Constants  
 
  Diff. equations  
 
  Hilbert space  
 
  HandS Pictures  
 
  Pauli matrices  
 
  home  
 
  ask us  
 

  CGC  
 
  Units   
 
  Jacobians  
 
  Angular momenta  
 
  Elliptic coordinates  
 

 

Quantum Mechanics



   Schrodinger equation


   Quantum Mechanics



   Propagators : Pg




Quantum Simple Harmonic
Oscillator QSHO




Quantum Mechanics
Simulation With GNU Octave




© The scientific sentence. 2010

elliptic coordinates



Quantum Mechanics enables us to understands the existence and properties of chemical bonds that are responsible for the formation of molecules from isolated atoms. The simplest molecule possible is the H2+ ion, which is composed of two protons and a single electron.

ρ = R/ao
ρ1= r1/ao
ρ2= r2/ao


R is the internuclear distance P2 nad P2

The normalized gound state wave function 1s of the hydrogen atom formed around P1 is:
φ1 = 1/(πao3)1/2 exp{- ρ1}

It is convenient to use the system of elliptic coordinates. The point Mof space (electron) is defined by:
μ = (r1 + r2)/R = (ρ1 + ρ2)/ρ
ν = = (r1 - r2)/R = (ρ1 - ρ2)/ρ

r12 = x2 + y2 + (z - R/2)2
r22 = x2 = Y2 = (z + R/2)2
tan φ = y/x

We will calculate the Jacobian of the transformation: {x,y,z} → {μ, ν, φ}; in order to write:
d3r = dxdydz = J dμdνdφ
It is more easy to calculate first J-1, then set J according to JJ-1 = 1.

Therefore:
∂μ/∂x = (1/R) (∂r1/∂x + ∂r2/∂x ) = (1/R) (x/r1 + x/r2) = μx/r1r2
∂ν/∂x = (1/R) (∂r1/∂x - ∂r2/∂x ) = (1/R) (x/r1 - x/r2) = - νx/r1r2
∂μ/∂y = μy/r1r2
∂ν/∂y = - νy/r1r2
∂μ/∂z = (1/R) (∂r1/∂z + ∂r2/∂z ) = (1/R) ((z - R/2)r1 + (z + R/2)/r2) = (μz + νR/2)/r1r2
∂ν/∂z = (1/R) (∂r1/∂z - ∂r2/∂z ) =
(1/R) ((z - R/2)r1 - (z + R/2)/r2) = - (νz + μR/2)/r1r2
∂φ/∂x = - y/(x2 + y2)
∂φ/∂y = x/(x2 + y2)
∂φ/∂z = 0

Therefore:
J-1 = (1/(r1r2)2)
| μx μy μz + νR/2|
|- νx - νy - (νz + μR/2)|
|- y/(x2 + y2) x/(x2 + y2) 0|


= μx (νz + μR/2)(x/(x2 + y2)) + μy (νz + μR/2)(y/(x2 + y2)) - (μz + νR/2) =
(R/2)(μ2 - ν2). Then:
J-1 = (R/2)(μ2 - ν2) (1/(r1r2)2)
Since
μ2 - ν2 = 4r1r2/R2, we have then:
(r1r2)2 = R42 - ν2)2/16
Hence:
J-1 = 8/R32 - ν2)
Then:
J = R32 - ν2)/8


Jacobian of the transformation:
{x,y,z} → {μ, ν, φ}:
d3r = dxdydz = J dμdνdφ
J = R32 - ν2)/8



12 > =
∫∫ d3r φ1* (r)φ2 (r) = (1/πao3) ∫∫∫ d3r exp{-( r1 + r2)/ao}
= (1/πao3) ∫∫∫ J dμdνdφ exp{- μρ}

We have:
μ = (r1 + r2)/R
μmin = R/R = 1
μmax = ∞/R = ∞

And
ν = (r1 - r2)/R
νmin = -R/R = -1
νmax = R/R = 1;

Then:
12 > = (R3/8)(1/πao3) ∫∫∫ (μ2 - ν2) dμdνdφ exp{- μρ}
[μ: 1 → ∞, ν: -1 → +1, φ: 0 → 2π ]
= 2π(R3/8)(1/πao3) ∫∫(μ2 - ν2) dμdν exp{- μρ}
[μ: 1 → ∞, ν: -1 → +1,]
Using ρ = R/ao yields: R3 = ρ3ao 3 = ρ3/4




