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I can understand the reason why ∫ψψ*dx < ∞ But do not understand how quadratic integrability implies that.

I would be very thankful to anybody who can give me some idea.

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- Thread starter relativist
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In summary, the conversation discusses the reason why a normalizable wave function decays faster than 1/sqrt(x) as x approaches infinity. The concept of quadratic integrability is also mentioned, with the understanding that this does not necessarily imply the faster decay rate. The conversation also touches on the difference between physicists and mathematicians when it comes to math, and concludes with a thank you note for the helpful explanation.

- #1

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I can understand the reason why ∫ψψ*dx < ∞ But do not understand how quadratic integrability implies that.

I would be very thankful to anybody who can give me some idea.

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Can anybody please answer my question

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I took a quick look at Griffiths. It's footnote 8 on page 11 in the copy I could get hold of quickly, so maybe the claim is different in your edition. In the text I'm looking at, he's talking about square integrable

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Physicists are usually sloppy when it comes to math as compared with mathematicians. Suffice it to say that the 1/sqrt(x) rule is at the very least a good rule of thumb for square integrability.

1/x, when integrated over the part of the real line where x>1 (giving you a natural log function) diverges due to it not "decaying fast enough". That's why functions must decay FASTER than 1/x in order that this integral does not diverge. This corresponds to a wave-function which must decay faster than 1/sqrt(x) (since it will be squared).

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Matterwave said:

Physicists are usually sloppy when it comes to math as compared with mathematicians. Suffice it to say that the 1/sqrt(x) rule is at the very least a good rule of thumb for square integrability.

1/x, when integrated over the part of the real line where x>1 (giving you a natural log function) diverges due to it not "decaying fast enough". That's why functions must decay FASTER than 1/x in order that this integral does not diverge. This corresponds to a wave-function which must decay faster than 1/sqrt(x) (since it will be squared).

Thanks for a nice answer. I fact I stumbled upon a problem in chapter 3 (problem 3.2) which got me thinking in the right direction after working on it. Your reply has helped to gain a good deal of confidence that I am making some progress in learning qm.

Thanks again,

Relativist.

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