Introduction To Quantum Mechanics
Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 4, Problem 4.49P

(a)

To determine

Show that J satisfies the continuity equation J=t|Ψ|2 express local conservation of probability which follows from divergence theorem.

(b)

To determine

The value of J for hydrogen in the state n=2.

(c)

To determine

Calculate Lz for the state ψ211 .

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Divergence theorem. (a) Use the divergence theorem to prove, v = -478 (7) (2.1) (b) [Problem 1.64, Griffiths] In case you're not persuaded with (a), try replacing r by (r² + e²)2 and watch what happens when ɛ → 0. Specifically, let 1 -V². 4л 1 D(r, ɛ) (2.2) p2 + g2 By taking note of the defining conditions of 8°(7) [(1) at r = 0, its value goes to infinity, (2) for all r + 0, its value is 0, and (3) the integral over all space is 1], demonstrate that 2.2 goes to 8*(F) as ɛ → 0.
Problem 2.11 (a) Compute (x). (p). (x²), and (p²), for the states yo (Equation 2.60) and 1 (Equation 2.63), by explicit integration. Comment: In this and other problems involving the harmonic oscillator it simplifies matters if you introduce the variable = √mo/hx and the constanta (m/h)¹/4 (b) Check the uncertainty principle for these states. (c) Compute (T) and (V) for these states. (No new integration allowed!) Is their sum what you would expect?
Problem 2.14 In the ground state of the harmonic oscillator, what is the probability (correct to three significant digits) of finding the particle outside the classically allowed region? Hint: Classically, the energy of an oscillator is E = (1/2) ka² = (1/2) mo²a², where a is the amplitude. So the “classically allowed region" for an oscillator of energy E extends from –/2E/mw² to +/2E/mo². Look in a math table under “Normal Distribution" or "Error Function" for the numerical value of the integral, or evaluate it by computer.
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