A Quantum Problem Every Physics Student Should Attempt
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- Опубликовано: 5 авг 2024
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You are a real reference when it comes to combining programming with cutting edge physics! Could you please create a series (if it does not conflict with your PhD) where you teach and explain Physics from a really beginner level (Newtonian, momentum, torque…etc.) and work your way up to the concepts of quantum physics? I believe the reason why people face challenges in understanding math and physics is that text books and teachers are not successful in explaining the link between the different concepts and treat each subject as a stand alone idea.
It's refreshing to have those physics videos back!
You said “comprehensive exam” and my spine shivered. We call ours the “qualifying exam” and it’s an oral exam (1/2 research talk + questions, 1/2 standing at a chalk board while your committee asks you random physics questions). I studied all summer, and passed two weeks ago! I wish you the best of luck on yours, whenever it may be!
Incredible teaching skills. One of your best yet!
What a great resources, so glad the algorithm lead me here today! Stoked to watch other python videos and get some learnin in!
Thank you for all the videos! You really make it understandable and comprehensive
I really like physics and programing and you combine both. Keep going!
Dude you broke the algorithm i only did high school physics but somehow RUclips recommended me this am not complaining
There is a closed form for this using the erf function. The gaussian wave packet in x-space transforms to a gaussian in k-space, without any potential |psi(x,t)|^2 stays gaussian, and the integral from -L to L can then be expressed with the erf function: p(t)=(1-erf(L/sqrt(1+t^2/m^2)) )/2
for the solution see below.
Note that this is the probability of finding the particle somewhere beyond L with a measurement at time t. The probability of the particle hitting a detector at L before time t should actually be higher, as once a particle is detected it can't be "undetected", but the propagator exp(iHt) will transport parts of the wave function beyond L back across L. In effect one would have to clip psi(x,t) at L, and determine the part that was detected, then apply the propagator to get to t+dt and repeat.
Since the initial wave function is a gaussian, there is a closed form when it is transformed to k-space using the units from the video and for now not bothering with normalisation:
psi(k,0) = exp(-k^2/2)
Applying the Hamiltonian is then easy as H=k^2/2m:
psi(k,t) = exp(-k^2/2 + i tk^2/2m)
transform back into real space (integrate over all k) using the quadratic completion trick:
psi(x,t) = integral exp ( (it/m-1)/2 k^2 + ikx ) dk
= integral exp ( (it/m-1)/2 * ( k^2 + 2*i/(it/m-1) kx + (i/(it/m-1)*x)^2 ) + (1/2(it/m-1)*x^2) ) ) dk
= C(t)* exp (-1/2 * (1+it/m)/(1+t^2/m^2)x^2 )
where C(t) is some constant resulting from the integral after a simple substitution k+2i/(it/m-1)x -> k' so only the exp(...x^2) term added for the quadratic completion depends on x. As we need to normalize anyways let's just normalize the gaussian now:
Then |psi(x,t)|^2 = 1/sqrt( pi*(1+t^2/m^2) ) exp (-x^2/(1+t^2/m^2))
Then integral (-L,L) |psi(x,t)|^2 dx = 2/sqrt(pi) integral(0,L/sqrt(1+t^2/m^2) exp(-y^2) dy
= erf(L/sqrt(1+t^2/m^2))
using the substitution x/sqrt(1+t^2/m^2) = y
The result is then (1-erf(L/sqrt(1+t^2/m^2)) )/2
One gets pretty much the same graph using L=5, m=1 and:
t=np.arange(0,30,.01)
T=np.array([0,1,2,3,5,8,11,15,20,30])
plt.plot(t,.5-erf(5/np.sqrt(1+t**2))/2)
plt.plot(T,.5-erf(5/np.sqrt(1+T**2))/2,'o')
erf is scipy.special.erf, np is numpy
Worked it thru with you and just shows the important differences and advantages of taking the time to actually type the code vs just watching. One issue is use of magenta esp for minus signs. Magenta doesnt provide sufficient contrast for the all important minus signs in terms or limits, just hard to see for some of us.
Learning math can be so much fun, well done
Awesome video! The problem wording is super confusing though. If (A) is interpreted as "at what time is p > 0.5", then the answer is "never." And for part (B), the probability of observing the electron _exactly_ at L is of course zero. I suspect those aren't the answers they wanted, but I can't figure out what they _did_ want.
Really interesting approach! How did you fix the equation numbering on jupyter's notebook?
Amazing way to explain 👍🏻
Great work although I think you're missing a Planck constant in the momentum operator !
As always, another outstanding video.
Admittedly, as a beginner in quantum dynamics I have to watch this video at least 3 times to follow along, but that is not meant as criticism. Nevertheless, your videos have been a complete game changer for my understanding of quantum dynamics.
From my first watch through I got the impression that you use some approaches, techniques, assumptions over and over again as a toolbox to get to your final result. I'd like to suggest a video where you show off this toolbox in more detail, tool by tool. This would help beginners like me tremendously :)
Albeit your style is different, I feel you're approaching @3blue1brown level of impact on people's understanding. Keep up the amazing work!
Appreciate the fact you've watched it multiple times :). Thank you for the kind words. Perhaps I will need to put together a video like you've mentioned in the future ..
@@MrPSolver Im sure you pull it off, but preparation for your exams is ofc higher priority. However, could you maybe share some literature that you use to put together these videos? Especially with the focus on ther numerical solutions. I just can't suffer the the analytical textbooks :(
Is it possible to bring the microphone down while still keep the good quality?
Nice explanation btw.
Can you please show, how to solve for the collisoin time of two objects in space due to gravitation?
Amazing video
But what should I know to understand this video completely?)
9:40 Shouldn’t we have a minus in the exponent?
Beauuutiful physics equation❤❤
I have gotten into your videos recently, as I am a physics student that is expected to know programming without ever having been taught anything. But there is something I really want to know. How do you create these jupyter notebooks that not only contain python programs, but also text and lateX text? I would be very interested to know how I can smoothly and efficiently combine all of that into one document.
use can use markdown option in jupyter notebooks to do that
Great video! Now let's solve it in 2D! :)
you lure in the physics and math majors with the funny meme thumbnails on the brain dead (but also genius and hilarious) niche physics and math major jokes, and then trap them with information (unlike what is given to me at the university i am paying 30k every year to even after a decent scholarship) that is actually useful
Bruh I suck at physics but enjoy your video about it
I downloaded soft soft but uninstalled later because I don't know how to work it or anytNice tutorialng. I'm actually tNice tutorialnking of downloading it again
my wave function collapsed
nice
What if d-o-g really spelled 'cat'?
fr
You have a very nice beard 🤭
tNice tutorials actually helped