Question

Consider a three dimensional rectangular infinite potential well with sides of length L, 2L and 3L.

What is the energy of the first excited state relative to the
energy of the ground state?

What is the energy of the second excited state relative to the
energy of the ground state?

What is the energy of the third excited state relative to the
energy of the ground state?

What is the energy of the fourth excited state relative to the
energy of the ground state?

Answer #1

For one dimensional rectangular box of length,

E = k n^{2} /
L^{2}
{ eq. 1 }

where, k = (hbar)^{2}^{2} /
2m

inserting the value of k in above eq.

E = (hbar)^{2}^{2}
n^{2} / 2m L^{2} {
eq.2 }

For three dimensional box,

E = k [n^{2}/L^{2} +
m^{2}/(2L)^{2} +
p^{2}/(3L)^{2}]
where, n = 1,2,3,.. m = 1,2,3,... p = 1,2,3,...

At n = 1, m = 1 & p = 1,

on the ground state, energy is given as ::

E = k [1/ L^{2} + 1/4 L^{2} + 1/9
L^{2}]

E = k [(36 + 9 + 4) / (36
L^{2})]

E = 49 k / 36
L^{2}
{ eq.3 }

(a) the energy of the first excited state relative to the energy of the ground state which is given as ::

At n=m=1, p = 2, energy of first excited state -

E_{1} = k [1/ L^{2} + 1/4 L^{2} + 4/9
L^{2}]

E_{1} = 61 k / 36
L^{2}
{ eq.4 }

and E_{1} / E = (61 k / 36 L^{2}) x
(36 L^{2} / 49 k)

E_{1} / E = 61 / 49

**E _{1} / E = 1.24**

(b) the energy of the second excited state relative to the energy of the ground state is given as ::

At n=1, m=2 , p=1

energy of the second excited state, E_{2} = k [1/
L^{2} + 4/4 L^{2} + 1/9
L^{2}]

E_{2} = 76 k / 36
L^{2}
{ eq.5 }

and E_{2} / E = (76 k / 36 L^{2}) x
(36 L^{2} / 49 k)

E_{2} / E = 76 / 49

**E _{2} / E = 1.55**

(c) the energy of the third excited state relative to the energy of the ground state is given as :

At n=1, m=1 , p=3

energy of the second excited state, E_{3} = k [1/
L^{2} + 1/4 L^{2} + 9/9
L^{2}]

E_{3} = 81 k / 36
L^{2}
{ eq.6 }

and E_{3} / E = (81 k / 36 L^{2}) x
(36 L^{2} / 49 k)

E_{3} / E = 81 / 49

**E _{3} / E = 1.65**

(d) the energy of the fourth excited state relative to the energy of the ground state which is given as :

At n=1, m=2 , p=2

energy of the second excited state, E_{4} = k [1/
L^{2} + 4/4 L^{2} + 4/9
L^{2}]

E_{4} = 88 k / 36
L^{2}
{ eq.7 }

and E_{4} / E = (88 k / 36 L^{2}) x
(36 L^{2} / 49 k)

E_{4} / E = 88 / 49

**E _{4} / E = 1.79**

Exercise
3. Consider a particle with mass m in a
two-dimensional infinite well of length L, x, y
∈ [0, L]. There is a weak potential in the well
given by
V (x,
y) = V0L2δ(x −
x0)δ(y − y0)
.
Evaluate the first order correction to the energy of the ground
state.
Evaluate the first order corrections to the energy of the first
excited states for x0 =y0 = L/4.
For the first excited states, find the points...

For the infinite square-well potential, find the probability
that a particle in its fourth excited state is in each third of the
one-dimensional box:
a) (0 ≤ x ≤ L/3)
b) (L/3 ≤ x ≤ 2L/3)
c) (2L/3 ≤ x ≤ L)

An electron is trapped in an infinite one-dimensional well of
width = L. The ground state energy for this electron is 3.8
eV.
a) Calculated energy of the 1st excited state.
b) What is the wavelength of the photon emitted between 1st
excited state and ground states?
c) If the width of the well is doubled to 2L and mass is halved
to m/2, what is the new 3nd state energy?
d) What is the photon energy emitted from the...

For the infinite square-well potential, find the probability
that a particle in its third excited state is in each third of the
one-dimensional box:
(0 ≤ x ≤ L/3)
(L/3 ≤ x ≤ 2L/3)
(2L/3 ≤ x ≤ L)

A particle is in the ground state of an infinite square well.
The potential wall at x = L suddenly (i.e., instantaneously) moves
to x = 3L. such that the well is now three times its original size.
(a) Let t = 0 be at the instant of the sudden change in the
potential well. What is ψ(x, 0)?
(b) If you measure the energy of the particle in the new well,
what are the possible energies?
(c) Estimate the...

A particle is confined to the one-dimensional infinite potential
well of width L. If the particle is in the
n=2 state, what is its probability of detection between a) x=0, and
x=L/4; b) x=L/4, and x=3L/4; c) x=3L/4,
and x=L? Hint: You can double check your answer if you calculate
the total probability of the particle being
trapped in the well.
Please answer as soon as possible.

For a particle trapped in a one-dimensional infinite square well
potential of length ?, find the probability that the particle is in
its ground state is in
a) The left third of the box: 0 ≤ ? ≤ ?/3
b) The middle third of the box: ?/3 ≤ ? ≤ 2?/3
c) The right third of the box: 2?/3 ≤ ? ≤ L
After doing parts a), b), and c):
d) Calculate the sum of the probabilities you got for...

consider a square well if infinite sides of width L:
a)
Calculate the energy and wavelength of a photon emitted when
a
transition
between the n=5 and ground state is made.
b)
Write down the expression for the probability that a particle in
the nth
state will be found in the first 1/3 of a well of width

Consider a particle trapped in an infinite square well potential
of length L. The energy states of such a particle are given by the
formula: En=n^2ℏ^2π^2 /(2mL^2 ) where m is the mass of the
particle.
(a)By considering the change in energy of the particle as the
length of the well changes calculate the force required to contain
the particle. [Hint: dE=Fdx]
(b)Consider the case of a hydrogen atom. This can be modeled as
an electron trapped in an infinite...

Consider a two-dimensional squared-well of dimensions L
× L. The length L is such, that the ground state energy of
one electron confined in this box is 0.1eV.
(a) Write down the 5 lowest energy states and their
corresponding degeneracy (your energy values must all be
different!) and label them E1 · · · E5
(b) If the electron finds itself in one of the states with
energy E5, how much energy would be required to lift the
electron from...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 26 minutes ago

asked 26 minutes ago

asked 50 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 1 hour ago

asked 1 hour ago