Complex Gaussians and the uncertainty product
Consider the gaussian wavefunction
1
ψ(x) = N exp
−
2
x
2
, ∆ ∈ C , Re(∆2
) > 0 , (1) ∆2
where N is a real normalization constant
and ∆ is now a complex number: ∆∗ 6= ∆.
The integrals in Problem 1 are also useful here and so is thefollowing relation, valid
for any nonzero complex number z,
Re1
z
=
Re(z)
(prove it!) |z|
2
(a) Use the position space representation (1) of thewavefunction to calculate the
uncertainties ∆x and ∆p. Leave your answer in terms of |∆| andRe(∆2
). (∆x
will depend on both1
, while ∆p will depend only on Re(∆2
)).
(b) Calculate the Fourier transform φ(p) of ψ(x). Use Parsevalto confirm your
answer and then recalculate ∆p using momentum space.
(c) We parameterize ∆ using a phase φ∆ ∈ R as follows
∆ = |∆| e
iφ∆ .
Calculate the product ∆x∆p and confirm that the answer can beput in terms of
a trigonometric function of φ∆ and that |∆| drops out. Is youranswer reasonable
for φ∆ = 0 and for φ∆ =
π
?4
(d) Consider the free evolution of a gaussian wave packet inProblem 3 of Home-
work 4. What is ∆p at time equal zero? Examine the timeevolution of the
gaussian (from the solution!) and read the value of thetime-dependent (com-
plex) constant ∆2
. Confirm that ∆p, found in (a), gives a time-independent
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2. Complex Gaussians and the uncertainty product [10 points] Consider the gaussian wavefunction 4(x) = N exp(-2²), AEC, Re(A²)>0, (1) where N is a real normalization constant and A is now a complex number: A* # A. The integrals in Problem 1 are also useful here and so is the following relation, valid for any nonzero complex number 2, Re(²) = Re(z) 121² (prove it!) (a) Use the position space representation (1) of the wavefunction to calculate the uncertainties Ar and Ap. Leave your answer in terms of A and Re(A²). (Ar will depend on both¹, while Ap will depend only on Re(A²)). (b) Calculate the Fourier transform (p) of (x). Use Parseval to confirm your answer and then recalculate Ap using momentum space. (c) We parameterize A using a phase A ER as follows A = Aleis Calculate the product ArAp and confirm that the answer can be put in terms of a trigonometric function of A and that A drops out. Is your answer reasonable for A = 0 and for a = ? (d) Consider the free evolution of a gaussian wave packet in Problem 3 of Home- work 4. What is Ap at time equal zero? Examine the time evolution of the gaussian (from the solution!) and read the value of the time-dependent (com- plex) constant A². Confirm that Ap, found in (a), gives a time-independent result.