Download Problem set 2 A - De Broglie wavelength B

Problem set 2
Due date january 17, 2012
A - De Broglie wavelength
1. What is the de Broglie wavelength of a rubidium 87 atom (m = 1.45 × 10−25 kg)
at room temperature?
2. At which temperature does the de Broglie wavelength of a rubidium 87 atom
coincide with the wavelength of optical transition at 780 nm? What is the velocity
of the atoms?
3. Quantum degeneracy is obtained when the condition nλ3 ≈ 1 is achieved, with λ
the de Broglie wavelength and n the spatial density. Calculate n in at/cm3 for the
atoms of the previous question.
4. The coldest temperature ever measured is 500 pK in a cloud of sodium atoms.
What is the de Broglie wavelength of the atoms. Compare it to the size of a
microbe.
B - Effective potential
The elastic scattering of a particle of mass m by a potential V (r) is calculated using
perturbation theory. One writes the wave function ψ(r, θ, ϕ) of the particle as a sum of
an incident wave with wave vector k (modulus k) and of a scattered spherical wave:
ψ(r, θ, ϕ) = eikz + f (θ, ϕ)
eikr
when r → +∞ .
r
(1)
To first order perturbation theory, called the Born approximation, the scattering amplitude f is given by
Z
m
V (r) eiq·r d3 r ,
(2)
f (θ, ϕ) = −
4πh̄2
with q = k′ −k, k′ the wavevector scattered in a direction (θ, ϕ) and with k = k ′ because
the scattering is elastic.
When the energy of the collision is very small, k → 0 and therefore q → 0. The scattering amplitude f is then independent of the energy of the particle and isotropic. The
scattering length is defined in this low energy limit as a = −f . In order to describe the
interaction between two particles, it is useful to introduce an effective contact potential
Veff (r) = gδ(r), so that this contact potential gives the same scattering length than the
real potential V (r). Show that in this case
4πh̄2
g=
a.
m
1
(3)
C - Scattering length
One models the interaction between two atoms separated by a distance r by the potential
shown in the figure below. In the center-of-mass frame of the two atoms the scattering
V(r)
0
R
r
-V
of one atom of mass m by the other is governed by the time-independent Schrödinger
equation
h̄2
∆ψ(r) + V (r)ψ(r) = Eψ(r) ,
(4)
−
2 m/2
with ψ(r) the wave function of the center of mass of the two atoms. We consider the lowd2
energy scattering (also called s-wave scattering). In this case E ≈ 0 and ∆ψ = 1r dr
2 (rψ)
as the orbital angular momentum l = 0. In order to solve the problem we use, as usual,
the function u(r) = rψ(r) and we define K 2 = mV
.
h̄2
1. Why is the potential shown in the figure a reasonable approximation of the interaction between two atoms ?
2. Write the equation governing u(r) for r > R and r < R.
3. The solution for r > R can be written u(r) = r − a. Use the boundary conditions
in r = R and r = 0 to show that
a = R(1 −
tan KR
).
KR
(5)
4. Plot a as a function of KR.
5. Write the expression of ψ(r) for r > R, and show that the quantity a is the
scattering length introduce in the previous problem B when k → 0.
2