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with the \"generalized uncertainty principle\": if [AB] = iC, then Delt A Delta

ID: 1915691 • Letter: W

Question

with the "generalized uncertainty principle": if [AB] = iC, then Delt A Delta B >= 1/2 || (no need to prove) and then invoked Dirac's quantum condition [x, p] = i (in reduced units) to prove Delta x Delta p >= 1/2 (in reduced units). After the class meeting, one student asked: how about the uncertainty principle for angular momentum? Let's help answer this good question. For the common eigenkets of L2 and Lz, as denoted by |j,m>, please calculate , , and ALX. Recall that uncertainty (Delta A)2 = - 2 by definition. Please show algebraic details. Based on xy-symmetry of this state | ,m>, Delta Lx=Delta Ly (no need to prove). Please calculate the uncertainty product Delta Lx, Delta Ly for the eigenket In CM, orbital angular momentum L (a vector) is defined as L = r Times p where vector L = Lxi + Lyj + Lzk, vector r = x i + y j + z k, and vector p = pxi + pyj + pzk. Therefore, the x, y, z components of L are: Lx= ypz - zpy; Ly = zpx - xpz ; Lz = xpy - ypx. (no need to prove this) you have proved [Lx, Ly] = iLz (in reduced units). Based on [Lx, Ly] = iLz and the "generalized uncertainty principle", please calculate the uncertainty product Delta Lx Delta Ly for the eigenket |l, m>, Is your answer in [4](a) consistent with that in [4](c)? Please prove your answer by algebra.

Explanation / Answer

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