{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 486_HW5 - Physics 486 FA07 Homework Set 5 Due Oct 4 2007 1...

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

Physics 486 FA07 Homework Set 5 Due: Oct 4, 2007 1) Based on Griffiths 2.22 . A free particle has the initial Gaussian wave function Ψ (x,0) = A e -ax ² , where A and a are constants (a is real and positive). (a) Determine the constant A. (b) Find the time-dependent wavefunction Ψ (x,t). Hint: Integrals of the function e - (ax ² +bx) (limits - to + ) can be performed by “completing the square”: Let y = a (x+(b/2a)) and note that ax ² +bx = y ² - (b ² /4a). (c) Find | Ψ (x,t)| ² . Express your answer in terms of the quantity w = [a/(1+(2at /m) ² )] 1/2 . Sketch | Ψ (x,t)| ² as a function of x at t = 0, and once again at some very large time t. Qualitatively, what happens to | Ψ | ² as time goes on? 2) Double well potential. Consider the “double square well” potential model of an H 2 or NH 3 molecule. Suppose the depth, V 0 , and the width, a , are fixed and large enough that several bound states occur. The separation, b , can be varied. (a) Sketch the ground state wave function, ψ 1 , and the first excited state, ψ 2 , for the following cases: (i) b=0; (ii) b a; and (iii) b >>a.

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.

{[ snackBarMessage ]}