In spherical coordinates

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Orbital angular momentum, the spherical harmonics and rotations

Problem

In spherical coordinates, with θ\theta the polar angle from the zz axis and ϕ\phi the azimuth, the orbital angular momentum operators areL^z=−iℏ∂∂ϕ,L^2=−ℏ2[1sin⁡θ∂∂θ ⁣(sin⁡θ∂∂θ)+1sin⁡2θ∂2∂ϕ2],\hat L_z = -i\hbar\frac{\partial}{\partial\phi},\qquad \hat L^2 = -\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right],and a rotation of a function through an angle ϕ0\phi_0 about the zz axis is carried out by R^(ϕ0)=e−iϕ0L^z/ℏ\hat R(\phi_0) = e^{-i\phi_0\hat L_z/\hbar}, defined by its exponential series. ((a)) At one instant a particle, treated classically as a point with a definite position and momentum, has position r=(1.0, 2.0, −1.0)\mathbf r = (1.0,\,2.0,\,-1.0) nm and momentum p=(2.0, −1.0, 3.0)×10−24\mathbf p = (2.0,\,-1.0,\,3.0)\times10^{-24} kg m s−1^{-1}. Calculate the three components of L=r×p\mathbf L = \mathbf r\times\mathbf p, in J s. ((b)) Verify that Y10(θ,ϕ)=3/4π cos⁡θY_1^{0}(\theta,\phi) = \sqrt{3/4\pi}\,\cos\theta is an eigenfunction of both L^2\hat L^2 and L^z\hat L_z, and give the eigenvalue of L^2\hat L^2 as a multiple of ℏ2\hbar^2 and the eigenvalue of L^z\hat L_z as a multiple of ℏ\hbar. ((c)) This same Y10Y_1^{0} is rotated about the zz axis through an arbitrary angle ϕ0\phi_0. Give the constant factor by which the function is multiplied, and explain the result in terms of the quantum number found in (b).

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