Page 55 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

46 Schro Èdinger wave mechanics

1 1 ÿ" @Ø " @Ø
2
(ÄxÄp) ˆ (xØ )(xØ)dx dx
i @x i @x
ÿ1 ÿ1
Applying Schwarz's inequality (A.56), we obtain

2
1 " 1 @Ø @Ø 2 " 1 @ 2
2

(ÄxÄp) > xØ ‡ xØ dx ˆ x (Ø Ø)dx

4 i ÿ1 @x @x 4 ÿ1 @x

2 1 1 2
"
ˆ xØ Ø ÿ Ø Ø dx
4
ÿ1 ÿ1
p 
The integrated part vanishes because Ø goes to zero faster than 1= jxj,as x
approaches ( ) in®nity and the remaining integral is unity by equation (2.9).
Taking the square root, we obtain an explicit form of the Heisenberg uncer-
tainty principle
"
ÄxÄp > (2:26)
2
This expression is consistent with the earlier form, equation (1.44), but relation
(2.26) is based on a precise de®nition of the uncertainties, whereas relation
(1.44) is not.
2.4 Time-independent Schro Èdinger equation
The ®rst step in the solution of the partial differential equation (2.6) is to
express the wave function Ø(x, t) as the product of two functions
Ø(x, t) ˆ ø(x)÷(t) (2:27)
where ø(x) is a function of only the distance x and ÷(t) is a function of only
the time t. Substitution of equation (2.27) into (2.6) and division by the product
ø(x)÷(t)give
2
1 d÷(t) " 2 1 d ø(x)
i" ˆÿ ‡ V(x) (2:28)
÷(t) dt 2m ø(x) dx 2
The left-hand side of equation (2.28) is a function only of t, while the right-
hand side is a function only of x. Since x and t are independent variables, each
side of equation (2.28) must equal a constant. If this were not true, then the
left-hand side could be changed by varying t while the right-hand side
remained ®xed and so the equality would no longer apply. For reasons that will
soon be apparent, we designate this separation constant by E and assume that
it is a real number.
Equation (2.28) is now separable into two independent differential equations,
one for each of the two independent variables x and t. The time-dependent
equation is