द्रव्यमान m के दो विभेद्य (distinguishable) गैर-परस्पर क्रिया करने वाले कण, अंतराल [0, a] में एक-आयामी अनंत वर्ग कुएँ (infinite square well) में हैं। यदि x1 और x2 दो कणों के स्थिति ऑपरेटर हैं, तो उस स्थिति में प्रत्याशा मान (expectation value) ⟨x1x2⟩ कितना होगा जिसमें एक कण मूल अवस्था (ground state) में है और दूसरा प्रथम उत्तेजित अवस्था (first excited state) में है?

Consider a particle in a symmetric one-dimensional infinite square well potential V(x) = 0 for |x| ≤ L/2 and ∞ for |x| > L/2. At time t = 0, the particle is in an equal superposition state ψ(x, 0) = 1√2(φ1(x) + φ2(x)), where φ1(x) and φ2(x) are normalized ground and first excited states. If τ is the minimum time at which the particle is equally likely to be found in either half of the well, calculate the value of τ hm L2 where h is Planck's constant and m is mass.

A particle in a central potential has an orbital angular momentum l = 2 and a spin s = 1. Find the energy level shift and total degeneracy associated with the highest energy state arising from a spin-orbit interaction of the form Hso = A L · S, where A is a positive constant.