- 4 Eigenvalue Problem For L The Z Component Of Orbital Angular Momentum Is Given In Position Representation By Ha N L 1 (26.39 KiB) Viewed 142 times
4. Eigenvalue problem for L₂. The z component of orbital angular momentum is given in position representation by ħa (n|L₂|y) = -(n|u) i ap so the eigenvalue problem for L₂ is of form ħa (1) Tap() = q (hy) where q is the eigenvalue. For simplicity, let the eigenfunction be (p) = (nly) and let the eigenvalue be q = mħ. Find P(q) by solving the differential equation (1) and apply the boundary condition (p) = (0 + 2π) to obtain the allowed values of m. The physical significance of the boundary condition is that p and p + 2π represent the same physical position and so the value of must be same whether we take or Q + 2π or p plus any integer multiple of 2π.