- Anelectron in the Coulomb field of a protonis in the state described by the wave
function^16 (4ψ 100 + 3ψ 211 −ψ 210 +
√
10 ψ 21 − 1 ). Find the expected value of the Energy,L^2 and
Lz. Now find the expected value ofLy.
5.*Write out the (normalized)hydrogen energy eigenstateψ 311 (r,θ,φ).
- Calculate theexpected value ofrin the Hydrogen stateψ 200.
- Write down the wave function of the hydrogen atom stateψ 321 (r).
- A Hydrogen atom is in its 4Dstate (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 ineV.
- A Hydrogen atom is in the state:
ψ(r) =
1
√
30
(ψ 100 + 2ψ 211 −ψ 322 − 2 iψ 310 + 2iψ 300 − 4 ψ 433 )
For the Hydrogen eigenstates,〈ψnlm|^1 r|ψnlm〉=a 0 Zn 2. Find the expected value of the potential
energy for this state. Find the expected value ofLx.
- A Hydrogen atom is in its 3Dstate (n= 3,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 ineV.
- The hydrogen atom is made up of a proton and an electron boundtogether by the Coulomb
force. The electron has a mass of 0.51 MeV/c^2. It is possible to make a hydrogen-like atom
from a proton and a muon. The force binding the muon to the protonis identical to that for
the electron but the muon has a mass of 106 MeV/c^2.
a) What is the ground state energy of muonic hydrogen (in eV).
b) What is the“Bohr Radius” of the ground state of muonic hydrogen.
- A hydrogen atom is in the state:ψ(r) =√^110 (ψ 322 + 2ψ 221 + 2iψ 220 +ψ 11 − 1 ) Find the possible
measured energies and the probabilities of each. Find the expectedvalue ofLz.
- Find the difference in frequency between light emitted from the 2P→ 1 Stransition in Hydro-
gen and light from the same transition in Deuterium. (Deuterium is an isotope of Hydrogen
with a proton and a neutron in the nucleus.)
- Tritium is an isotope of hydrogen having 1 proton and 2 neutronsin the nucleus. The nucleus
is unstable and decays by changing one of the neutrons into a proton with the emission of
a positron and a neutrino. The atomic electron is undisturbed by thisdecay process and
therefore finds itself in exactly the same state immediately after the decay as before it. If the
electron started off in theψ 200 (n= 2,l= 0) state of tritium, compute the probability to find
the electron in the ground state of the new atom with Z=2.
- Att= 0 a hydrogen atom is in the stateψ(t= 0) =√^12 (ψ 100 −ψ 200 ). Calculate the expected
value ofras a function of time.
Answer
ψ(t) =
1
√
2
(ψ 100 e−iE^1 t/ ̄h−ψ 200 e−iE^2 t/ ̄h) =e−iE^1 t/ ̄h
1
√
2
(ψ 100 −ψ 200 ei(E^1 −E^2 )t/ ̄h)
〈ψ|r|ψ〉=
1
2
〈ψ 100 −ψ 200 ei(E^1 −E^2 )t/ ̄h|r|ψ 100 −ψ 200 ei(E^1 −E^2 )t/ ̄h〉
