Question

An electron confined to a one-dimensional box has energy levels given by the equation

*E**n*=*n*2*h*2/8*m**L*2

where *n* is a quantum number with possible values of
1,2,3,…,*m* is the mass of the particle, and *L* is
the length of the box.

Calculate the energies of the *n*=1,*n*=2, and
*n*=3 levels for an electron in a box with a length of 180
pm .

Enter your answers separated by a comma.

Calculate the wavelength of light required to make a transition from n=1→n=2 and from n=2→n=3. |

Enter your answers separated by a comma.

In what region of the electromagnetic spectrum do these wavelengths lie?

Answer #1

Length of box = 180 pm = 1.80x10^{-10} m

Mass of electron = 9.1x10^{-31} kg

En = n^{2}h^{2} /
8mL^{2}

Fon n = 1

E1 = 1^{2}*(6.626x10^{-34})^{2} /
8*9.1x10^{-31} * (1.80x10^{-10})^{2}

**E1 = 1.86x10 ^{-18} J**

Fon n = 2

E2 = 2^{2}*(6.626x10^{-34})^{2} /
8*9.1x10^{-31} * (1.80x10^{-10})^{2}

**E2 = 7.44x10 ^{-18} J**

Fon n = 3

E3 = 3^{2}*(6.626x10^{-34})^{2} /
8*9.1x10^{-31} * (1.80x10^{-10})^{2}

**E3 = 1.67x10 ^{-17} J**

n =1 n = 2

We know that,

1 / = R
[1/(n1)^{2} - 1/(n2)^{2}]

where,

= wavelength

R = Rydberg's constant (1.097 x 10^{7}
m^{−1})

1 / = (1.097 x
10^{7} m^{−1}) [1/(1)^{2} -
1/(2)^{2}]

= (1.097 x 10^{7} m^{−1}) [1 - 1/4]

= (1.097 x 10^{7} m^{−1}) * 0.75

= 0.82275 x10^{7} m^{-1}

= 1.215 x
10^{-7} m

** = 121.5 nm
(ultraviolet region)**

For, n =2 n = 3

1 / = (1.097 x
10^{7} m^{−1}) [1/(2)^{2} -
1/(3)^{2}]

= (1.097 x 10^{7} m^{−1}) [1/4 - 1/9]

= (1.097 x 10^{7} m^{−1}) * 0.138

= 0.15138 x10^{7} m^{-1}

= 6.605 x
10^{-7} m

** = 660.5 nm
(visible region)**

An electron confined in a one-dimensional box is observed, at
different times, to have energies of 32 eV , 72 eV , and 128 eV
.
What is the length of the box? Hint:
Assume that the quantum numbers of these energy levels are less
than 10.
Answer should be in nm

An electron is confined in a one-dimensional box 1nm long. How
many energy levels are there with energy between 10 eV and 100
eV?

Particle-in-a-box energies take the form (Fig 3):
En = εn2 =
(h2/8meL2)n2,
where h = 6.6×10-34 J·s, me = electron mass,
and n = quantum number starting at n = 1 for the
ground state of an isolated electron. Using this formula, estimate
the HOMO→LUMO transition energy ∆Ein kcal/mol for
1,3,5-hexatriene.
Fig 3. Particle-in-box energies.
You can do this in many different unit systems. Atomic units are
particularly convenient; in these units, h = 2π and
me = 1. The...

An electron is contained in a one-dimensional box of length
0.562 nm. (a) Draw an energy-level diagram for the electron for
levels up to n = 4. (b) Photons are emitted by the electron making
downward transitions that could eventually carry it from the n = 4
state to the n = 1 state. Find the wavelengths of all such photons:
λ4 → 3, λ4 → 2, λ4 → 1, λ3 → 2, λ3 → 1, λ2 → 1 {149,...

Find the energy of a ground state (n=1) of a proton in a one
dimensional box of length 6 nm. (meV)
Calculate the wavelength of electromagnetic radiation when the
photoon makes a transition from n=2 to n=1, and from n=3 to n=2
(micrometers)

1. Find the first three energy levels En (n = 1,2,3) that an electron can have in a quantum well structure with a well thickness of 120 Angstroms and an infinite potential barrier.
2. Also find the wave function, ?(x) (solve Schrodinger's equation)
3. Draw the wave function when n = 1 ~ 3.
(?=√2mE/ℏ)

II(20pts). Short Problems
a) The lowest energy of a particle in an infinite one-dimensional
potential well is 4.0 eV. If the width of the well is doubled, what
is its lowest energy?
b) Find the distance of closest approach of a 16.0-Mev alpha
particle incident on a gold foil.
c) The transition from the first excited state to the ground
state in potassium results in the emission of a photon with = 310
nm. If the potassium vapor is...

When an electron trapped in a one-dimensional box transitions
from its
n = 2
state to its
n = 1
state, a photon with a wavelength of 636.4 nm is emitted. What
is the length of the box (in nm)?
What If? If electrons in the box also occupied
the
n = 3
state, what other wavelengths of light (in nm) could possibly be
emitted? Enter the shorter wavelength first.
shorter wavelength nmlonger
wavelength nm

1) The Pauli Exclusion Principle tells us that
no two electrons in an atom can have the same four quantum
numbers.
Enter ONE possible value for each quantum number of an electron
in the orbital given.
Orbital
n
l
ml
ms
1s
There are a total of values possible for
ml.
2s
There are a total of values possible for
ml.
2) The Pauli Exclusion Principle tells us that
no two electrons in an atom can have the same four...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 17 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago