Am I doing this right? Finding the time derivative of expectation value of momentum

Question

I am currently working through the problems in Griffiths Intro to Quantum mechanics, with the goal of hopefully understanding how magnets work.

At the moment I am a little stuck on one of the early problems in the first chapter. And I would like a Hint about how to proceed.

The problem is to solve the time derivative of the expectation value of momentum of a particle. ie: $$ \frac{d\left<p\right>}{dt}. $$

My working so far

Just prior to that question the text gives an expression for momentum, $$\int\limits_{-\infty}^{+\infty}{\Psi^*\left[-i\hbar \frac{\partial}{\partial x}\right]\Psi}dx, $$ (Where $\Psi$ is a function satisfying Schrödinger's equation, $i$ is the square root of negative 1, and $\hbar$ is Planck's constant divided by $2\pi$.)

So my initial approach was to try and differentiate this with respect to $t$. $$ \frac{d}{dt}\int\limits_{-\infty}^{+\infty}{\Psi^*\left[-i\hbar \frac{\partial}{\partial x}\right]\Psi}dx. $$ However I didn't know how to do that, so I asked around in the maths chatroom, and got a lot of help. Specifically this strategy from @RobJohn: $$ \begin{align} \int vv'\,\mathrm{d}x &=vv-\int vv'\,\mathrm{d}x\\ &=\frac12vv+C \end{align} $$

So I suppose it would follow that, $$ \begin{align} \int\limits_{-\infty}^{+\infty}{ \left( \Psi^* \left[ \frac{\partial}{\partial x} \right] \Psi \right) }\,\mathrm{d}x &= \left[\Psi^* \cdot \Psi\right]_{-\infty}^{+\infty} - \int\limits_{-\infty}^{+\infty} {\Psi\frac{\partial \Psi^*}{\partial x} }\,\mathrm{d}x\\ &= \frac{1}{2}|\Psi|^2 \Big|_{-\infty}^{+\infty}, \end{align} $$ And then, $$ \begin{align} &\frac{d}{dt}\int\limits_{-\infty}^{+\infty}{\Psi^*\left[-i\hbar \frac{\partial}{\partial x}\right]\Psi}dx \\&=-i\hbar \frac{d}{dt}\left( \frac{1}{2}|\Psi|^2 \Big|_{-\infty}^{+\infty} \right). \end{align} $$ But this feels wrong. I'm not sure exactly why. (Maybe it's just because it is the limits of my maths)

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For reference, regarding what I understand, my maths is pretty low-level, I took a course in undergrad calculus over 10 years ago and only just passed that at the time, and my physics knowledge is not much further than that of an enthusiastic lay person.

Answer 1

The strategy you got from @RobJohn is not correct - specifically,

$\begin{align} \int\limits_{-\infty}^{+\infty}{ \left( \Psi^* \left[ \frac{\partial}{\partial x} \right] \Psi \right) }\,\mathrm{d}x \neq \frac{1}{2}|\Psi|^2 \Big|_{-\infty}^{+\infty}, \end{align}$

Since you just asked for hints, here are a couple useful things to know to solve this problem:

(1) Since the variables $t$ and $x$ are independent, you can bring the time derivative into the integral. You can also use the derivative product rule to get two terms in the integral.

(2) Do you know how to evaluate the time derivative of the wavefunction? Hint: Shrodinger equation

The problem involves a bit of algebra, but that should be a start.