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My question is above. Firstly, I don't actually know whether it is true or not (!). Secondly, if I were to try to prove it, then I have very little idea how to. The potential steps that I have always done are steps from a constant level to another constant level (Heavyside), whereas this is different.

I would imagine the answer is yes, but I'm not sure how to show it.

Can I approximate the curve by small steps (/sums of Heavyside functions), and then show that a larger Heavyside step gives a larger probability?

Thanks, Sam

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migrated from math.stackexchange.com May 25 at 20:58

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Seeing that this is not Physics, you should target the question at mathematical audience. What is the mathematical relation between $p_i$ and $V_i$? –  words that end in GRY May 25 at 4:50
    
I realise that this isn't physics, hence being in the maths part! =P The issue is that I can't determine the relation. :( –  Smiley Sam May 25 at 6:58
    
I mean this site is not Physics.SE: it's Mathematics.SE. And you don't have a clear mathematical question here. –  words that end in GRY May 25 at 17:16
    
Oh, I see what you mean. But no, this is a maths QM question, not a physics QM question. "Justify your answer carefully, either by giving a rigorous proof or by presenting a counterexample with explicit calculations of $p_1$ and $p_2$." That's maths, no physics. =P –  Smiley Sam May 25 at 20:27
    
Also, it's a past paper from the maths university course that I'm doing. –  Smiley Sam May 25 at 20:28

1 Answer 1

up vote 1 down vote accepted

It is not necessarily true. For a zero potential $V_2$ you have $p_2=0$, whereas if $V_1$ is a rectangular pit, in general, $p_1>0$.

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So the natural question is: is the reverse inequality true? –  ramanujan_dirac May 25 at 22:26
    
Ah yes, I thought that it was $V_i \ge 0$! It's clearly not true as said. How about the reverse inequality then, or when $V_1(x) \ge V_2(x) \ge 0$? –  Smiley Sam May 25 at 22:29
    
@ramanujan_dirac: I don't think the reverse inequality is necessarily true. I don't have a proof, but I would think that a smoother but a bit deeper pit would reflect less than an "inscribed" rectangular pit. –  akhmeteli May 25 at 22:35
    
Ok yeah, thank you. Also, sorry that this is on physics.SE. It's definitely a maths question, not a physics question, but someone decided to migrate it ¬_¬. I don't have the possibility to migrate it back, otherwise I would. Still, at least it meant that you saw the question! :) –  Smiley Sam May 25 at 22:36
    
@Smiley Sam: I am not sure what you have in mind, but the considerations in my previous comment can be somewhat relevant. –  akhmeteli May 25 at 22:37

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