Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

4.P: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

  1. Demonstrate directly from the fundamental commutation relations for angular momentum, ???, that [L±,Lz]=L±, and Lx
  2. Ly Lz θ, ψ(θ,φ)=Ylm(θ,φ). Evaluate Ly, L2y.
  3. Derive Equations ??? and ??? from Equation ???.
  4. Find the eigenvalues and eigenfunctions (in terms of the angles φ ) of l=1. A measurement of . What values will be obtained by a subsequent measurement of Lx yields the results 0 and .
  5. The Hamiltonian for an axially symmetric rotator is given by H=L2x+L2y2I1+L2z2I2. What are the eigenvalues of H ?
  6. The expectation value of f(x,p) in any stationary state is a constant. Calculate 0=ddt(xp)=i[H,xp] for a Hamiltonian of the form H=p22m+V(r). Hence, show that p22m=12rdVdr in a stationary state. This is another form of the Virial theorem. (See Exercise 8.)
  7. Use the Virial theorem of the previous exercise to prove that 1r=1n2a0 for an energy eigenstate of the hydrogen atom.
  8. Demonstrate that the first few properly normalized radial wavefunctions of the hydrogen atom take the form:
    1. R10(r)=2a3/20exp(ra0).
    2. R20(r)=2(2a0)3/2(1r2a0)exp(r2a0).
    3. R21(r)=13(2a0)3/2ra0exp(r2a0).

Contributors

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)


This page titled 4.P: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

  • Was this article helpful?

Support Center

How can we help?