{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture13_final

Lecture13_final - CHEM 356 Lecture 13 Fall 2009 1 The Bohr...

This preview shows pages 1–2. Sign up to view the full content.

CHEM 356, Lecture 13, Fall 2009 1 The Bohr Correspondence Principle. A. For the particle–in–a–box model. (i) Classical mechanics: The position 𝑥 , of a classical particle, is given by 𝑥 = 𝑥 0 + 𝑣𝑡 , so that for a uniform speed 𝑣 , we obtain 𝑣𝑇 = 𝐿 for motion along a frictionless string of fixed length 𝐿 , with a period of length 𝑇 . Let us denote by 𝑃 cl 𝑑𝑥 the probability that the particle is in the interval ( 𝑥, 𝑥 + 𝑑𝑥 ) at some later time. If the time to traverse a distance 𝑑𝑥 is 𝑑𝑡 , then the probability that the particle is in the interval ( 𝑥, 𝑥 + 𝑑𝑥 ) is given by the fraction of time that the particle has spent in the interval, i.e., by 𝑃 cl 𝑑𝑥 = 𝑑𝑡 𝑇 = 𝑣𝑑𝑡 𝐿 = 𝑑𝑥 𝐿 , so that 𝑃 cl is thus 𝑃 cl ( 𝑥 ) = 1 𝐿 . (ii) Quantum mechanics: We have seen that the probability density 𝑃 𝑛 ( 𝑥 ) is de- fined by 𝑃 𝑛 ( 𝑥 ) 𝜓 𝑛 ( 𝑥 ) 𝜓 𝑛 ( 𝑥 ) = 2 𝐿 sin 2 ( 𝑛𝜋𝑥 𝐿 ) , which attains a maximal value 2 /𝐿 whenever sin( 𝑛𝜋𝑥/𝐿

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 2

Lecture13_final - CHEM 356 Lecture 13 Fall 2009 1 The Bohr...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online