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

**on** **Chapter** 4

**1.** **If** J x , J y **and** J z **are** **the** **Cartesian** comp**on**ents of some angular momentum,

what commutati**on** relati**on**s do **the**y satisfy?

2. The spherical harm**on**ic Yl

m (θ, φ) is a product of a functi**on** which depends

**on**ly **on** θ **and** **on**e which depends **on**ly **on** φ. Write down **the** factor that

depends **on**ly **on** φ

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

**the** three indices n, l, m? (st**and**ard notati**on**). What about in **the** case

of a n**on**-Coulomb, but still spherically symmetric, potential V (r)?

4. Write down **the** ground state wave functi**on** of Hydrogen, apart from

normalizati**on**.

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 gravit**on** (s = 2) have? Write down **the**

matrix corresp**on**ding to S z in this case.

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

functi**on** 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** electr**on**)

9. How many nodes does **the** radial part of **the** Hydrogen wave functi**on** 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

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

for six Hydrogen-atom wave functi**on**s. The corresp**on**ding indices **are**, in

a r**and**om 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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