Summary of concepts you need to be familiar with to solve most GRE problems in this category:
The Bohr model of the hydrogen atom
In particular En = -μe4/(2ħ2n2) = -13.6 eV/n2.
rn = n2ħ2/(μe2) = n2a0 = n2 * (53 pm). a0 is called the Bohr radius.
Here μ ~me, the reduce mass m of the hydrogen atom is very close to the electron mass me.
The Hydrogen atom:
The ground state energy of the hydrogen atom is -EI.
EI = e2/(2a0) = μe4/(2ħ2), En = -EI/n2
Given n, l can take on n possible values l = 0, 1, 2, ..., (n-1). n characterizes an electron shell, which contains n subshells characterized by l. Each subshell contains 2l + 1 distinct states.
We also write En = -hcRH/n2, where RH is the Rydberg constant.
En' - En = hcRH(1/n2 - 1/n'2).
If J denotes an angular momentum operator, then the eignvalues of J2 ore j(j + 1)ħ2 and the eigenvalues of Jz are mħ.
Adding angular momenta: Let J = J1 + J2. The possible values of j range from |j1 – j1| to j1 + j2 in integer steps. For each j, m can take on values between –j and +j in integer step. When adding angular momenta, the m values add algebraically, m = m1 + m2.
Photons are most likely emitted or absorbed if these selection rules are satisfied. If these selection rules are not satisfied a transition is less likely and is said to be forbidden.
Δl = ±1, Δs = 0, Δm = 0, ±1.
With spin-orbit coupling,
Δj = 0, ±1, (except ji = jf = 0), Δs = 0, Δl = ±1, Δmj = 0, ±1.
Example Problems: (Solutions)
In the Bohr model, the radius of an excited hydrogen atom in the n = 100 state is closest to __?
(A) 100 μm (B) 10 μm (C) 0.5 μm (D) 0.1 μm (E) 5 nm
A muon is a heavy electron with a mass of about 200 times
the electron mass. If the binding energy of a hydrogen atom is E0,
the binding energy of a muon-proton atom is __?
(A) E0, (B) 14 E0, (C) 200 E0, (D) 4000 E0, (A) 40,000 E0