PHY4221Fall05_Prob3

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

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

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-

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.

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

## 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
Ask a homework question - tutors are online