**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 E_{n} = -μe^{4}/(2ħ^{2}n^{2})
= -13.6 eV/n^{2}.

r_{n} = n^{2}ħ^{2}/(μe^{2}) = n^{2}a_{0}
= n^{2} * (53 pm). a_{0} is called the Bohr radius.

Here μ ~m_{e}, the reduce mass m of the hydrogen atom is very close to
the electron mass m_{e}.

**The Hydrogen atom:**

The ground state energy of the hydrogen atom is -E_{I}.

E_{I} = e^{2}/(2a_{0}) = μe^{4}/(2ħ^{2}),
E_{n} = -E_{I}/n^{2}

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 E_{n} = -hcR_{H}/n^{2}, where R_{H}
is the Rydberg constant.

E_{n'} - E_{n} = hcR_{H}(1/n^{2} - 1/n'^{2}).

### Hydrogenic atoms (two particles with charge –q_{e}
and charge Zq_{e} orbiting each other):

To find the eigenfunctions and eigenvalues of the Hamiltonian of a hydrogenic
atom we replace in the eigenfunctions of the Hamiltonian of the hydrogen atom a_{0}
by a_{0}' = ħ^{2}/(μ'Ze^{2}) = a_{0}(μ/μ')(1/Z),
and in the eigenvalues of the Hamiltonian of the hydrogen atom we replace E_{I}
by E_{I}' = μ'Z^{2}e^{4}/(2ħ^{2}) = E_{I}(μ'/μ)Z^{2}.

Here μ' denotes the reduced mass of the hydrogenic atom, μ' = m_{1}m_{2}/(m_{1}
+ m_{2}).

**Angular momentum**

If **J** denotes an angular momentum operator, then the eignvalues of J^{2}
ore j(j + 1)ħ^{2} and the eigenvalues of J_{z} are mħ.

Adding angular momenta: Let **J** = **J**_{1} + **J**_{2}.
The possible values of j range from |j_{1} – j_{1}| to j_{1}
+ j_{2} 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 = m_{1} + m_{2}.

**Selection rules**

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 j_{i }= j_{f }= 0), Δs = 0, Δl = ±1, Δm_{j
}= 0, ±1.

Example Problems: (Solutions)

**Problem 1:**

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

**Problem 2:**

**Problem 3:**

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 E_{0},
the binding energy of a muon-proton atom is __?

(A) E_{0}, (B) 14 E_{0}, (C) 200 E_{0}, (D)
4000 E_{0}, (A) 40,000 E_{0}

**Problem 4:**

**Problem 5:**

**Problem 6:**

**Problem 7:**

**Problem 8:**

**Problem 9:**

**Problem 10:**