Download Introduction to Quantum Mechanics: Homework #1 (Due by Sep

Introduction to Quantum Mechanics
Homework #2 (Due by April 8, 2015)
1. Consider a box (-L ≦ x ≦ L) -normalized free particle wave function
Ψ= N [(e3ikx + 2e-2ikx)].
1) Express the normalization constant N in terms of L.
2) If one measure px, what value will be resulted? Repeat the same problem
for E.
3) Since it is a mixed state, it is neither an eigenfunction of the momentum
operator or of the energy operator Ĥ. Calculate the expectation values of
the momentum and energy, respectively.
2. Consider a 1-D particle (mass m) in a box (size L).
1) Calculate the probability P of finding the particle in the region 0 <x< L/4 for
the ground state.
2) Repeat the problem to calculate Pn for a state with a translational quantum
number n.
3) Show that the ‘correspondence principle’ holds true as n approaches ∞.
3. The average translational energy of N2 is <Etrans> = 3/2 kT.
1) Calculate the quantum number n = (nx2 + ny2+ nz2)1/2 corresponding to
<Etrans> = 3/2 kT for N2 at 298 K.
2) What can you learn from the magnitude of n about the energy gap between
two adjacent energy levels.
4. An electron is approaching a rectangular potential barrier from the left, where
L = 4 Å and V0 = 15 eV. The De Broglie wavelength of the electron is 8 Å.
1) Calculate the energy of the electron in eV.
2) Calculate the tunneling probability through the barrier.
5. The valence electrons of a conjugated linear hydrocarbon molecule with
alternating double and single c-c bonds (ex: CH2=CH-CH=CH-CH=CH2) can
be considered a 1-D particle in a box because the electron is delocalized
over the entire carbon chain. The bond length of c=c is 1.3 Ǻ and that of c-c
is 1.54 Ǻ. Neglect the edge effect. Calculate the wavelength of the light in Ǻ
emitted in the electronic transition n=2→ n=1 for the molecule CH2=CHCH2=CH-CH2=CH2.
6. 1) Solve the differential equation y’ = 2xy by an ordinary method.
2) Repeat the problem by using the power series method.
7. The ground state wavefunction of a 1D H.O. is given by
Ψ(x) = N0 exp (- x2/2α2), where α = (h2/4π2mk)1/4
1) Determine the normalization constant N0 .
2) Show that Ψ(x) is an eigenfunction of the Hamiltonian operator with an
eigenvalue of E0 = hν/2
3) Show that <T> = <V> = E /2 for the first excited state.
8. The vibrational motion of 1H19F can be approximated as a harmonic oscillator.
The force constant k = 966 N/m.
1) Calculate the reduced mass of 1H19F.
2) Calculate ω
3) Calculate the wavenumber in cm-1 unit of the infrared light emitted by a
vibrationally excited HF* molecule. The selection rule is that only Δv = -1 is
allowed in emission.
9. Show that the operator Lz =( h/2πi) ∂/∂Φ
10. Suppose that Ĥ1 (x ) Φ1 (x) = E1 Φ1 (x) and Ĥ2 (y) Φ2 (y) = E2 Φ2 (y).
Let Ĥ(x,y)= Ĥ1(x) +Ĥ2(y). Then, show that Ψ(x,y) = Φ1(x)Φ2(y) is an
eigenfunction of Ĥ and the corresponding eigenvalue is E = E1 + E2.
11. The bond length of H2 is 0.74 Ǻ. Assuming H2 is a rigid rotor,
1) Calculate the moment of inertia of H2 molecule.
2) Which rotational quantum number l is the most probable for H2 in thermal
equilibrium at 300 K? The 3D rotational wavefunction Y2,1 = (15/8π) cosθ sinθ
exp (iΦ)
1) Show that Y2,1 is an eingenfunction of the operators L2 and Lz.
2) What are the eingenvalues of E, L2, and Lz , respectively.
3) What is the magnitude of the angular moment for this state?
4) Calculate the angle between L and the z-axis?
12. A rotational state of a 3D rigid rotor is represented by
Ψ = N (Y3,0 + 3 Y2,-1 + 2 Y1,1 + Y0,0)
1) Calculate the normalization constant N.
Hint: <Yl,ml│Yl’,ml’> = δl,l’ δml, ml’ (orthonormal)
2) Show the results (probability) of the E, L2, and Lz measurements by a
Histograms, respectively. Assume that the measurements are made many
times.
3) What are the expectation values of E, L2, and Lz, respectively.