Quantum Mechanics Homework 2: Uncertainty Principle and Angular Momentum, Assignments of Quantum Mechanics

Download Quantum Mechanics Homework 2: Uncertainty Principle and Angular Momentum and more Assignments Quantum Mechanics in PDF only on Docsity! Quantum Mechanics A (PHY 5645): Homework 2 DUE: Friday Sept 21 P1(20 points): Consider a wavepacket: ψ(x) = const× e ih̄ p0x− α2h̄ x2 . Show that the product of uncertainty in the position and momentum of a quantum particle described by this wavefunction satisfies: √〈 (∆px) 2 〉 〈 (∆x)2 〉 = h̄ 2 . Prove that the same quantity for any other wavefunction satisfies the following inequality: √〈 (∆px) 2 〉 〈 (∆x)2 〉 ≥ h̄ 2 . P2(20 points): A quantum particle obeying Schrodinger equation and moving in one dimen- sion experiences a potential V (x). In a stationary state of this system show that 1 2 〈 x ∂ ∂x V̂ (x) 〉 = 〈 p̂2 2m 〉 Hint: Analyze time dependence of 〈xp〉. P3(20 points): The components of the angular momentum satisfy: [Lx, Ly] = ih̄Lz; [Lz, Lx] = ih̄Ly; [Ly, Lz] = ih̄Lx; Prove that the operator L2 ≡ L2x + L2y + L2z commutes with all three components of ~L. Demonstrate that the following commutation relations hold for the operator of the dipole moment of a system of N electrons, ~d ≡ −e ∑Ni=1 ~ri and the operator of the system’s total angular momentum ~L ≡ ∑Ni=1 ~Li: [Lx, dx] = 0; [Lx, dy] = ih̄dz; [Lx, dz] = −ih̄dy; + cyclic permutations ofx, y, z (1) Show that [L2, ~d] = 2h̄2~d + 2ih̄~d× ~L. (2) Recall that ~Li = ~ri × ~pi P4(20 points): Given that [Â, B̂] = c, where c is a c-number, find eλÂB̂e−λ =?