Problem Set 3: Quantum Mechanics: The Basic Principles
1. A photon-powered spacecraft of mass 10 kg emits radiation of wavelength 225
nm with a steady power of 1.5 x 103 Watts entirely in the backward direction. To
what speed will this spacecraft have accelerated 10 years after being released into
space ? (Assume that the spacecraft experiences no friction while traveling in
space).
2. The work function for metallic cesium is 2.14 eV. Calculate the kinetic energy,
and speed, of electrons ejected from cesium by light of wavelength (i) 700 nm,
and (ii) 300 nm.
3. What is the minimum uncertainty in the position of a bullet of mass 5 gm that is
known to have a speed in between 350.0001 m s-1 and 350.0000 m s-1 ?
4. The ground state wavefunction for a particle confined to a one-dimensional box of
length L is:
x
2
 ( x) 
Sin ( ).
L
L
Assume the “box” is of length L = 10 nm. Calculate the probability that the
particle is located (i) between the range x = 1.95 nm and x = 2.05 nm, and (ii) in
the central one-third of the box.
5. Consider a particle confined to a one-dimensional box of length L. What are the
most likely locations for the particle to be found if the particle is described by the
wavefunction with quantum number n = 3 ? (In other words, at which points in
the box does the probability distribution for the position of the particle have
maxima).
6. Calculate the percentage change in a given energy level of a particle confined to a
three-dimensional, cubic, box (with sides of length = L in each direction) when
the length of each edge of the cube is reduced by 10 %.
7. Calculate the wavelength of a photon that can excite a transition between
neighboring energy levels of a simple harmonic oscillator with mass equal to that
of a proton (1.673 x 10-27 kg), and force constant = 850 N m-1. What would the
wavelength for such a photon have to be if the mass of the particle were doubled
with the force constant left unaltered ?
8. Assume that the vibrations of a 35Cl2 molecule are equivalent to those of a simple
harmonic oscillator with force constant = 330 N m-1. Estimate the zero-point
vibrational energy of this molecule.
9. (MORE DIFFICULT:) Consider a particle confined to the interior of a onedimensional box of length = L. (i) If the particle is classical (i.e., follows classical
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mechanics and has a positional probability distribution that accords with classical
notions), show that the average value of x equals (L / 2), and that the root-meansquared value of x =
x 2  L . (ii) Next, show that for a particle that follows
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quantum mechanics instead, the average and root-mean-squared values of x
approach the results for the classical particle in the limit that the quantum number
n becomes large.
10. (MORE DIFFICULT:) Consider a particle of mass m, that experiences a
potential energy given by: U (x) = k x4, where x is the location of the particle, and
k is the analog of the “spring constant” for a simple harmonic oscillator. The
particle can more in only one direction, along the x-axis. Use the Uncertainty
Principle to estimate the ground state energy level (or zero-point energy) in terms
of k, m, and h for this “quartic oscillator”.
Useful Mathematical Results:
Sin 2 x  2Sin x Cos x , 2 Sin 2 x  1 Cos 2 x ,
Sin  a x 
 Cos a x dx  a ,
Cos a x  x Sin  a x 
 x Cos a x dx  a 2  a ,
2 x Cos a x  a 2 x 2  2 Sin  a x 
2


x
Cos
a
x
dx


,

a2
a3
Sin zero 0, Cos  zero  1,
Sin 2n 0, Cos2n 1
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