### Derivation of (3.44)

(3.44) ${\frac {d}{dt}}\langle {\hat {\theta }}\rangle _{t}={\frac {1}{i\hbar }}\langle {[{\hat {\theta }},H]}\rangle _{t}+\langle {\frac {\partial {\hat {\theta }}}{\partial {t}}}\rangle _{t}$

${\frac {d}{dt}}\langle {\hat {\theta }}\rangle _{t}={\frac {d}{dt}}\int _{\tau }\psi ^{*}{\hat {\theta }}\psi \,d\tau$

- $=\int _{\tau }{\frac {d}{dt}}\left(\psi ^{*}{\hat {\theta }}\psi \right)\,d\tau$ since space and time commute
- $=\int _{\tau }\left({\frac {\partial {\psi ^{*}}}{\partial {t}}}{\hat {\theta }}\psi +\psi ^{*}{\frac {\partial {\hat {\theta }}}{\partial {t}}}\psi +\psi ^{*}{\hat {\theta }}{\frac {\partial {\psi }}{\partial {t}}}\right)\,d\tau$ by Liebniz's rule for differentiation (mathworld.wolfram.com)
- $=\int _{\tau }{\frac {\partial {\psi ^{*}}}{\partial {t}}}{\hat {\theta }}\psi \,d\tau +\int _{\tau }\psi ^{*}{\frac {\partial {\hat {\theta }}}{\partial {t}}}\psi \,d\tau +\int _{\tau }\psi ^{*}{\hat {\theta }}{\frac {\partial {\psi }}{\partial {t}}}\,d\tau$
- $=\int _{\tau }{\frac {\partial {\psi ^{*}}}{\partial {t}}}{\hat {\theta }}\psi \,d\tau +\int _{\tau }\psi ^{*}{\hat {\theta }}{\frac {\partial {\psi }}{\partial {t}}}\,d\tau +\int _{\tau }\psi ^{*}{\frac {\partial {\hat {\theta }}}{\partial {t}}}\psi \,d\tau$
- $=\int _{\tau }{\frac {\partial {\psi ^{*}}}{\partial {t}}}{\hat {\theta }}\psi \,d\tau +\int _{\tau }\psi ^{*}{\hat {\theta }}{\frac {\partial {\psi }}{\partial {t}}}\,d\tau +\langle {\frac {\partial {\hat {\theta }}}{\partial {t}}}\rangle _{t}$
- Schrodinger's time dependent equation states ${\hat {H}}\psi =i\hbar {\frac {\partial \psi }{\partial {t}}}$
- Therefore, ${\frac {\partial \psi }{\partial {t}}}={\frac {1}{i\hbar }}{\hat {H}}\psi$ and ${\frac {\partial \psi ^{*}}{\partial {t}}}=({\frac {\partial \psi }{\partial {t}}})^{*}=({\frac {1}{i\hbar }}{\hat {H}}\psi )^{*}=-{\frac {1}{i\hbar }}({\hat {H}}\psi )^{*}$

- $=-{\frac {1}{i\hbar }}\int _{\tau }({\hat {H}}\psi )^{*}{\hat {\theta }}\psi \,d\tau +{\frac {1}{i\hbar }}\int _{\tau }\psi ^{*}{\hat {\theta }}({\hat {H}}\psi )\,d\tau +\langle {\frac {\partial {\hat {\theta }}}{\partial {t}}}\rangle _{t}$
- $={\frac {1}{i\hbar }}\int _{\tau }\left(\psi ^{*}{\hat {\theta }}({\hat {H}}\psi )-({\hat {H}}\psi )^{*}{\hat {\theta }}\psi \right)\,d\tau +\langle {\frac {\partial {\hat {\theta }}}{\partial {t}}}\rangle _{t}$
- $={\frac {1}{i\hbar }}\int _{\tau }\left(\psi ^{*}({\hat {\theta }}{\hat {H}})\psi -({\hat {H}}\psi )^{*}{\hat {\theta }}\psi \right)\,d\tau +\langle {\frac {\partial {\hat {\theta }}}{\partial {t}}}\rangle _{t}$

