deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 4d9fe71..19df3b1 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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s^+s|n>=\sqrt{n+\frac{1}{2}}|n> \\  r^+r|n>=\sqrt{n}|n> \\  ss^+|n>=\sqrt{n+1}|n> \\  rr^+|n>=\sqrt{n+\frac{1}{2}}|n> rr^+|n>=\sqrt{n+\frac{1}{2}}|n>\\  \\  sr^+=\sqrt[4]{n+\frac{1}{2}}\sqrt[4]{n+1} \\  rs^+=sr^+, Herm.. \\  r^+s=\sqrt[4]{n}\sqrt[4]{n+\frac{1}{2}} \\  s^+r=r^+s, Herm...  \end{equation}  So we have mixed hermitian eigenoperators of state |n>.  Which means a double application of r^+r is equal to the number operator N=a^+a, a double application of ss^+ is equivalent to the operator $aa^+$ and it implies that $s^+s \equiv rr^+$. This is interesting, we can measure the number of particles in an oscillator by only taking half of one out and putting it back (r then r^+ is the operation r^+r), but there are two ways to take the particle out and two ways to put it back and each appear as different actions with different 'compensations'.  These sub-operations work in line with the conjugate operation. if we introduce a basic ladder opperator with no coefficients \begin{equation}