Quiz on Chapter 4 1. If Jx, Jy and Jz are the Cartesian ... - dirac

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Quiz on Chapter 4 1. If Jx, Jy and Jz are the Cartesian ... - dirac

ong>Quizong> on Chapter 4

1. If J x , J y and J z are the Cartesian components of some angular momentum,

what commutation relations do they satisfy?

2. The spherical harmonic Yl

m (θ, φ) is a product of a function which depends

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

depends only on φ

3. The Hydrogen atom energies that we have studied depend on which of

the three indices n, l, m? (standard notation). What about in the case

of a non-Coulomb, but still spherically symmetric, potential V (r)?

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

normalization.

5. For a spin-half particle, what are the eigenvalues of the operators S x , S y

and S z ?

6. How many spin states does a graviton (s = 2) have? Write down the

matrix corresponding to S z in this case.

7. What is the result of applying the operator L + ≡ L x + iL y to the wave

function ψ 211 ?

8. If the ground state energy of Hydrogen is -13.6 eV, what is the numerical

value of the first excited state? What is its degeneracy? (ignoring the

spin of the electron)

9. How many nodes does the radial part of the Hydrogen wave function ψ 531

have?

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10. A Hydrogen atom is in the state 2 3 ψ 210 + 2 3 ψ 31−1 + 1 3 ψ 42−2. Write down

the probability that a measurement of

• energy will give the value −3.4eV

• L z will give the value −2¯h

• L z will give the value 4¯h

• L 2 will give the value 2¯h 2

11. For a challenge and for fun: The figure shows plots of probability density

for six Hydrogen-atom wave functions. The corresponding indices are, in

a random order, (4,0,0), (4,1,0), (4,2,0), (2,0,0), (3,1,0), (4,3,0). Write

the appropriate set of indices under each picture.

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