## Transcribed Text

Quantum Mechanics and Spectroscopy
1. The speed of a 2.0 g projectile is known to within 10-6 m s
-1
. Calculate the minimum uncertainty
in its position.
2. When UV light of wavelength 165 nm strikes a metal surface, electrons are ejected with a speed
of 1.24x106 m s-1
.
(i) Calculate the speed of electrons ejected by radiation of wavelength 247 nm.
(ii) Would electrons be ejected when light of wavelength 400 nm is shone on the metal?
3. What is the wavelength of electrons that have been accelerated from rest through a potential
difference of 1.2 kV?
4. The two lowest energy wavefunctions for a particle confined to a box of length L are:
L
x
L L
x
L
2
sin 2
sin
2
2
1
2
1
1 2
(i) Sketch the wavefunctions and corresponding probability densities.
(ii) Show that the wavefunctions are normalised.
(iii) Show that the wavefunctions are orthogonal.
(iv) Find the expectation value of the momentum (
dx
d
x
ˆ
ip
).
You may find the following relations useful for question 4:
cossin22sin
2cos1sin2
2
5. Derive an expression for the energy of a particle in a box model using the de Broglie relation.
6. Derive the same energy expression as in question 5 using the same wavefunction, but using the
Schrödinger equation.
7. The wavelength of the absortption band of CH3-(CH≡CH)n-CH3 varies as follows. Explain this
observation, giving any relevant equations. Hint: think particle in a box!
n λ / nm
2 225
3 275
4 310
5 342
6 373
8. Consider the harmonic oscillator model of vibrational motion, with V = ½kx2
.
(i) Show that ψ = sin(πx / L) is not a valid wavefunction.
(ii) For each of the wavefunctions,
2
ax
e
and
2
ax
xe
, find the value of a, and also
the value of the energy (in terms of ħ, k and m). What energy levels do each of these
energies correspond to?
9. Where is an electron in a 1s orbital of the hydrogen atom most likely to be found? Show that
the electron in the He+
ion is more likely to be found closer to the nucleus than that of hydrogen.
10. By considering the radial wavefunction, show that the 1s orbital of hydrogen has no nodes.
11. By considering the radial wavefunctions of the 2s and 3s orbitals of hydrogen, deduce the
number of nodes for each, and their location(s).
12. The wavefunction for the ground state of a hydrogen atom is
0
2
1
3
0
1 a
r
e
a
Calculate the probability that the electron will be found within a sphere of radius a0 centred at
the nucleus, given the following integral:
2
2 2 22
aa
r
r
a
e
er dr
ar
ar
Weighting of marks
Question %
1 2.5
2(i) 5
2(ii) 2.5
3 2.5
4(i) 2.5
4(ii) 5
4(iii) 5
4(iv) 7.5
5 5
6 7.5
7 7.5
8(i) 5
8(ii) 12.5
9 5
10 5
11 7.5
12 12.5

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