A Derivation of Ehrenfest's Theorem in a particular case

Question

What are the missing lines in the integration?

$$\frac{\text d \langle {p} \rangle}{ \text{d} t} $$ $$= \frac{\text d}{\text d t} \int\limits_{-\infty}^{\infty} \Psi^* \left( \frac{\hbar}{i}\frac{\partial}{\partial x} \right) \Psi ~\text d x$$ $$= \frac{\hbar}{i}\int\frac{\partial }{\partial t} \left(\Psi^*\frac{\partial\Psi}{\partial x}\right)~\text d x $$ $$= \frac{\hbar}{i}\int \frac{\partial \Psi^*}{\partial t}\frac{\partial \Psi}{\partial x} +\Psi^*\frac{\partial }{\partial x}\frac{\partial \Psi }{ \partial t} ~\text {d} x$$ $$= \frac{\hbar}{i}\int\left( -\frac{i\hbar}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}+\frac{i}{\hbar} V\Psi^*\right)\frac{\partial \Psi}{\partial x}+\Psi^* \frac{\partial}{\partial x} \left(\frac{i\hbar}{2m} \frac{\partial^2 \Psi }{\partial x^2} -\frac{i}{\hbar}V\Psi \right)~\text d x$$ $$=\int\left(V\Psi^*-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^*}{\partial x^2}\right) \frac{\partial \Psi}{\partial x}+\Psi^*\frac{\partial}{\partial x}\left(\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}-V\Psi\right)\text d x$$

$$=\left. \left(V\Psi^*-\frac{\hbar^2}{2m}\frac{\partial^2 \Psi^*}{ \partial x^2} \right) \Psi \right|_{-\infty}^{\infty}-\int\left(\frac{\partial}{\partial x} (V\Psi^*)-\frac{\hbar^2}{2m}\frac{\partial^3 \Psi^*}{ \partial x^3}\right)\Psi \text d x+ \left.\Psi^*\left(\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} - V\Psi\right)\right|_{-\infty}^\infty-\int\frac{\partial\Psi^*}{\partial x} \left( \frac{\hbar^2}{2m}\frac{\partial ^2 \Psi}{\partial x^2} - V \Psi \right)\text d x$$

$$=0 + \int\left(\frac{\hbar^2}{2m}\frac{\partial^3 \Psi^*}{ \partial x^3}-\frac{\partial}{\partial x} (V\Psi^*)\right) \Psi \text d x+0+\int\frac{\partial\Psi^*}{\partial x}\left(V\Psi - \frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2}\right) \text d x$$ $$=\int\frac{\hbar^2}{2m} \left(\frac{\partial^3\Psi^*}{\partial x^3}\Psi -\frac{\partial\Psi^*}{\partial x} \frac{\partial^2\Psi}{\partial x^2}\right)+\frac{\partial\Psi^*}{\partial x}(V\Psi )-\frac{\partial}{\partial x}(V\Psi^*)\Psi\text d x $$

$$\vdots$$ $$=\int \frac{\hbar^2}{2m}\left( \Psi^* \frac{\partial^3 \Psi}{\partial x^3} -\frac {\partial^2 \Psi^*}{\partial x^2} \frac{\partial \Psi}{\partial x} \right)+\left( V \Psi^* \frac{\partial \Psi}{\partial x}-\Psi^* \frac{\partial}{\partial x} (V \Psi) \right)~\text d x$$ $$\vdots $$ $$= \int \left( V \Psi^* \frac{\partial \Psi}{\partial x}- \Psi^* \frac{\partial V}{\partial x} \Psi-\Psi^*V \frac{\partial \Psi}{\partial x} \right)~\text d x$$ $$= \int\limits_{-\infty}^{\infty} -\Psi^* \frac{\partial V}{\partial x} \Psi ~\text {d} x$$ $$=\left\langle - \frac{ \partial V }{\partial x} \right\rangle $$

Answer 1

That's kind of horrible. But in the first three dots, the changed quantity is at least not wrong, as $$\Psi^*_{,x}(V\Psi) - (V\Psi^*)_{,x}\Psi = V\Psi\Psi^*_{.x} - (\Psi^*V_{,x} + V\Psi^*_{,x})\Psi = -\Psi^* V_{,x} \Psi$$ is definitely the same as $$V\Psi^*\Psi_{,x} - \Psi^*(V\Psi)_{,x} = \Psi^*V\Psi_{,x} - \Psi^*(V_{,x}\Psi + V\Psi_{,x}) = - \Psi^*V_{,x}\Psi.$$ The global conjugation flip over the entire integral is itself valid if that integral is real (since $\langle a|b \rangle = \langle b|a \rangle^*$), and we should expect it to be real on physical grounds, being $d\langle p\rangle/dt$. Though why would one want to do anything of the sort instead of directly proceeding to the goal, I really have no idea. In any case, this makes it clear that what's really missing is a proof of $$\int\left(\frac{\partial^3\Psi^*}{\partial x^3}\Psi -\frac{\partial\Psi^*}{\partial x} \frac{\partial^2\Psi}{\partial x^2}\right)~\mathrm{d}x = 0 $$ or $$\int \left( \Psi^* \frac{\partial^3 \Psi}{\partial x^3} -\frac {\partial^2 \Psi^*}{\partial x^2} \frac{\partial \Psi}{\partial x} \right)~\mathrm{d}x = 0$$ These are also correct for nice enough boundary conditions, and can be shown using integration by parts twice on the first term.

