Schmidt Orthogonalization Technique

Starting from a set of linearly independent eigenfunctions {f₁, f₂, ....., f_n} of a Hermitian operator with a degenerate eigenvalue, the application of the Schmidt Orthogonalization Technique gives a new set {F₁, F₂, ....., F_n} of eigenfunctions with the same eigenvalue, such that the new set is an orthogonal set.

According to this procedure, the first new function F₁ is taken to be equal to f₁. F₂ is taken to be a linear combination of f₂ and F₁, i.e., to be f₂ + c₂₁F₁ (where c₂₁ is a constant). Similarly, the other functions are defined. Thus we get:

F₁ = f₁
F₂ = f₂ + c₂₁F₁
F₃ = f₃ + c₃₂F₂ + c₃₁F₁
..........
F_j = f_j + _i=1S^j-1 c_ji F_i
..........
F_n = f_n + _i=1S^n-1 c_ni F_i

Orthogonality of F₁ & F₂ gives <F₁ | F₂> = 0, which means <F₁ | f₂> + c₂₁<F₁ | F₁> = 0, thus giving
c₂₁ =  – <F₁ | f₂> / <F₁ | F₁>
Orthogonality of F₁ & F₃ gives <F₁ | F₃> = 0, which means <F₁ | f₃> + c₃₂<F₁ | F₂> + c₃₁<F₁ | F₁> = 0, thus giving c₃₁ =  – <F₁ | f₃> / <F₁ | F₁> (as <F₁ | F₂> = 0 because of orthogonality of F₁ & F₂).
 Similarly, <F₂ | F₃> = 0 implies c₃₂ =  – <F₂ | f₃> / <F₂ | F₂>

Thus we arrive at a general formula for the coefficients c_ji (for i < j) as c_ji =  – <F_i | f_j> / <F_i | F_i>. As the new functions F_j gets defined (as F_j = f_j + _i=1S^j-1 c_ji F_i) when these coefficients are defined, the new orthogonal set of functions has been fully obtained.
 

Schwartz Inequality and Uncertainty Principle

For any two arbitrary well-behaved functions f and g the following relation is obeyed:
4 <f | f><g | g>  ≥  (<f | g> + <g | f>)²  This relation is known as the Schwartz inequality.
Proof: Let I = <(f + sg) | (f + sg)> where s is an arbitrary real parameter. In the integral I, the integrand
|(f + sg)|² is everywhere non-negative (i.e., positive or zero), and so I is positive, unless it happens that f = - sg (in which case I is zero as the integrand becomes everywhere zero). So we have two possible cases: (1) f = -sg in which case I = 0 and (2) f ≠ -sg in which case I > 0. However, by expanding the expression for I, we get
I = <f | f> + s <f | g> + s* <g | f> + ss* <g | g> = <f | f> + s (<f | g> + <g | f>) + s² <g | g>
(as s is by definition real). Calling <g | g> = a,  (<f | g> + <g | f>) = b and <f | f> = c, we get I = as² + bs + c with a, b, c being some constant integrals. Now considering the more general case (2) of  f ≠ -sg and I > 0, we get as² + bs + c > 0 where s is real. This means that no real root for s exists for the equation as² + bs + c = 0, meaning that (b² - 4ac) is negative giving only non-real complex roots for this quadratic equation. So we get, for f ≠ - sg, b² - 4ac < 0 i.e., 4ac > b² i.e.,
4<f | f><g | g>   >  (<f | g> + <g | f>)²     --------------- (i)
For the more specific case (1) of  f = -sg, < f | f > = < (-sg) | (-sg) > = s² < g | g > whereas
<f | g> = -s* <g | g> = -s <g | g> and <g | f> = -s <g | g>. These relations give:
4<f | f><g | g> = (<f | g> + <g | f>)²     --------------- (ii)
Thus, combining the two possible cases, we arrive at the Schwartz inequality:
4<f | f> <g | g>  ≥  (<f | g> + <g | f>)²

In the quantitative formulation of the (generalized) uncertainty principle stated as
DA. DG  ≥   ½ |<y | [Â, Ĝ] | y>|, the stated uncertainties DA and DG are nothing but the standard deviations in measurement of the physical observables A & G, where  & Ĝ are the corresponding Hermitian operators. This relation can be derived starting from the Schwartz inequality as follows:
From postulates of quantum mechanics, it is obvious that the standard deviations DA is given by:
DA = (<y | ² | y> - <y< |  | y>²)^1/2 (square root of the difference between average of square and square of average). In the Schwartz inequality, let us choose the arbitrary functions f & g as f = ( - <Â>)Y and g = i (Ĝ - <Ĝ>)Y, where i = √(-1) and Y is the normalized system wavefunction (with <Y | Y> = 1). Now
<f | f> = <(Â - <Â>)Y | (Â - <Â>)Y>
= <ÂY | ÂY> - <ÂY | <Â>Y> - < <Â>Y |  ÂY> + < <Â>Y | <Â>Y>
= <Y | ²Y> - <Â> <Y | ÂY> - <Â>^* < Y |  ÂY> + <Â> <Â>^* < Y | Y>       (as  is Hermitian)
= <Y | ² | Y> - <Â> <Y | ÂY> - <Â> < Y |  ÂY> + <Â> <Â>          (as <Â>^* = <Â>)
= <Y | ² | Y> - <Y > |  | Y>² = (DA)²
while <g | g> = <i(Ĝ - <Ĝ>)Y | i(Ĝ - <Ĝ>)Y>
= i(-i) <(Ĝ - <Ĝ>)Y | (Ĝ - <Ĝ>)Y>
=  <(Ĝ - <Ĝ>)Y | (Ĝ - <Ĝ>)Y>
= (DG)²       (similarly)
Combining, it gives <f | f> <g | g> = (DA)² (DG)² . Now let us look for <f | g> and <g | f>:
<f | g> = <(Â - <Â>)Y | i(Ĝ - <Ĝ>)Y>
= i <ÂY | ĜY> - i <Ĝ> <ÂY | Y> - i <Â>^* < Y | ĜY> + i <Â>^* <Ĝ> < Y | Y>
=  i <Y | ÂĜ | Y> - i <Ĝ> <Y | Â | Y> - i <Â>^* < Y | Ĝ | Y> + i <Â>^* <Ĝ> < Y | Y>   (as Â, Ĝ Hermitian)
=  i <Y | ÂĜ | Y> -  i <Â> <Ĝ>    (as <Â>, <Ĝ> real)
while <g | f> = -i <(Ĝ - <Ĝ>)Y | (Â - <Â>)Y> = (-1)(i <Y | ĜÂ | Y> -  i  <Ĝ> <Â>)    (similarly)
So, adding them we get, <f | g> + <g | f> = i <Y | ÂĜ | Y> - i <Y | ĜÂ | Y>
= i <Y | (ÂĜ - ĜÂ) | Y>  =  i <Y | [ Â, Ĝ ] | Y>
Now application of the Schwartz inequality 4 <f | f><g | g>  ≥  (<f | g> + <g | f>)² gives
4 (DA)² (DG)²  ≥  (i <Y | [ Â, Ĝ ] | Y>)²   i.e., 4 (DA)² (DG)²  ≥  - <Y | [ Â, Ĝ ] | Y>²
Taking modulus of square root on both sides we get
2 (DA) (DG)  ≥  |i <Y | [ Â, Ĝ ] | Y>| or, 2 (DA) (DG)  ≥  |<Y | [ Â, Ĝ ] | Y>|  i.e.,
DA  DG  ≥  ½ |<Y | [ Â, Ĝ ] | Y>| , which is the (generalized) uncertainty principle.