{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Discussion11 - (a Rewrite the radial wavefunction as R(r =...

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

View Full Document Right Arrow Icon
Physics 486 Fall 2007 Discussion Problems 11 1) Superposition of spherical harmonics. Consider a particle with the following wavefunction: ( ) ( ) 2 2 2 2 / ( , , ) x y z a x y z N x y z e ψ - + + = + + , where N is a normalization constant. (a) Rewrite the wavefunction in terms of spherical co-ordinates. (b) Express the angular dependence in terms of (as a superposition of) the appropriate spherical harmonics m l Y . (c) What are the possible results of measurements of L 2 and L z , and what are the probabilities of these measurements? (Remember that in a superposition, if the co-efficient preceding an eigenstate is complex, the probability of being in that state is proportional to the square of the magnitude of the complex number.)
Background image of page 1

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

View Full Document Right Arrow Icon
Physics 486 Fall 2007 2) Infinite spherical well. The radial part of Schrodinger’s equation for central potentials is given by 2 2 2 2 ( 1) ( ) , 2 2 R l l r R V r R ER m r mr + - - + = where the entire wavefunction in the product ψ = R(r)Y( θ , ϕ ), and l is the quantum number associated with the net angular momentum.
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: (a) Rewrite the radial wavefunction as R(r) = u(r)/r, and by substituting into the equation above, obtain a differential equation for u. What is the advantage of doing this transformation? (b) The infinite spherical well is give by the potential V(r) = 0 for r a, V(r) = for r > a. Draw the effective potential appropriate for the Schrodinger equation describing the function u for the quantum values l = 0, l = 1 and l =2. For the values l = 0 and l = 1, draw the corresponding eigenstate wavefunctions. How are these wavefunctions related to the quantum number n for the radial wavefunction? (c) Find the eigenstates and eigenenergies associated with u for l = 0 (the familiar case). Keep in mind that the actual wavefunction is R = u/r and that it should not blow up at r = 0. (The solutions for general l are known as Bessel functions See Griffiths 4.1.3)...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

Discussion11 - (a Rewrite the radial wavefunction as R(r =...

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

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