Oct 30 (Fri) Uncertainty Principle [Physics Intranet]

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Oct 30 (Fri) Uncertainty Principle

Responsible party: Esquire, David Hilbert's Hat

To go back to the lecture note list, click Lecture Notes
previous lecture note: Oct 28 (Wed) 3.4 generalized probability, delta-func normalization
next lecture note: Nov 02 (Mon) Main Topics in Chap 4, Separation variables for Spherical coordinate

Main class wiki page: Physics 4101.001 QM wiki page

Generalized Uncertainty Principle:

$\sigma_{A}^2\sigma_B^2\geq|\frac{[\hat{A},\hat{B}]}{2i}|^2$

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

$[\hat{x},\hat{p}]=i\hbar$ ⇒ $\sigma_{x}^2\sigma_p^2\geq|\frac{i\hbar}{2i}|^2=(\frac{\hbar}{2})^2$ ⇒ $\sigma_{x}\sigma_p\geq\frac{\hbar}{2}$

Derivation:

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

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

Assuming $\hat{A}$ is Hermitian,

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

Now Let f and g be functions of A and B and psi respectively,

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

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

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

This works if you think of |f> and |g> as vectors with an angle ( $\theta$ ) separating them, in this case

$|f|^2|g|^2\geq|f|^2|g|^2cos^2\theta$

Keep in mind, f and g can be (and often are) complex. Any complex number can be expressed as follows:

$Z=x+iy$ so $|z|^2 \geq x^2+y^2 => y=\frac{z-z^*}{2i}$

This is analogous to $\frac{<f|g>-<g|f>}{2i}\geq\frac{[\hat{A},\hat{B}]}{2i}$

To go back to the lecture note list, click Lecture Notes
previous lecture note: Oct 28 (Wed) 3.4 generalized probability, delta-func normalization
next lecture note: Nov 02 (Mon) Main Topics in Chap 4, Separation variables for Spherical coordinate