## Problem 4, Practice Exam I

In the problem, we are given a physical system characterized by a potential $V(x)=\left\{{\begin{matrix}\infty &x\leq 0\\qx&x>0\end{matrix}}\right.$
, where q is positive. Thus we know that the particle can never exist on the negative x axis. Considering this, we can rewrite our potential in a more manipulatable form as $V(x)=\left\{{\begin{matrix}-qx&x<0\\\delta (x)&x=0\\qx&x>0\end{matrix}}\right.$ where it is obvious that a particle starting on the left side exactly mirrors a particle starting on the right side.
To compute $\langle {p_{x}}\rangle _{t}$, we use ${\frac {d}{dt}}\langle {p_{x}}\rangle _{t}={\frac {1}{i\hbar }}\langle {[p_{x},H]}\rangle _{t}+\langle {\frac {\partial {p_{x}}}{\partial {t}}}\rangle _{t}$
$={\frac {1}{i\hbar }}\langle {[p_{x},{\frac {p_{x}^{2}}{2m}}+V]}\rangle _{t}$
$={\frac {1}{i\hbar }}\langle {[p_{x},{\frac {p_{x}^{2}}{2m}}]+[p_{x},V]}\rangle _{t}$
$={\frac {1}{i\hbar }}\langle {[p_{x},V]}\rangle _{t}$
$={\frac {1}{i\hbar }}\int _{-\infty }^{\infty }\,\psi ^{*}[p_{x},V]\psi \,dx$
$={\frac {1}{i\hbar }}\int _{-\infty }^{0}\,\psi ^{*}[p_{x},-qx]\psi \,dx+{\frac {1}{i\hbar }}\int _{0}^{0}\,\psi ^{*}[p_{x},\delta (x)]\psi \,dx+{\frac {1}{i\hbar }}\int _{0}^{\infty }\,\psi ^{*}[p_{x},qx]\psi \,dx$
$={\frac {-q}{i\hbar }}\int _{-\infty }^{0}\,\psi ^{*}[p_{x},x]\psi \,dx+{\frac {1}{i\hbar }}\int _{0}^{0}\,\psi ^{*}[p_{x},\delta (x)]\psi \,dx+{\frac {q}{i\hbar }}\int _{0}^{\infty }\,\psi ^{*}[p_{x},x]\psi \,dx$
$={\frac {-q}{i\hbar }}\int _{-\infty }^{0}\,\psi ^{*}(-i\hbar )\psi \,dx+{\frac {1}{i\hbar }}\int _{0}^{0}\,\psi ^{*}[p_{x},\delta (x)]\psi \,dx+{\frac {q}{i\hbar }}\int _{0}^{\infty }\,\psi ^{*}(-i\hbar )\psi \,dx$
$=q\int _{-\infty }^{0}\,\psi ^{*}\psi \,dx+{\frac {1}{i\hbar }}\int _{0}^{0}\,\psi ^{*}[p_{x},\delta (x)]\psi \,dx-q\int _{0}^{\infty }\,\psi ^{*}\psi \,dx$
$=-2q\int _{0}^{\infty }\,\psi ^{*}\psi \,dx+{\frac {1}{ih}}\int _{0}^{0}\,\psi ^{*}[p_{x},\delta (x)]\psi \,dx$
$=-2q\int _{0}^{\infty }\,\psi ^{*}\psi \,dx+{\frac {1}{ih}}\int _{0}^{0}\,\psi ^{*}[-i\hbar \nabla ,\delta (x)]\psi \,dx$
$=-2q\int _{0}^{\infty }\,\psi ^{*}\psi \,dx-\int _{0}^{0}\,[\nabla ,\delta (x)]\,dx$
$=-2q\int _{0}^{\infty }\,\psi ^{*}\psi \,dx-\int _{0}^{0}\,\psi ^{*}[{\frac {\partial {}}{\partial {x}}},\delta (x)]\psi \,dx$

Now $[{\frac {\partial {}}{\partial {x}}},\delta (x)]\psi ={\frac {\partial {}}{\partial {x}}}(\delta (x)\psi )-\delta (x){\frac {\partial {}}{\partial {x}}}(\psi )={\frac {\partial {}}{\partial {x}}}(\delta (x))\psi +\delta (x){\frac {\partial {}}{\partial {x}}}(\psi )-\delta (x){\frac {\partial {}}{\partial {x}}}(\psi )$

So, $[{\frac {\partial {}}{\partial {x}}},\delta (x)]={\frac {\partial {\delta (x)}}{\partial {x}}}$

The probability of a particle being in one half or the other is exactly one half, therefore the integral from 0 to infinity of the probability amplitude squared must be one half. Thus $-2q\int _{0}^{\infty }\,\psi ^{*}\psi \,dx=-2q({\frac {1}{2}})=-q$

## Theta function

Hi, I answered your question on theta functions. (Also, when you post,please sign your notes with four tilde's in a row, that automatically adds a timestamp and name). linas 04:24, 2 Mar 2005 (UTC)

## Induction & forces

Hello Ub3rm4th

We were talking originally about induced emf and inductance. So having accepted this idea of the necessity of the field inducing the emf, I postulated the notion that every force in the universe howsoever caused, must 'induce' an equal and opposite one (to be in balance). This was quickly realised to be nothing other than a restatement of Newtons 3rd law, but with applicability to ALL forces and combinations of forces. We then concluded that the sum of all forces is the universe, or any closed system must be zero. We thought that this might be a usefule rule to bear in mind when discussing other promlems of an electrodynamical nature. Hope this explains the coversation we had.!!--Light current 17:13, 28 September 2005 (UTC)

## Homework

Can I just remind you that WP is not a place for you to discuss your assignments/homework (unless it has to do with WP articles) even on your own page!--Light current 00:36, 29 September 2005 (UTC)

## Page name for temperature articles

To avoid flip-flopping between 'degree Fahrenheit' and 'Fahrenheit' or 'degree Celsius' and 'Celsius', I propose that we have a discussion on which we want. I see you have contributed on units of measurement, please express your opinion at Talk:Units of measurement. Thanks. bobblewik 22:07, 12 February 2006 (UTC)