130_notes.dvi

=

− ̄h^2

m^2

4

(

Z

a 0

) 3 ∫∞

0

r^2 e

−Zr

a 0

(

Z^2

a^20

−

2 Z

a 0 r

)

e

−Zr

a (^0) dr

=

− ̄h^2

m^2

4

(

Z

a 0

) 3 (

Z^2

a^20

2

(a 0

2 Z

) 3

−

2 Z

a 0

(a 0

2 Z

) 2 )

=

̄h^2

m^2

(

Z

a 0

) 2

Sincea 0 =αmc ̄h , we get

〈ψ 100 |(vr)^2 |ψ 100 〉=Z^2 α^2 c^2

ForZ= 1, the RMS velocity isαcor

β=α=

1

137

We can compute the expected value of the kinetic energy.

K.E.=

1

2

mv^2 =

̄h^2

2 m

Z^2

a^20

=

1

2

Z^2 α^2 mc^2 =−E 100

This is what we expect from the Virial theorem.

16.5 Sample Test Problems

1. A Hydrogen atom is in its 4D state (n= 4,ℓ= 2). The atom decays to a lower state by emitting
   a photon. Find the possible photon energies that may be observed.Give your answers in eV.
   Answer
   Then= 4 state can decay into states withn= 1, 2 ,3. (Really then= 1 state will be
   suppressed due to selection rules but this is supposed to be a simple question.) The energies
   of the states are
   En=−

13. 6

n^2

eV.

The photon energy is given by the energy difference between the states.

Eγ= 13. 6

(

1

n^2

−

1

42

)

For then= 1 final state,E=^151613 .6 = 12.8 eV.

For then= 2 final state,E= 16313 .6 = 2.6 eV.

For then= 3 final state,E= 144713 .6 = 0.7 eV.

2. Using theψnℓm notation, list all then= 1, 2 ,3 hydrogen states. (Neglect the existence of
   spin.)
   Answer
   The states are,ψ 100 ,ψ 200 ,ψ 211 ,ψ 210 ,ψ 21 − 1 ,ψ 300 ,ψ 311 ,ψ 310 ,ψ 31 − 1 ,ψ 322 ,ψ 321 ,ψ 320 ,ψ 32 − 1 ,
   ψ 32 − 2.


3. Find the difference in wavelength between light emitted from the 3P → 2 S transition in
   Hydrogenand light from the same transition in Deuterium. (Deuterium is an isotope of
   Hydrogen with a proton and a neutron in the nucleus.) Get a numerical answer.