Another method: by differentiation the definition of expectation value and substituting the Schrödinger equation for the time derivative of the state, $$\begin{eqnarray*} \frac{d\langle A\rangle}{dt} &=& \langle\dot{\psi}|A|\psi\rangle + \langle\psi|\dot{A}|\psi\rangle + \langle\psi|A|\dot{\psi}\rangle\\ &=& -\frac{1}{i\hbar}\langle\psi|H^\dagger A|\psi\rangle + \left\langle\frac{\partial A}{\partial t}\right\rangle + \frac{1}{i\hbar}\langle\psi|AH|\psi\rangle\\ &=& \left\langle\frac{\partial A}{\partial t}\right\rangle + \frac{1}{i\hbar}\left\langle[A,H]\right\rangle \end{eqnarray*}$$ Since $[p,H] = [p,V]$ is obvious, it is then easy to verify that $[p,V] = -i\hbar(\nabla V)$, and $p$ is not explicitly dependent on time, so that term doesn't even appear in that case. This scheme can be done without explicit bra-ket notation, but even the position-space integral representation is simpler if one leaves the Hamiltonian as an abstract operator for a while.

Answer 2

$$ \newcommand \ps [1] {\left(#1\right)} % parentheses (auto) \newcommand \bs [1] {\left[#1\right]} % brackets (auto) \newcommand \expt [1] {\left\langle#1\right\rangle} % expectation value \newcommand \de [2] {\frac{\mathrm d #1}{\mathrm d#2}} % first order derivative \newcommand \den [3] {\frac{\mathrm d^#3 #1}{\mathrm d#2^#3}} % n-th derivative \newcommand \pa [2] {\frac{\partial #1}{\partial #2}} % partial derivative \newcommand \pan [3] {\frac{\partial^#3 #1}{\partial#2^#3}} % n-th partial derivative \newcommand \inti [2]{\int\limits_{-\infty}^\infty{#1}\,\mathrm d #2} % integral -oo -> oo $$

\begin{align*} \de{\expt p}t &= \de{}t\inti{\Psi^*\ps{\frac\hbar i\pa{}x}\Psi}x\\ &= \frac\hbar i\inti{\pa{}t\ps{\Psi^*\pa\Psi x}}x\\ &= \frac\hbar i\inti{\left(\pa{\Psi^*}t\pa\Psi x+\Psi^*\pa{}x\pa\Psi t\right)}x % &\hspace{3cm} \text{from the Schr\"odinger Equation}\\ &i\hbar\pa\Psi t &= -\frac{\hbar^2}{2m}\pan\Psi x2+V\Psi\\ && \pa\Psi t &= \frac{i\hbar}{2m}\pan\Psi x2-\frac i\hbar V\Psi\\ && \pa{\Psi^*}t &= -\frac{i\hbar}{2m}\pan{\Psi^*}x2+\frac i\hbar V\Psi^*\\[-4ex] &= \frac\hbar i\inti{\ps{\bs{-\frac{i\hbar}{2m}\pan{\Psi^*}x2+\frac i\hbar V\Psi^*}\pa\Psi x +\Psi^*\pa{}x \bs{\frac{i\hbar}{2m}\pan{\Psi}x2-\frac i\hbar V\Psi}}}x\\ % &\hspace{3cm} \text{rearranging}\\ &= \frac\hbar i\inti{\frac{i\hbar}{2m}\ps{\Psi^*\pan\Psi x3-\pan{\Psi^*}x2\pa\Psi x} + \frac i\hbar\ps{V\Psi^*\pa\Psi x-\Psi^*\pa{}x(V\Psi)}}x\\ &= \frac{\hbar^2}{2m}\inti{\pa{}x\ps{\Psi^*\pan\Psi x2-\pan{\Psi^*}x2\Psi}}x + \inti{\ps{V\Psi^*\pa\Psi x-\Psi^*\pa Vx\Psi-\Psi^*V\pa\Psi x}}x\hspace{-4em}\\ &= \frac{\hbar^2}{2m}\ps{\Psi^*\pan\Psi x2-\pan{\Psi^*}x2\Psi}\Big|_{-\infty}^\infty - \inti{\Psi^*\pa Vx\Psi}x &\hspace{-5em} &\Psi(\pm\infty),\Psi^*(\pm\infty)=0\\[-2ex] &= 0-\expt{\pa Vx}\\ &= \expt{-\pa Vx} \end{align*}