Physics 2D Quiz #4
Department of Physics, UCSD
Summer Session II - 2009
Prof. Pathria
28 August 2009
Some Useful Data
Speed of light:
c = 2.998 ×108 m/s
Planck’s constant:
h = 6.626 ×10−34 J·s
Rest mass of an electron:
me = 9.109 ×10−31 kg = 0.511M eV /c2
1 eV = 1.602 ×10−19 J
Schrödinger wave equation:
2
2
h̄ d ψ
− 2m
dx2 + U (x)ψ = Eψ
Potential energy of a simple
U (x) = 21 Kx2 = 12 mω 2 x2
harmonic oscillator:
Some Useful Formulae
sin2 x = 12 (1 − cos 2x)
R∞
0
2
e−ax dx =
1
2
pπ
a
cos2 x = 12 (1 + cos 2x)
R∞
0
2
x2 e−ax dx =
1
4
pπ
a3
R∞
0
2
x4 e−ax dx =
3
8
pπ
a5
Instructions
Please write your answers in your blue book, and make sure your secret code
number is written on all pages in indelible ink.
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1. An electron is contained in a one-dimensional box of width 0.1 nm.
(a) Draw an energy level diagram for the electron for states up to n = 3, and
show all the transitions that would eventually get the electron from the n
= 3 state to the n = 1 state.
(b) Find the wavelengths of the photons emitted during these transtions.
2. An electron is trapped in a one-dimensional rigid-walled box of width 1.0
nm.
(a) Sketch the wavefunctions and the probability densities for the n = 1 and
the n = 2 states.
(b) For the n = 1 state, calculate the probability of finding the electron between x = 0.15 nm and x = 0.35 nm, where x = 0 is at the left end of the
box.
(c) Repeat the calculation in (b) for the n = 2 state.
(d) If the electron absorbs a photon and, as a result, makes a transition from
the n = 1 state to the n = 2 state, what would the wavelength of the
photon have to be?
3. The wavefunction:
2
ψ(x) = Cxe−ax
describes one of the eigenstates of the simple harmonic oscillator, provided that
the constant α is chosen appropriately.
(a) Using the Schrödinger wave equation for the oscillator, obtain an expression for α in terms of the oscillator mass m and the angular frequency
ω.
(b) What is the energy associated with this state?
(c) Determine the root-mean-square values of the deviations ∆x and ∆px
for the oscillator in this state, and verify that they are consistent with
Heisenberg’s uncertainty principle.
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