Quiz on Chapter 4

1. If Jx, Jy and Jz are the Cartesian components of some angular momen-tum, what commutation relations do they satisfy?

2. The spherical harmonic Y ml (θ, φ) is a product of a function which depends

only on θ and one which depends only on φ. Write down the factor thatdepends only on φ

3. The Hydrogen atom energies that we have studied depend on which ofthe three indices n, l,m? (standard notation). What about in the caseof a non-Coulomb, but still spherically symmetric, potential V (r)?

4. Write down the ground state wave function of Hydrogen, apart fromnormalization.

5. For a spin-half particle, what are the eigenvalues of the operators Sx, Sy

and Sz?

6. How many spin states does a graviton (s = 2) have? Write down thematrix corresponding to Sz in this case.

7. What is the result of applying the operator L+ ≡ Lx + iLy to the wavefunction ψ211 ?

8. If the ground state energy of Hydrogen is -13.6 eV, what is the numericalvalue of the first excited state? What is its degeneracy? (ignoring thespin of the electron)

9. How many nodes does the radial part of the Hydrogen wave function ψ531

have?

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10. A Hydrogen atom is in the state 23ψ210 + 2

3ψ31−1 + 13ψ42−2. Write down

the probability that a measurement of

• energy will give the value −3.4eV

• Lz will give the value −2h̄

• Lz will give the value 4h̄

• L2 will give the value 2h̄2

11. For a challenge and for fun: The figure shows plots of probability densityfor six Hydrogen-atom wave functions. The corresponding indices are, ina random order, (4,0,0), (4,1,0), (4,2,0), (2,0,0), (3,1,0), (4,3,0). Writethe appropriate set of indices under each picture.

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