Download Problem set #1 - U.C.C. Physics Department

PY3102 - Quantum Mechanics
Problem Set 1
1) Wavevectors and wavelength Consider the complex wave of amplitude
ψ(x, y, t) = ei(kx x+ky y−ωt) ,
(1)
where kx = k cos θ and ky = k sin θ. Calculate the wavelength λ, the phase
velocity vφ and the direction of motion of this wave. Consider the square
region
0<x<L
(2)
0<y<L
and let kx = 6π/L; ky = 4π/L. Draw out (i) for t = 0 and (ii) for t = π/ω,
the lines along which ψ(x, y, t) = 1. Calculate the repeat distance of the
wave along the x-direction, the y-direction, and its direction of motion.
2) Bohr’s atomic model
Recall that Bohr derived Rydberg’s constant by assuming (1) that the electrons move around the nucleus in discrete orbits; and (2) that the angular
momentum is quantized. Here, I ask you to repeat Bohr’s basic arguments.
a) Compare the Coulomb attractive force acting on the electron with its
centripetal force (me v 2 /r), and apply the condition of quantization of
angular momentum: |L| = |r| × |p| = nh̄ (n = 1, 2, 3, ...), to find the
allowed radii of the electron’s orbit around the nucleus.
b) Calculate the electron’s kinetic energy, T and its potential energy, V (from
which the Coulomb force was derived). When calculating the potential
energy, assume that when the electron is at rest at r → ∞, its energy is
E = 0.
c) The electron’s total energy is obviously T + V . Using Bohr’s frequency
relation, hν = Eb − Ea , obtain Rydberg’s formula,
1
1
me q 4
−
.
νab = 2 3
8ǫ0 h
n2a
n2b
(q is the electron’s charge, me is the electron’s mass, c is the speed of light,
ǫ0 is the permittivity of space, h is Planck’s constant and h̄ = h/2π).
3) Using the appropriate units, one finds that the innermost orbit of the electron
when it circulates the nucleus of the Hydrogen atom to be
r=
(4πǫ0 )h̄2
= 5.3 × 10−9 cm.
me q 2
(This is known as “Bohr’s radius”).
Let us assume that the electron’s motion around the nucleus is circular. The
electron is therefore subject to centripetal acceleration, v 2 /r. Classically,
when a particle is accelerated, it emits radiation, thereby loses its energy.
The dominant energy loss is from electric dipole radiation, which obeys the
Larmor formula,
2q 2 a2
dE
=−
.
dt
3c3
Here, q = 4.8×10−10 is the electron’s charge (in esu units), c = 3×1010 cm/s
is the speed of light and a is the electron’s acceleration. Calculate the time
it takes the electron to lose its energy and fall into the nucleus. Hint: this
time can be approximated by t ≈ E/(dE/dt). Recall that the electron mass
(in cgs units) is me = 9.1 × 10−28 gr, and h = 6.626 × 10−27 erg s.
4) Let us assume that a light bulb emits a monochromatic yellow light, at
wavelength λ = 600 nm. You are standing 3 meters away from a 60 W light
bulb, and looks at it. Calculate the number of photons that enter your eyes
each second. You can assume that the light bulb’s emission is spherical, and
the (combined) size of your pupils is ≈ 0.1 cm2 .
5) Prove Euler’s formula,
eiθ = cos θ + i sin θ.
Big hint. In my lecture notes, you will find a derivation of a very important
mathematical tool known as Taylor expansion. Read it and make sure you
are familiar with using it.
To be handed on or before Thursday, Feb. 2nd