Download Homework Set 1

University of Massachusetts Amherst
Department of Chemistry
Chem 584 — Advanced Physical Chemistry I
Quantum Mechanics and Spectroscopy
Problem Set #1
1. Classical Mechanics of the Harmonic Oscillator
The low energy vibration of HCl is well approximated by a classical harmonic oscillator
with the Hamiltonian:
H(x, p) =
p2
1
+ kx2 ,
2m 2
(1)
where m is the vibrational mass and k is the spring constant.
a. Write down Hamilton’s equations for the time derivatives of x(t) and p(t).
q
b. Solve Hamilton’s equations using the following idea. By noticing that ω = k/m
is the only “time scale” in the system, we can write the most general solution as:
x(t) = A cos ωt + B sin ωt
d
p(t) = mv(t) = m x(t).
dt
(2)
(3)
Use the initial conditions x(t = 0) ≡ x0 and p(t = 0) ≡ p0 to obtain the constants
A and B in terms of m, ω, x0 and p0 .
c. Now show that, although x(t) and p(t) clearly vary with time, the energy function
H[x(t), p(t)] is actually a constant, i.e., a conserved quantity. Do this by plugging
in the solutions for x(t) and p(t) found above into the function for H(x, p).
Culture: The continuum of energy levels allowed by the classical harmonic oscillator
fails to predict that heat capacities of solids go to zero in the limit of zero absolute
temperature. The quantum description (first proposed in 1907 by Einstein) is needed
to reproduce this, and most other chemical phenomena, correctly.
2. Semiclassical Mechanics of Orbiting Systems
The de Broglie wavelength is defined by λ ≡ h/p, where h is Planck’s constant and
p is linear momentum. By calculating (λ/r) for an orbiting system, where r is the
radius of the orbit, we can determine if the system behaves “classically” or quantum
mechanically (a.k.a. “quantally”).
a. Calculate (λ/r) for the Earth orbiting the Sun assuming a circular orbit. Use p
= mv and v = 2πr/t, where m = 6×1024 kg, r = 1011 m, t = 3×107 sec, and h
= 6.6×10−34 J·sec. The result indicates what sort of (λ/r) values are typical for
very classical systems.
1
b. Now calculate (λ/r) for an electron orbiting a proton in the Bohr model of the
hydrogen atom. The angular momentum of the electron is given by L = pr. Using
Bohr’s quantization of angular momentum, Ln = nh/2π where n is some sort of a
“quantum number,” show that (λ/r) = 2π/n for the Bohr model of the hydrogen
atom. (This formula foreshadows the fact that, in general, the ground state of
any system is the most in need of a quantum description.)
c. Taking λ/r ≤ 0.1 as the (arbitrary) cut-off when classical mechanics begins to
be valid as Bohr’s quantum number n increases, calculate the lowest (smallest n)
classical Bohr orbit.
d. Using the Bohr theory, calculate the ionization energies in electron volts (eV) of
hydrogen (H → H+ ) and of singly ionized helium (He+ → He+2 ).
Culture: The Bohr model gets these ionization energies exactly, but totally fails
to predict the electronic properties of neutral helium, or any other element for that
matter. The full wave-particle duality is needed to treat these more complex elements.
3. Mathematics of Waves
a. Prove Euler’s formula, eiθ = cos θ + i sin θ.
Hint: Use the following power series expansions:
ex =
∞
X
xn
n=0
cos θ =
∞
X
n!
=1+x+
sin θ =
(4)
(−1)n
θ2n
θ2 θ4
=1−
+
+ ···
(2n)!
2!
4!
(5)
(−1)n
θ2n+1
θ3 θ5
=θ−
+
+ ···
(2n + 1)!
3!
5!
(6)
n=0
∞
X
x2 x3
+
+ ···
2!
3!
n=0
b. Show that e−iθ = cos θ − i sin θ.
c. Consider the wavefunction in space and time: W (x, t) = e2πi(x/λ−t/τ ) where λ is
the wavelength and ν = 1/τ is the frequency. Show that W (x, t) is periodic in
space and time, i.e., show that
W (x + λ, t) = W (x, t)
W (x, t + τ ) = W (x, t).
(7)
(8)
4. Free-Particle Schrödinger Equation
The wavefunction above can be written as: W (x, t) = eikx ·e−iωt , where k = 2π/λ is
called the “wavenumber,” and ω = 2π/τ is the “angular frequency.”
a. Show that W (x, t) solves the time-dependent Schrödinger Equation for a free
particle, i.e., with V (x) = 0.
2
b. Show that the following four spatial wavefunctions (eikx , e−ikx , sin(kx), cos(kx))
all solve the time-independent Schrödinger Equation for a free particle, with all
four wavefunctions giving the same energy. Explain the differences in the physical
states described by these four wavefunctions.
5. Hermitian Operators
An operator  is Hermitian on a space of functions F if (i) the operator  has real
eigenvalues, and (ii) if operating  on functions in F gives back functions in F. For
the problem below, we shall consider F to comprise all smooth, real and bounded
functions in 1d space. (Bounded simply means the function vanishes as x → ±∞.)
a. Prove that all (non-degenerate) eigenstates of  are orthogonal, i.e.,
that hψn |ψm i = 0 if n 6= m.
b. The Heisenberg Uncertainty Principle is really a mathematical relation satisfied
by any combination of two Hermitian operators. Using the Uncertainty Principle
in its commutator form, prove the following “space-energy” uncertainty relation
for the function space F:
∆x∆E ≥ 0.
3
(9)