∫ x2exp{- ax} dx = x2 (exp{- ax}/(-a)) - ∫ (exp{- ax}/(-a)) (2xdx) =
(- 1/a)x2 (exp{- ax}) + ( 2/a) ∫ x (exp{- ax}) dx

∫ x (exp{- ax}) dx = x ((exp{- ax})/(-a)) - ∫ ((exp{- ax})/(-a))dx =
-(x/a)(exp{- ax}) +(1/a) ∫ (exp{- ax})dx
= -(x/a)(exp{- ax}) - (1/a2)(exp{- ax}) = - (exp{- ax}/a) [x + 1/a]

∫ x (exp{- ax}) dx = - (exp{- ax}/a) [ x + 1/a]


With a = 1:
∫ x exp{- x} dx = exp{-x} [ - x - 1]

Then:
∫ xexp{- ax} dx = -(x/a)(exp{- ax}) +(1/a) ∫ (exp{- ax})dx

∫ x2exp{- ax} dx =
(- 1/a)x2 (exp{- ax}) + (2/a) [-(x/a)(exp{- ax}) +(1/a) ∫ (exp{- ax})dx] =
(- 1/a)x2 (exp{- ax}) + (2/a) [-(x/a)(exp{- ax}) - (1/a2) exp{- ax}] =
(exp{- ax}/a) [- x2 - 2 [(x/a) + (1/a2)]]

∫ x2exp{- ax} dx = (exp{- ax}/a) (- x2 -2x/a - 2/a2)




12 > =
ρ3/4[ ∫ dμ exp{- μρ} ∫(μ2 - ν2) dν]
[μ: 1 → ∞, ν: -1 → +1,]

= ρ3/4 [ ∫ dμ exp{- μρ} (2μ2 - 2/3) ]
[μ: 1 → ∞]

= ρ3/2 [ ∫ dμ exp{- μρ} (μ2 - 1/3)]
[μ: 1 → ∞]

= ρ3/2 [ ∫ dμ exp{- μρ} μ2 - 1/3 ∫ dμexp{- μρ}]
[μ: 1 → ∞]

= ρ3/2 [ (exp{- ρμ}/ρ) [- μ2 - 2 [(μ/ρ) + (1/ρ2)]] +( 1/3ρ) exp{- μρ}]
[μ: 1 → ∞]

= ρ3/2 [ (exp{- ρμ}/ρ) [- μ2 - 2 [(μ/ρ) + (1/ρ2)] + 1/3]]
[μ: 1 → ∞]

= ρ3/2 [ (exp{- ρ}/ρ) [ 1 + 2 [(1/ρ) + (1/ρ2)] - 1/3]]
= ρ3/2 (exp{- ρ}/ρ) [2/3 + 2/ρ + 2/ρ2]
= ρ2 (exp{- ρ}) [1/3 + 1/ρ + 1/ρ2]
= exp{- ρ} (ρ2/3 + ρ + 1)


12 > = exp(- ρ) (ρ2/3 + ρ + 1)
ρ = R/ao



--------------
I = ρ2EI ∫∫∫dμdν(μ + ν) exp{-(μ + ν)ρ} dφ [μ: 1 → ∞, ν: -1 → +1, φ: 0 → 2π ] Let's write:
x = ρμ and y = ρν

I = (EI/ρ) ∫∫dx;dy;(x + y) exp{-(x + y)}
[x: ρ → ∞, y: -ρ → +ρ ]

I = I = (EI/ρ) J

J = ∫∫dx;dy;(x + y) exp{-(x + y)} = ∫ exp{-(x} dx; ∫dy;(x + y) exp{- y}
[x: ρ → ∞, y: -ρ → +ρ ]
J = ∫ exp{-(x} dx; K

K = ∫dy;(x + y) exp{- y} = x ∫dy; exp{- y} + ∫dy;y exp{- y} =
[y: -ρ → +ρ ]

