Introduction to Quantum Mechanics · Problem 5.7

Identical Particles — David J. Griffiths

Physics — Quantum Mechanics · exchange_forces_harmonic_oscillator

Problem

Two noninteracting particles (of equal mass) share the same harmonic oscillator potential, one in the ground state and one in the first excited state. (a) Construct the wave function, ψ(x1,x2)\psi(x_1, x_2), assuming (i) they are distinguishable, (ii) they are identical bosons, (iii) they are identical fermions. Plot ∣ψ(x1,x2)∣2|\psi(x_1, x_2)|^2 in each case (use, for instance, Mathematica's Plot3D). (b) Use Equations 5.23 and 5.25 to determine ⟨(x1−x2)2⟩\langle (x_1 - x_2)^2 \rangle for each case. (c) Express each ψ(x1,x2)\psi(x_1, x_2) in terms of the relative and center-of-mass coordinates r≡x1−x2r \equiv x_1 - x_2 and R≡(x1+x2)/2R \equiv (x_1 + x_2)/2, and integrate over RR to get the probability of finding the particles a distance ∣r∣|r| apart: P(∣r∣)=2∫∣ψ(R,r)∣2 dRP(|r|) = 2 \int |\psi(R, r)|^2 \, dR (the 2 accounts for the fact that rr could be positive or negative). Graph P(r)P(r) for the three cases. (d) Define the density operator by n(x)=∑i=12δ(x−xi);n(x) = \sum_{i=1}^{2} \delta(x - x_i); ⟨n(x)⟩ dx\langle n(x) \rangle \, dx is the expected number of particles in the interval dxdx. Compute ⟨n(x)⟩\langle n(x) \rangle for each of the three cases and plot your results. (The result may surprise you.)

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