4.P: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
- Demonstrate directly from the fundamental commutation relations for angular momentum, ???, that [L±,Lz]=∓ℏL±, and Lx
- Ly Lz θ, ψ(θ,φ)=Ylm(θ,φ). Evaluate ⟨Ly⟩, ⟨L2y⟩.
- Derive Equations ??? and ??? from Equation ???.
- Find the eigenvalues and eigenfunctions (in terms of the angles φ ) of l=1. A measurement of ℏ. What values will be obtained by a subsequent measurement of Lx yields the results 0 and −ℏ .
- The Hamiltonian for an axially symmetric rotator is given by H=L2x+L2y2I1+L2z2I2. What are the eigenvalues of H ?
- The expectation value of f(x,p) in any stationary state is a constant. Calculate 0=ddt(⟨x⋅p⟩)=iℏ⟨[H,x⋅p]⟩ for a Hamiltonian of the form H=p22m+V(r). Hence, show that ⟨p22m⟩=12⟨rdVdr⟩ in a stationary state. This is another form of the Virial theorem. (See Exercise 8.)
- Use the Virial theorem of the previous exercise to prove that ⟨1r⟩=1n2a0 for an energy eigenstate of the hydrogen atom.
- Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:
- R10(r)=2a3/20exp(−ra0).
- R20(r)=2(2a0)3/2(1−r2a0)exp(−r2a0).
- R21(r)=1√3(2a0)3/2ra0exp(−r2a0).
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)