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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