Question

Calculate the probability that a particle will be found between 0.49a and 0.5a in a box of length a when it has:

a) n=1

b) n=2

Take the wavefunction to be constant in this small range.

Answer #1

Calculate the probability that a particle will be found in a
tiny slice of space between 0.59L and 0.61L in a box of length L
(defined in the interval (0,L) ) when it is in quantum state n = 3.
For simplicity of integration, take the wavefunction to have a
constant value equal to its midpoint value in the range given.

In class, we are discussing a free particle trapped inside the
box. Keeping this discussion in mind, please answer the following
questions. (a) Calculate the probability of finding the particle in
the first one third of the box (0 to a/3). The particle is residing
in the first excited state. (b) Show that the ground state
wavefunction is orthogonal to the first excited state wavefunction.
(c) Uncertainty is defined as the square root of variance ( a 2 =
-...

The particle in a 1-D box is sometimes used as a model for
electrons in a conjugated pi-bond system (alternating double and
single bonds).
a. The molecule has four pi electrons. Assume that two are in
the state corresponding to n=1 and that two are in the state
corresponding to n=2. Find the frequency and wavelength of the
light absorbed if an electron makes a transition from n=2 to
n=3.
b. Calculate the probability that a particle in a 1-D...

Calculate the amplitude change of the wavefunction of a particle in
a wall over the the distance of twice its decay length. Assume a
particle in a finite energy box

For a particle in a one-dimensional box with the length of 30 Å,
its wavefunction is ψ1+ψ3. What is the
location (except x=0 and x =30 Å) where the probability to find
this particle is 0?

For a particle in a one-dimensional box of width a, determine
the probability of finding the particle in the right third of the
box (between ‘2/3 a’ and ‘a’) if the particle is in the ground
state. ( Given: Y(x)= sqrt(2/a) sin(npix/a) )

The normalized wave functions for the particle is in a 1D box of
length L., with limits on x = 0 and x = L. V (x) = 0 for 0 <= x
<= L and V (x) = Infinity elsewhere. The probability of a
particle being between x = 0 and x = L / 8 in the ground quantum
state (n = 1) should be calculated.

Find the probability of finding a particle in a box of
length L in a region 0.45L to 0.55L to ground state and its excited
state.

For a particle in the first excited state of harmonic oscillator
potential,
a) Calculate 〈?〉1, 〈?〉1, 〈? 2〉1, 〈? 2〉1.
b) Calculate (∆?)1 and (∆?)1.
c) Check the uncertainty principle for this state.
d) Estimate the length of the interval about x=0 which
corresponds to the classically allowed domain for the first excited
state of harmonic oscillator.
e) Using the result of part (d), show that position uncertainty
you get in part (b) is comparable to the classical range of...

I want you to compare particle in a box with the Bohr atom. For
the particle in the box you can assume the size is .5x10^-10 m and
the particle that is trapped is an electron. For the Bohr atom
consider a hydrogen atom.
a.) What is the energy of photon emitted when the electron drops
from 3->2 and 2->1 in the particle in a box?
b.) What is the energy of photon emitted when the electron drops
from 3->2...

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