===== Oct 30 (Fri) Uncertainty Principle =====
** Responsible party: Esquire, David Hilbert's Hat   ** 

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===Generalized Uncertainty Principle:=== 
The general formula is given by Griffiths as:
<math>\sigma_{A}^2\sigma_B^2\geq ( \frac{<[\hat{A},\hat{B}]>}{2i} )^2</math>

For example, use the canonical commutation relation [x, p] = i<math>\hbar</math>. Then, 
<math>\sigma_{x}^2\sigma_{p}^2\geq (\frac{i\hbar}{2i})^2 </math> => <math>\sigma_{x}\sigma_{p}\geq\hbar/2</math>. In general, the commutation relation between any non-commuting operators has an imaginary term, which will cancel the imaginary part of the 2i term in the generalized uncertainty principle formula. Recall that this only works with x and <math> p_{x} </math>; for example, <math> [x, p_{y}] = 0 </math> so the uncertainty in x and <math> p_{y} </math> is zero. That is, you can measure x and <math> p_{y} </math> simultaneously. 

1-Dimensional Example: Let <math>\hat{A}=\hat{x}=x</math> and <math>\hat{B}=\hat{p}=-i\hbar(\frac{\partial}{\partial x})</math>

<math>[\hat{x},\hat{p}]=i\hbar </math> => <math>\sigma_{x}^2\sigma_p^2\geq|\frac{i\hbar}{2i}|^2=(\frac{\hbar}{2})^2</math> => <math>\sigma_{x}\sigma_p\geq\frac{\hbar}{2}</math>

===Derivation:===
Start with the general form for any observable A with operator <math> \hat{A} </math>. 

<math>\sigma_A^2=<(\hat{A}-<A>)^2>_{\psi} </math>

<math><\psi|(\hat{A}-<A>)^2\psi>=<(\hat{A}-<A>)\psi|(\hat{A}-<A>)\psi> </math>

Which will be true for any observable A since all observables are Hermitian. 

Assuming <math>\hat{A}</math> is Hermitian,

<math>(\hat{A}-<A>)(\hat{A}-<A>)|\psi>=(|(\hat{A}-<A>)\psi>)^2 </math>

Now Let f and g be functions of A, <math> \psi </math> and B, <math> \psi </math> respectively,

<math>\sigma_A^2=<f|f> </math> where <math>|f>=(\hat{A}-<A>)\psi </math>

<math>\sigma_B^2=<g|g> </math> where <math>|g>=(\hat{B}-<B>)\psi </math>

Then invoking the Schwartz Inequality, which is true for any vectors in an inner product space such as Hilbert space:

<math>\sigma_A^2\sigma_B^2=<f|f><g|g> </math> <math>\geq</math>  <math> |<f|g>|^2 </math>

Geometrically this is similar to a simple case of |f> and |g> as vectors in 3D space with an angle (<math>\theta </math>) separating them, such as

<math>|f|^2|g|^2\geq|f|^2|g|^2cos^2\theta </math>

Keep in mind, f and g can be (and often are) complex when you have <f|g>. Any complex number can be expressed as follows:

<math> z=x+iy </math> 

As having a real part and an imaginary part, where x and y are both real. 

<math>|z|^2 = x^2+y^2 \geq y^2 </math>

Now we know that the eigenvalues of any observable <math> \hat{A} </math> and <math> \hat{B} </math> are real, and that the commutation relation between A and B is complex. So we can solve for the imaginary part of <math> |z|^2 </math> to obtain y:

<math> x + iy - (x - iy) = 2iy = z - z* </math> and solving for y, <math> y = \frac{z - z*}{2i} </math>.

Now we can let z be the inner product of our two vectors, and y will be the commutation relation between them; 

<math> z = <f|g> </math>, <math> z* = <g|f> </math>, and <math> \frac{z - z*}{2i} = \frac{<f|g> - <g|f>}{2i} </math>

and this numerator is nothing but <math> [\hat{A}, \hat{B}] </math>, 

and 

<math>\sigma_A^2\sigma_B^2 </math>  <math>\geq</math>  <math> |<f|g>|^2 </math>  <math> \geq </math> <math> y^2 = (\frac{z - z*}{2i})^2 </math>, so


<math>\sigma_{A}^2\sigma_B^2\geq ( \frac{[\hat{A},\hat{B}]}{2i} )^2</math>

This relation will hold for any general pair of observables A and B which do not commute - these are the so called "incompatible observables." A rather famous example outside of x and p is the pair E and t. Incompatible observables do not share a complete set of eigenfunctions, meaning that any two of these observables cannot be expressed with one complete set of eigenfunctions, so you can't measure the two observables at once. 




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