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PHYS 342
Fall Semester 2011
Solutions: Homework No. 4
Due: October 14, 2011
1.
In an experiment, the speed of an electron is measured to be 6.0x10
5
m/s. Since all
measurements have an uncertainty associated with them, in this case say the
electron speed is only measured to an accuracy of
1.0x10
4
m/s.
This implies that
the uncertainty in the speed of the electron is therefore 2x10
4
m/s. According to
quantum mechanics, what is the size of the smallest region in space in which the
electron could be found?
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2.
Consider the wave form shown below
Assume the wave can be represented by the formula
By inspection, what are your best guesses for B, C,
1
and
2
? Assume
2
>
1
.
12
22
sin
sin
x
x
AB
C
‐
3
‐
2
‐
1
0
1
2
3
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Amplitude,A
x (in m)
3
3.
a) Write down the solution to the angular Schrödinger’s Equation for the m=+1,
ℓ
=1
quantum state of atomic hydrogen.
b)
Verify that the spherical harmonic Y
11
(
ϑ
,
φ
) you wrote in a) satisfies the angular
Schrödinger’s Equation
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4.
The radial Schrödinger’s Equation for atomic hydrogen is
a) Write an equation for the effective potential energy that appears in this equation.
This effective potential energy will include the electrostatic interaction energy and any
energy associated with rotational motion.
b) Graph the function written in a) for three cases:
ℓ
=0,
ℓ
=1,
and
ℓ
=2.
Before
plotting, convert all energies to eV. Limit the plot of the effective potential to energies
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This note was uploaded on 12/08/2011 for the course PHYS 342 taught by Professor Staff during the Fall '08 term at Purdue University.
 Fall '08
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