## 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) = 13.6 eV,  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.

### 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:

For sodium the energy levels with the same principal quantum number n and with different angular momentum quantum numbers l are different, whereas for hydrogen the energy levels depend only on n.  The principal reason for this is that

(A)  states with different l values have different spin-orbit splittings.
(B)  in states with smaller l values, the electrons penetrate farther into the electron cloud shielding the nucleus.
(C)  the Pauli exclusion principle allows only two electrons to occupy each state.
(D)  the relativistic change in mass of the electron lowers its energy.
(E)  n is not a good quantum number.

### 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    (E)  40,000 E0

### Problem 4:

Which of the following states is possible for an atom with a closed core plus one d-electron (l = 2)?

(A)  3D5/2    (B)  4D3/2    (C)  2D1/2    (D)  2D5/2    (E)  3D1/2

### Problem 5:

If a weak electric field of magnitude E is applied to an atom in its ground state (Stark effect), what happens to the energy of the atom?

(A)  It changes by an amount proportional to E.
(B)  It changes by an amount proportional to E2.
(C)  It changes by an amount proportional to E3.
(D)  It changes by an amount proportional to E4.
(E)  It does not change.

### Problem 6:

Two spin 1/2 particles have spins in a singlet state with spin wave function

ψ(1,2) = 2-1/2[α(1)β(2) - α(2)β(1)]

where α and β refer to up and down spins, respectively, along any chosen axis.  The spin of particle 1 is measured along the z-axis and found to be up.
A simultaneous measurement of the spin of particle 2 along the z-axis would yield which of the following results?

(A)  Up with 100% probability
(B)  Down with 100% probability
(C)  Up with 25% probability and down with 75% probability
(D)  Up with 50% probability and down with 50% probability
(E)  Up with 75% probability and down with 25% probability

### Problem 7:

Refer to the previous problem.
A simultaneous measurement of the spin of particle 2 along the z-axis would yield which of the following results?

(A)  Up with 100% probability
(B)  Down with 100% probability
(C)  Up with 25% probability and down with 75% probability
(D)  Up with 50% probability and down with 50% probability
(E)  Up with 75% probability and down with 25% probability

### Problem 8:

A one dimensional Square well potential with infinitely high sides is shown.  In the lowest energy state the wave function is proportional to sin(kx).  if the potential is altered slightly by introducing a small bulge in the middle as shown, which of the following is true of the ground state?

(A)  The energy of the ground state remains unchanged.
(B)  The energy of the ground state is increased .
(C)  The energy of the ground state is decreased.
(D)  The original ground state splits into two states of lower energy.
(E)  The original ground state splits into two states of higher energy.

### Problem 9:

X and Y are two stationary states of a particle ina spherically symmetric potential.  In which pf the following situations will the wave function of the two states be orthogonal?

I.  X and Y correspond to different energies.
II.  X and Y correspond to different total orbital angular momentum L.
III.  X and Y correspond to the same L but  different Lz.

(A)  Not necessarily in any of these situations.
(B)  In situation I, but not necessarily in II or III.
(C)  In situation I and II, but not necessarily in III.
(D)  In situation II and III, but not necessarily in I.
(E)  In all three situations.

### Problem 10:

A portion of the energy level diagram for sodium is shown below.
Which of the following optical transition is NOT allowed?

(A)  5s - 3p    (B)   4p - 3s    (C)   4p - 4s   (D)   4d - 3s    (E)   3d - 3p