Problem 2 in "Quantum Theory for Mathematicians", Solving for the travel time of a particle in a potential

In summary, @BvU has been trying to solve problem 2 in Hall’s QM book for ages, but is having trouble. He thinks that it might be because the book is titled Quantum Theory for Mathematicians and he’s not a mathematician, so he is looking for help from someone who could provide a formal proof. @BvU is unsure of what to do next.
  • #1
haziq
3
1
Homework Statement
Problem 2 in Chapter 2 of Hall’s QM book. See pictures below
Relevant Equations
See photo below
9DC9B877-BAD4-4B03-943A-F785EF133E35.jpeg
910F8033-9470-490C-9F50-8329C5AAAADC.jpeg
4ADAC378-42EC-4A12-88CA-53E329483451.jpeg

I’ve been trying to solve this for ages. Would really appreciate some hints. Thanks
 
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  • #2
Hello @haziq ,
:welcome: ##\qquad ## !​

haziq said:
Homework Statement:: Problem 2 in Chapter 2 of Hall’s QM book. See pictures below
Relevant Equations:: See photo below

solve this for ages

And you are aware that with 'this' you mean the step from $$\dot x(t)=\sqrt{{2\left (E_0-V\left ( x\left (t \right )\right) \right) \over m}} $$ to ##t= ...## by separation ?

PF guidelines require that you actually post your best attempt before we are allowed to assist ...

PS notice how much sharper it looks with ##\LaTeX## ?
##\ ##
 
  • #3
Hi @BvU, sorry for the confusion. That’s not the problem I was referring to. I just included that for context. That’s actually problem 1 and it was pretty easy to solve. With regards to problem 2, I’m completely lost. Probably because the book is titled Quantum Theory for Mathematicians and I’m not a mathematician. Perhaps I could share my attempt after someone gives me some hints?
 
  • #4
BvU said:
PF guidelines require that you actually post your best attempt before we are allowed to assist .
 
  • #5
@BvU Fair enough :smile:. Here’s what I’ve deduced so far (for part a)
  • Assuming ##V’(x_1) \neq 0##, we need to show that ##t=lim_{h \rightarrow x_1} \int_{x_0}^{h} {\sqrt{\frac {m} {E_0 - V(y)}} dy} \in \mathbb{R}## since the upper bound is a vertical asymptote.
  • I *think* ##V’(x_1) \neq 0## implies that ##lim_{x \rightarrow x_1}{E_0 - V(x) \neq 0}##. Not sure how to prove this formally. I just applied the definition of derivative and did some sketchy algebra and thought this is plausible.
Not sure what to do next...
 
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  • #6
Ad 2a) It's just that the singularity of the integrand at ##y=x_1## is integrable. If ##V'(x_1) \neq 0##, then you have
$$V(y)=V(x_1) + (y-x_1) V'(x_1) + \mathcal{O}[(y-x_1)^2],$$
and thus around the singularity the integral behaves like
$$\Delta t_{\epsilon}=\int_{x_1-\epsilon}^{x_1} \mathrm{d} y \sqrt{\frac{m}{-2V'(x_1)(y-x_1)}}.$$
Since ##V'(x_1)>0## you have
$$\Delta t_{\epsilon}=-\sqrt{2mV'(x_1) (x_1-y)}|_{y=x_1-\epsilon}^{x_1}= \sqrt{\frac{2 m}{V'(x_1)} \epsilon}.$$
The total time is
$$t=\int_{x_0}^{x_1-\epsilon} \mathrm{d} y \sqrt{\frac{m}{2 [E_0-V(y)]}} + \Delta t_{\epsilon}.$$
Since ##t## doesn't depend on ##\epsilon## and ##\Delta t_{\epsilon} \rightarrow 0## for ##\epsilon \rightarrow 0## the total time is finite.

If ##V'(x_1)=0##, the above Taylor expansion starts at best with the quadratic term, i.e.,
$$V(y)=V(x_1) + \frac{1}{2} (y-x_1)^2 V''(x_1) + \mathcal{O}[(y-x_1)^3],$$
and the above analysis shows that ##\Delta t_{\epsilon}## diverges logarithmically, i.e., even in this case the time ##t \rightarrow \infty##. If also ##V''(x_1)=0## the divergence gets worse, i.e., for ##V'(x_1)=0## the time always ##t \rightarrow \infty##.
 
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1. What is the purpose of solving for the travel time of a particle in a potential?

The purpose of solving for the travel time of a particle in a potential is to understand how quantum particles behave in different environments. This can help us predict the behavior of particles in various scenarios and can also provide insight into the fundamental principles of quantum mechanics.

2. How is the travel time of a particle in a potential calculated?

The travel time of a particle in a potential is calculated using the Schrödinger equation, which describes the time evolution of a quantum system. This equation takes into account the potential energy of the particle and can be solved using mathematical techniques such as eigenvalue problems and perturbation theory.

3. What factors affect the travel time of a particle in a potential?

The travel time of a particle in a potential is affected by several factors, including the shape and strength of the potential, the initial position and momentum of the particle, and any external forces acting on the particle. These factors can alter the trajectory and speed of the particle, ultimately affecting its travel time.

4. Can the travel time of a particle in a potential be measured experimentally?

Yes, the travel time of a particle in a potential can be measured experimentally using techniques such as time-of-flight measurements or interferometry. These methods involve tracking the position and velocity of the particle over time and can provide valuable data for validating theoretical predictions.

5. How does solving for the travel time of a particle in a potential contribute to our understanding of quantum theory?

Solving for the travel time of a particle in a potential is an important aspect of quantum theory as it allows us to make predictions and gain insight into the behavior of quantum particles. By studying the travel time, we can better understand the fundamental principles of quantum mechanics and how particles interact with their environment.

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