[Prepared for B.Sc. Chemistry 3rd year and M.Sc. Chemistry 1st semester students at Cotton College (under Gauhati University). Prepared by Rituraj Kalita]
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Schmidt Orthogonalization Technique

Starting from a set of linearly independent eigenfunctions {f1, f2, ....., fn} of a Hermitian operator with a degenerate eigenvalue, the application of the Schmidt Orthogonalization Technique gives a new set {F1, F2, ....., Fn} 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
..........
Fj = fj + i=1Sj1 cji Fi
..........
Fn = fn + i=1Sn1 cni Fi

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 cji (for i < j) as cji<Fi | fj> / <Fi | Fi>. As the new functions Fj gets defined (as Fj = fj + i=1Sj1 cji Fi) 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
D
A. 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 | Â2 | y> – <y | Â | 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 – <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>
= <Â
Y | ÂY> – <ÂY | <Â>Y> – < <Â>Y |  ÂY> + < <Â>Y | <Â>Y>
= <
Y | Â2Y> – <Â> <Y | ÂY> – <Â>* < Y |  ÂY> + <Â> <Â>* < Y | Y>       (as  is Hermitian)

= <Y | Â2 | Y> – <Â> <Y | ÂY> – <Â> < Y |  ÂY> + <Â> <Â>          (as <Â>* = <Â>)
= <Y | Â2 | Y> – <Y | Â | 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)
C
ombining, 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 px in place of G, we get 
D
x Dpx  ≥  ½ |<Y | [ ^x, ^px ] | Y>|. Now we know that 
[ ^x,
^px ] 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, ^px ] | Y> = <Y | i ħ Y> = i ħ <Y|Y> = i ħ (as Y is normalised). 
This gives |<
Y | [ ^x, ^px ] | Y>| = | i ħ | = ħ, so that Dx Dpx  ≥  ½ ħ = h/(4p), the well-known relation 
Note: the RHS of the last inequality is h/(4
p) 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 & py) 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, ^py] ) 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 & px) 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.