In: Physics

Consider a particle in an infinite square well, but instead of having the well from 0 to L as we have done in the past, it is now centered at 0 and the walls are at x = −L/2 and x = L/2.

(a) Draw the first four energy eigenstates of this well.

(b) Write the eigenfunctions for each of these eigenstates.

(c) What are the energy eigenvalues for this system?

(d) Can you find a general expression for the energy eigenstates for this well? (Hint: you should have a different form for even and odd solutions) (e) If instead, the well were from x = −L to x = L, how would these expressions change? Would the energy eigenvalues change? (And if so, how?)

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Consider a particle of mass ? in an infinite square well of
width ?. Its
wave function at time t = 0 is a superposition of the third and
fourth energy
eigenstates as follows:
? (?, 0) = ? 3i?3(?)+ ?4(?)
(Find A by normalizing ?(?, 0).)
(Find ?(?, ?).)
Find energy expectation value, <E> at time ? = 0. You
should not need to evaluate any integrals.
Is <E> time dependent? Use qualitative reasoning to
justify.
If you measure...

A particle in an infinite one-dimensional square well is in the
ground state with an energy of 2.23 eV.
a) If the particle is an electron, what is the size of the
box?
b) How much energy must be added to the particle to reach the
3rd excited state (n = 4)?
c) If the particle is a proton, what is the size of the box? d)
For a proton, how does your answer b) change?

A particle in the infinite square well (width a) starts out
being equally likely to be found in the first and last third of the
well and zero in the middle third.
What is the initial (t=0) wave function? Find A and graph the
initial wave function.
What is the expectation value of x? Show your calculation for
the expectation value of x, but then say why you could have just
written down the answer. Will you ever find the...

Find the energy spectrum of a particle in the infinite square
well, with potential U(x) → ∞ for |x| > L and U(x) = αδ(x) for
|x| < L. Demonstrate that in the limit α ≫ hbar^2/mL, the low
energy part of the spectrum consists of a set of closely-positioned
pairs of energy levels for α > 0. What is the structure of
energy spectrum for α < 0?

An infinite potential well in one dimension for 0 ≤ x ≤ a
contains a particle with the wave function ψ = Cx(a − x), where C
is the normalization constant. What is the probability wn for the
particle to be in the nth eigenstate of the innite potential well?
Find approximate numerical values for w1, w2 and w3.

Consider a semi-infinite square well: U(x)=0 for 0 ≤ x ≤ L,
U(x)=U0 for x > L, and U(x) is infinity otherwise.
Determine the wavefunction for E < Uo , as far as possible,
and
obtain the transcendental equation for the allowable energies E.
Find the necessary condition(s) on E for the solution to exist.

For the infinite square-well potential, find the probability
that a particle in its second excited state is in each third of the
one-dimensional box: 0?x?L/3 L/3?x?2L/3 2L/3?x?L There's already an
answer on the site saying that the wavefunction is equal to
?(2/L)sin(2?x/L). My professor gave us this equation, but also gave
us the equation as wavefunction = Asin(kx)+Bcos(kx), for specific
use when solving an infinite potential well. How do I know which
equation to use and when? Thanks

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

Consider a system of N particles in an infinite
square well fro, x=0 to x=N*a.
find the ground state wave function and
ground state energy for
A. fermions.
B. bosons.

Consider two non-interacting particles in an infinite square
well. One is in a state ψm, the other in a state
ψn with n≠m. Let’s assume that ψm and
ψn are the ground state and 1st excited state
respectively and that the two particles are identical fermions. The
well is of width 1Å. What is the probability of finding a particle
in the 1st excited state in a region of width 0.01Å?
Does this change if the particles are distinguishable?

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