Quantum Mechanics

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).

Hydrogenic atoms (two particles with charge qe and charge Zqe 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 a0 by a0' = ħ2/(μ'Ze2) = a0(μ/μ')(1/Z), and in the eigenvalues of the Hamiltonian of the hydrogen atom we replace EI by EI' = μ'Z2e4/(2ħ2) = EI(μ'/μ)Z2.
Here μ' denotes the reduced mass of the hydrogenic atom, μ' = m1m2/(m1 + m2).

Angular momentum
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.

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 ji = jf = 0), Δs = 0, Δl = 1, Δmj = 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 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 

Problem 4:

Problem 5:

Problem 6:

Problem 7:

Problem 8:

Problem 9:

Problem 10: