfer Matrix approach to curve crossing problems of two exponential potentials, Diwaker & A. Chakraborty, Mol. Phys., 113, 3406 (2015).
Exact Solution to curve crossing problems of two diabatic potentials by transfer matrix method: Diwaker & A. Chakraborty, Mol. Phys., 113, 3909 (2015).
Long range electron transfer reactions in solution: Diwaker & A. Chakraborty, Chem. Phys., 459, 19 (2015).
Exact Solution of time dependent Schrodinger equation for two state problem: Diwaker & A. Chakraborty, Chem. Phys. Lett., 638, 133 (2015).
Exact solution of Schrodinger equation for two state problem with time dependent coupling: Diwaker, B. Panda & A. Chakraborty, Physica A, 442, 380 (2016).
Curve crossing induced dissociation: An Analytical model, Diwaker & A. Chakraborty, Spectrochemica. Acta A, 151, 510 (2015).
Exact solution of Smoluchowski equation for linear potential with time dependent sink, Diwaker & A. Chakraborty, J. Exp. Theo. Phys., 149, 439 (2016).
Barrierless Chemical Reactions  in solution: An analytically solvable model. Diwaker & A. Chakraborty, Phys. Rev. E (under revision).
Exact solution of Schrodinger equation for two state problem with a moving Dirac Delta potential coupling: Diwaker & A. Chakraborty, Chem. Phys. (under revision).
Multichannel electron transfer reactions in solution: An analytically solvable model : Diwaker & A. Chakraborty, J. Stat. Phys. (under revision).
Seed Grant from IIT Mandi, Himachal Pradesh, India, Rs: 4,80,000/-.