Chem120A+Notes+10-1-08

Chem120A+Notes+10-1-08 - Chem120A Notes Single Particle...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chem120A Notes 10/1/08 Single Particle Subject to Central Force ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ + ∇ − = + = 2 2 2 2 2 2 2 2 2 ); ( 2 ˆ ˆ ˆ z y x r V m V T H h → is called the Laplacian 2 ∇ We can also express the Hamiltonian in terms of two components for the kinetic energy: a radial component, and an angular component. ) ( 2 ˆ 2 2 ˆ 2 2 2 2 2 r V mr L r r r m H + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − = h , is the angular momentum operator 2 ˆ L Particle subject to V(r) has angular momentum conserved. To be able to measure H ˆ and L ˆ simultaneously, we need [ ] ˆ , ˆ 2 = L H [ ] [ ] [ ] 2 2 2 ˆ , ˆ ˆ , ˆ ˆ , ˆ L V L T L H + = ( 1 ) [ ] ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − − ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ − = 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ˆ 2 2 ˆ ˆ 2 ˆ 2 2 ˆ , ˆ mr L r r r m L L mr L r r r m L T h h → Note: is a function of 2 ˆ L angles only i.e. ) , ( ˆ 2 φ θ f L = → Therefore, the order of operations for T and don’t matter, as T only affects radial portions of the wavefunction, and only affects the radial components i.e. only affects the radial components i....
View Full Document

This note was uploaded on 09/28/2009 for the course CHEM 120A taught by Professor Whaley during the Spring '07 term at Berkeley.

Page1 / 3

Chem120A+Notes+10-1-08 - Chem120A Notes Single Particle...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online