For an unbound (or “free”) particle having mass m in the complete absence of any potential energy (that is, V = 0 ), the acceptable one-dimensional wavefunctions are Ψ = A e i ( 2 m E ) 1 / 2 x / h + B e − i ( 2 m E ) 1 / 2 x / h , where A and B are constants and E is the energy of the particle. Is this wavefunction normalizable over the interval − ∞ < x < + ∞ ? Explain the significance of your answer.
Solution Summary: The author explains that the wavefunction is the function of the coordinates of particles and time.
For an unbound (or “free”) particle having mass
m
in the complete absence of any potential energy (that is,
V
=
0
), the acceptable one-dimensional wavefunctions are
Ψ
=
A
e
i
(
2
m
E
)
1
/
2
x
/
h
+
B
e
−
i
(
2
m
E
)
1
/
2
x
/
h
, where
A
and
B
are constants and
E
is the energy of the particle. Is this wavefunction normalizable over the interval
−
∞
<
x
<
+
∞
? Explain the significance of your answer.
For the system described in Exercise E7B.1(a) (A possible wavefunction for an electron in a region of length L (i.e. from x = 0 to x = L) is sin(2πx/L). Normalize this wavefunction (to 1)), what is the probability of finding the electron in the range dx at x = L/2?
The ground-state wavefunction for a particle confined to a one dimensional box of length L is
Ψ =(2/L)½ sin (πx/L)
Suppose the box 10.0 nm long. Calculate the probability that the particle is:
(a) between x = 4.95 nm and 5.05 nm (b) between 1.95 nm and 2.05 nm, (c) between x = 9.90 and 10.00 nm, (d) in the right half of the box and (e) in the central third of the box.
Determine the energetic difference between state n = 2 and n = 1 for a proton in a one-dimensional box of length 100 nm. At what wavelength does that energy correspond?
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