PHY4221Fall05_Prob3

PHY4221Fall05_Prob3 - mentum. (b) What are the normalized...

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

View Full Document Right Arrow Icon
PHY4221 Quantum Mechanics I Fall term of 2005 Problem Set No.3 (Due on November 2, 2005) 1. Apply the Wei-Norman theorem to calculate the coordinate-space propagator of a simple harmonic oscillator described by H = p 2 2 m + 1 2 2 x 2 . Hint : You might find these operators L + = i h x 2 , L - = i h p 2 , L 0 = i h ( xp + px ) useful. 2. Given Y 21 ( θ,φ ) = - s 15 8 π sin θ cos θ e , apply the raising operator ˆ L + to find Y 22 ( θ,φ ). 3. Determine graphically the allowed energies for the infinite spherical well when l = 1. Show that for large n , E nl π 2 ¯ h 2 2 ma 2 ± n + 1 2 2 . Hint : First show that j l ( x ) = 0 = x = tan x . Plot x and tan x on the same graph and locate the points of intersection. 4. Two particles of mass m are attached to the ends of a massless rigid rod of length a . The system is free to rotate in three dimensions about the centre (but the centre point itself is fixed). (a) Show that the allowed energies of this rigid rod are E n = ¯ h 2 n ( n + 1) ma 2 , for n = 0 , 1 , 2 ,..... Hint : First express the (classical) energy in terms of the total angular mo-
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: mentum. (b) What are the normalized eigenfunctions for this system? What is the de-generacy of the n th energy level? 5. (a) Find h r i and h r 2 i for an electron in the ground state of hydrogen. Express your answers in terms of the Bohr radius. (b) Find h x i and h x 2 i for an electron in the ground state of hydrogen. (c) Find h x 2 i in the state n = 2, l = 1, m = 1. 6. Suppose we perturb the infinite cubical potential well of side a by putting a delta function “bump” at the point ( a/ 4 ,a/ 2 , 3 a/ 4): H = a 3 V δ ( x-a/ 4) δ ( y-a/ 2) δ ( z-3 a/ 4) . Find the energy (up tp second-order corrections) and eigenfunction (up to first-order corrections) of the ground state and the (triply degenerate) first excited states. —— End ——...
View Full Document

Page1 / 2

PHY4221Fall05_Prob3 - mentum. (b) What are the normalized...

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