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 timedependent 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
ℏ
/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.
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 Fall '08
 Staff
 Work, Quantum Physics, NH3, 2k, harmonic oscillator potential, equilibrium position req

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