The generalised uncertainty principle states for any two Hermitian operators

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · Symmetry, angular momentum and spin

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Spin one-half, the Pauli matrices and two-level systems

Problem

The generalised uncertainty principle states

\Delta A\,\Delta B \ge \tfrac12|\langle[\hat A,\hat B]\rangle|

for any two Hermitian operators. Using

[S_x,S_y] = i\hbar S_z

from this chapter, write the bound this gives for

\Delta S_x\,\Delta S_y

. Then evaluate both sides in the state

\chi_+

, using

\langle S_x\rangle = 0

already found in this chapter and

S_x^2 = S_y^2 = (\hbar/2)^2\mathbb 1

, and say whether the bound is reached exactly or with room to spare.