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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 May 25 '14 at 20:58

This question came from our site for people studying math at any level and professionals in related fields.

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 '14 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 '14 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 '14 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 '14 at 20:27
Also, it's a past paper from the maths university course that I'm doing. – Smiley Sam May 25 '14 at 20:28
up vote 2 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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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 '14 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 '14 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 '14 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 '14 at 22:37
@SmileySam: This is very much a physics question. – user7757 May 25 '14 at 22:38

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