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> are 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.

For example, putting x in place of A and p_x in place of G, we get 
Dx Dp_x  ≥  ½ |<Y | [ ^x, ^p_x ] | Y>|. Now we know that 
[ ^x, ^p_x ] Y = x.(– i ħ ∂/∂x)Y – (– i ħ ∂/∂x) (xY) = – i ħ x ∂Y/∂x + i ħ ∂/∂x (xY) 
= – i ħ x ∂Y/∂x + i ħY + i ħ x ∂Y/∂x = i ħ Y, so that 
<Y | [ ^x, ^p_x ] | Y> = <Y | i ħ Y> = i ħ <Y|Y> = i ħ (as Y is normalised). 
This gives |<Y | [ ^x, ^p_x ] | Y>| = | i ħ | = ħ, so that Dx Dp_x  ≥  ½ ħ = h/(4p), the well-known relation 
Note: the RHS of the last inequality is h/(4p) i.e., 5.273 x 10^–35 J s, not h/(2p). 

The above concept is formulated also in the form of compatible observables and incompatible observables. A pair of mutually compatible observables are such a pair of observables (e.g., x & p_y) which can be simultaneously observed precisely, without any obstacle regarding the product of their uncertainties. For such a pair of observables, the commutator (e.g., [^x, ^p_y] ) of their corresponding quantum-mechanical operators is zero. The compatibility theorem states just this: "For a pair of mutually compatible observables, the commutator of their corresponding quantum-mechanical operators is zero". On the other hand, incompatible observables are such a pair of observables (e.g., x & p_x) which can't be simultaneously observed precisely, with the product of their uncertainties (i.e., of standard deviations) can't be possible to become smaller than a certain minimum limit (this limit being the half of the modulus of the expectation value of the commutator of their corresponding quantum-mechanical operators, as per the generalized uncertainty principle mentioned above). For them the commutator of their corresponding quantum-mechanical operators is not zero.