Problem 1.7 | Griffiths' Introduction to Quantum Mechanics | 3rd Edition

Problem 1.7
Calculate d{p}/dt. Answer: d{p}/dt = {-∂V/∂x} (1.38).
This is an instance of Ehrenfest’s theorem, which asserts that expectation values obey the classical laws.
In this video, we solve Problem 1.7 in Griffiths' Introduction to Quantum Mechanics (3rd Edition) as part of a series of solutions to the textbook's questions.

[@amilkyboi]

Griffiths goes through the process of getting to the time derivative of the expectation value of position on pages 14 - 17 in case anyone doesn't know (d/dt is given on page 17). I'm assuming it's done here as a first principles type of thing - this explanation was more explicit. Nice video.

[@photon7777777]

Excellent, thanks! How long, the first time you did it, did it take to figure that out? Two comments: 1) at 11:42 I had to stare a while to see why it was OK to take partial wrt x outside, because when it operates on the two terms inside I end up with 4 terms. However there is a plus and a minus (partial psi* wrt x partial times partial psi wrt x) and they cancel :) Comment 2) at 20:09 you have d/dt (psi* partial psi wrt x). Should that d/dt be partial wrt t? I thought when you take the d/dt across the integral, it becomes partial wrt time. Because in the next line you have d psi*/dt to start with, and later in that line you have partial wrt time. Soon after you plug in expressions (1) and (2). But you are plugging in (1) which gives partial wrt time for your earlier expression at 21:30 which has d/dt (not partial). Anyway, thanks again, it took me a good hour just to work through the video lol, I would NEVER have figured it out.

[@truphysics]

A little late here :) It probably took me 2 hours the first time. I was not great at quantum when I first started. For your second comment, yes, I totally jumped the gun there. I knew that the substitution was coming up to replace those partials wrt t, so I managed to do it a bit early by accident. Thanks for catching it! The issue is fixed a few lines later for those reading this comment in the future.

[@flaminghollows3068]

Thanks man! That last step with integration by parts caught me...

[@truphysics]

Absolutely, glad this helped!

[@porit1023]

Thank you that was really helpful!

[@splat752]

Cool, I just found this and I have been working through Griffiths book

[@truphysics]

Awesome, I'm glad these are useful to you!

[@fakhrealam3110]

Sir please solve all the problems of the Book.

[@truphysics]

Working on it! Check out tru-physics.org under the Solutions tab! I am uploading solutions there every day!

[@shumailamir4125]

Incredible explanation, thankyou

[@truphysics]

You're very welcome, thanks for the comment!

[@Mr-Halo-1]

Hey, considering this theorem that states expectation values obey classical laws, then we know that -∂V/∂x = Force = dp/dt, then wouldn't the expectation value of force be equal to the expectation value of momentum? Cause we know force is equal to the derivative of momentum wrt time and also derivative of PE wrt x? That would seem like an easy 'proof' considering expectation values obey classical laws. Please correct me if I'm wrong

[@hidethepainharold4256]

A quick note: I think you may have made a sign error when dividing by -ih_bar near around the 2 minute mark. It should be +h_bar/2mi not minus

[@rhodium8505]

it is not an error since -i = 1/i. he could've made the first term positive by putting the i in the numerator, but it would be the same thing

[@samuelhawksworth1923]

At 23:01 are we assuming psi is sufficiently smooth such that d^2 psi/ dtdx is the same as d^2 psi/ dx dt??

[@truphysics]

Great question. To be honest, I can't remember this problem well enough to answer, but that sounds reasonable. Comment again if you feel that you've found a solid answer to that please!

[@ebtisammuddatherhassoun]

When you were solving by integral by parts for example, at the minute 27, depending on what you chose u and dv ? Is it randomly or we follow a logic ?

[@truphysics]

There are acronyms online, but I honestly never remember them. For me, picking the correct one for u comes down to a matter of memory from previous problems solved as well as some good luck. Usually I can do a step or two in my head before writing anything down and that can help determine which choice to make.

[@strange_infinities]

Nice

[@luisalfredososadasilvapio6503]

at 27:34, why does that evaluate to 0?

[@johndoe-ow2ns]

yes I don't see that either.

[@truphysics]

Both the wave function and its derivative go to zero at these limits for a square -integrable wave function. We are requiring that the wave function is square-integrable because the probability over all space must add up to 1. If it didn't add up to 1, then the square of the wave function would not represent probability. Then in that case, doing this problem would be worthless since the result could not be interpreted physically.

[@amaljeevk3950]

❤

[@ryansliwinski]

aw hell nah. im in too deep

[@Hearthglow]

At 27:06, when I did this problem myself, I wondered why we can't (or don't) use ψ* ∂³ψ/∂x³ - ∂ψ/∂x ∂²ψ*/∂x² = ∂/∂x(ψ* ∂²ψ/∂x² - ψ ∂²ψ*/∂x²), and see it go to zero at the limits? This sort of move seems to be done repeatedly in the text and other problem solutions. I came here to see if there were an explanation here, but didn't see it. I appreciate getting used to using IBP as a tool in QM, but if what I wrote is valid, it ought to be also in the text and in a problem solution vid, if just as a passing comment. Of course, if I am wrong, I'd like a mathy explanation.