- x exp{- y} - y exp{- y} - exp{- y} = exp{- y}[ - x - y - 1} [y: -ρ → +ρ ]
=
exp{- ρ}[ - x - ρ - 1} + exp{ ρ }[ x - ρ + 1 } =
= exp{- ρ}[ - x - ρ - 1} + exp{ ρ }[ x - ρ + 1 }
x(exp{ρ} - exp{- ρ}) - exp{- ρ}[ + ρ + 1] + exp{ ρ }[ - ρ + 1 ]
= x A + B


A = exp{ρ} - exp{- ρ}
B = - exp{- ρ}[ + ρ + 1] + exp{ ρ }[ - ρ + 1 ]
A + B = exp{ ρ }[- ρ + 1 + 1] - exp{- ρ}[ + ρ + 1 + 1] =
exp{ ρ }[- ρ + 2] - exp{- ρ}[ + ρ + 2]

J = ∫exp{-(x} dx; (xA + B)
[x: ρ → ∞,]
= A ∫exp{-(x} x dx; + B ∫exp{-(x}dx; =
[x: ρ → ∞]

A(- x exp{-x} - exp{-x}) - B exp{-(x} = exp{-(x} [- xA - A -B ]
[x: ρ → ∞]

= exp{-(ρ} [ρA + A + B] =
exp{-(ρ} [ρ(exp{ρ} - exp{- ρ}) + exp{ ρ }[- ρ + 2] - exp{- ρ}[ + ρ + 2] ] = ρ(1 - exp{- 2ρ}) + [- ρ + 2] - exp{- 2ρ}[ + ρ + 2] =
- ρexp{- 2ρ} + 2 - exp{- 2ρ}[ + ρ + 2] = exp{- 2ρ} [- 2ρ - 2] + 2 = 2 [1 - (ρ + 1) exp{- 2ρ}]
Therefore:
I = (EI/ρ) 2 [1 - (ρ + 1) exp{- 2ρ}] =
(2EI/ρ) [1 - (ρ + 1) exp{- 2ρ}]


1(r1)|e2/r21(r1) > = (2EI/ρ) [1 - (ρ + 1) exp{- 2ρ}]



x exp{-x} = x / exp{x}

lim x exp{-x} = ∞/∞ :indetermined.
x → + ∞
De l'Hopital theorem:


lim x exp{-x} = lim x / exp{x} = lim 1/ exp{x} = 0
x → + ∞


lim x exp{- x} = 0
x → + ∞





1|e2/r12> = e2 ∫∫∫ φ1*(r1)(1/r1) φ2(r2) d3r

We have:
φ1(r1)= (1/πao3)1/2exp{-ρ1}
φ2(r2)= (1/πao3)1/2exp{-ρ2}
r1 = ρ ao(μ + ν)/2
ρ = R/ao
∫dφ = 2π
ρ1 + ρ2 = μρ
J = [R3/8] (μ2 - ν2)
= [(ρ ao)3/8] (μ2 - ν2)
EI = e2/2ao

Therefore:
1|e2/r12> =
e2 (1/πao3) [(ρ ao)3/8] (2/ρao) 2π ∫∫exp{-μρ} (1/(μ + ν))(μ2 - ν2) dμdν =
[μ: 1 → ∞, ν: -1 → +1]


(ρ)2EI ∫∫ exp{- μρ}(μ - ν) dμdν =
[μ: 1 → ∞, ν: -1 → +1]

(ρ)2EI ∫exp{- μρ}dμ ∫ (μ - ν) dν =
[μ: 1 → ∞, ν: -1 → +1]
=
(ρ)2EI ∫exp{- μρ}dμ (μ 2 - 0) =
[μ: 1 → ∞]

(ρ)2EI 2 ∫exp{- μρ}dμ = (2/ρ2)[- μρ - 1]exp{- μρ}
[μ: 1 → ∞]

= (ρ)2EI (2/ρ2)[ ρ + 1]exp{- ρ}

Hence:
1|e2/r12> = 2EI [ρ + 1] exp{- ρ}


1|e2/r12> = 2EI (ρ + 1) exp(- ρ)



  


chimie labs
|
Physics and Measurements
|
Probability & Statistics
|
Combinatorics - Probability
|
Chimie
|
Optics
|
contact
|


© Scientificsentence 2010. All rights reserved.