Physical Chemistry - Quantum in 3D Separable Systems_lecture16

5.61 Fall 2007 Separable Systems page 1




QUANTUM IN : SEPARABLE SYSTEMS

1D Systems 3D Systems
� � �
x̂ r̂ = ( x̂ ŷ ẑ ) = i x̂ + j ŷ + k ẑ
� d �� ∂ �� ∂ �� ∂
p̂ =
i dx (
p̂ = p̂x p̂y p̂z = i )
i ∂x
+j
i ∂y
+k
i ∂y
⎡⎣ x̂, p̂⎦⎤ = i� ⎡⎣ x̂, p̂x ⎤⎦ = i� ⎡ yˆ, pˆ y ⎤ = i� ⎡⎣ ẑ, p̂z ⎤⎦ = i�
⎣ ⎦
p̂2 −�2 d 2 p̂2 −�2 ∂ 2 −�2 ∂ 2 −�2 ∂ 2
Tˆ = = Tˆ = = + +
2m 2m dx 2 2m 2m ∂x 2 2m ∂y2 2m ∂y 2
ψ (x) ψ ( x, y, z )
∫ψ ( x ) Ô ψ ( x ) dx ∫ψ ( x, y, z ) Ô ψ ( x, y, z ) dx dy dz
* *
Ô = Ô =


By fiat, operators corresponding to different axes commute with one another.
ˆ ˆ = yx
xy ˆˆ pˆ z yˆ = yp
ˆˆz pˆ z pˆ x = pˆ x pˆ z etc.
Further, operators in one variable have no effect on functions of another:
ˆ ( y ) = f ( y ) xˆ
xf pˆ z f ( x ) = f ( x ) pˆ z f * ( z ) pˆ x = pˆ x f * ( z ) etc.

The Time Independent Schrödinger Equation becomes:
⎡ �2 ⎛ ∂ 2 ∂2 ∂2 ⎞ ⎤
⎢ − ⎜ 2 + 2
+ 2 ⎟
+ V ( x̂, ŷ, ẑ ) ⎥ψ ( x, y, z ) = Eψ ( x, y, z )
⎣ 2m ⎝ ∂x ∂y ∂z ⎠ ⎦

∇ 2 the Laplacian

⎡ �2 2 ⎤
⇒ ⎢ − 2m ∇ + V ( x̂, ŷ, ẑ ) ⎥ψ ( x, y, z ) = Eψ ( x, y, z )
⎣ ⎦

�2 2
Ĥ = − ∇ + V ( x̂, yˆ, zˆ ) Hamiltonian operator in 3D
2m

Hˆψ ( x, y , z ) = Eψ ( x, y , z ) 3D Schrödinger equation
(Time Independent)
Separation of variables

, 5.61 Fall 2007 Separable Systems page 2




IF V ( x̂, ŷ , ẑ ) = Vx ( x̂ ) + V y ( ŷ ) + Vz ( ẑ )


ˆ ⎡ �2 ∂ 2 ⎤ ⎡ �2 ∂ 2 ⎤ ⎡ �2 ∂ 2 ⎤
H ( x, y, z ) = ⎢ − 2
+ V x ( ˆ
x ) + −
⎥ ⎢ 2m ∂y 2 + V y ( ˆ
y ) + −
⎥ ⎢ 2m ∂z 2 + Vz ( ˆ
z ) ⎥
then ⎣ 2m ∂x ⎦ ⎣ ⎦ ⎣ ⎦
= Ĥ x + Ĥ y + Ĥ z
⇒ Schrödinger’s Eq. becomes:
⎡ Ĥ x + Ĥ y + Ĥ z ⎤ψ ( x, y, z ) = Eψ ( x, y, z )
⎣ ⎦

Then try solution of form ψ (x, y, z )= ψ x (x )ψ y (y )ψ z (z )
(separation of variables)
Where we assume that the 1D functions satisfy the appropriate 1D TISE:
Ĥ xψ x ( x ) = E xψ x ( x )
Ĥ yψ y ( y ) = E yψ y ( y )
Ĥ zψ z ( z ) = E zψ z ( z )

First term:
Ĥ xψ x ( x )ψ y ( y )ψ z ( z ) = ψ y ( y )ψ z ( z ) Ĥ xψ x ( x ) = ψ y ( y )ψ z ( z ) E xψ x ( x )
= E xψ x ( x )ψ y ( y )ψ z ( z )

Same for Ĥ y and Ĥ z ⇒

Ĥ ψ = E ψ
⎡ Hˆ x + Hˆ y + Hˆ z ⎤ ⎡⎣ψ x ( x )ψ y ( y )ψ z ( z )⎤⎦ = ( E x + E y + E z ) ⎣⎡ψ x ( x )ψ y ( y )ψ z ( z )⎦⎤
⎣ ⎦

E = Ex + E y + Ez
Thus, if the Hamiltonian has this special form, the eigenfunctions of the 3D
Hamiltonian are just products of the eigenfunctions of the 1D Hamiltonian and
the situation is equivalent to doing three separate 1D problems.


Conclusion: Wavefunctions multiply and the energies add if Ĥ is separable into