Question

Consider the full time-dependent wavefunctions Ψ(x, t) =
ψ(x)φ(t). For the case of an infinite

square well in 1D, these were

Ψn(x, t) = Sqrt (2/L) sin(nπx/L) e^(−i(En/h)t

In general, the probability density |Ψn|2 is time-independent. But suppose instead of being ina fixed energy state, we are in a special state Ψmix(x, t) = √12(Ψ1 − iΨ2). What is the time-dependent part of |Ψmix|2?

Answer #1

In this problem we are interested in the time-evolution of the
states in the infinite square potential well. The time-independent
stationary state wave functions are denoted as ψn(x) (n = 1, 2, . .
.).
(a) We know that the probability distribution for the particle
in a stationary state is time-independent. Let us now prepare, at
time t = 0, our system in a non-stationary state
Ψ(x, 0) = (1/√( 2)) (ψ1(x) + ψ2(x)).
Study the time-evolution of the probability...

Consider the wave function at t = 0, ψ(x, 0) = C sin(3πx/2)
cos(πx/2) on the interval 0 ≤ x ≤ 1.
(1) What is the normalization constant, C?
(2) Express ψ(x,0) as a linear combination of the eigenstates of
the infinite square well on the interval, 0 < x < 1. (You
will only need two terms.)
(3) The energies of the eigenstates are En =
h̄2π2n2/(2m) for a = 1. What is
ψ(x, t)?
(4) Compute the expectation...

A particle in a strange potential well has the following two
lowest-energy stationary states:
ψ1(x) = C1 sin^2 (πx) for − 1 ≤ x ≤ 1
ψ2(x) = C2 sin^2 (πx) for − 1 ≤ x ≤ 0
ψ2(x) = −C2 sin^2 (πx) for 0 ≤ x ≤ 1
The energy of ψ1 state is ¯hω. The energy of ψ2 state is 2¯hω.
The particle at t = 0 is in a superposition state Ψ(x, t = 0) = 1/√...

Show that
x(t) =c1cosωt+c2sinωt, (1)
x(t) =Asin (ωt+φ), (2) and x(t) =Bcos
(ωt+ψ) (3)
are all solutions of the differential equation d2x(t)dt2+ω2x(t)
= 0. Show that thethree solutions are identical. (Hint: Use the
trigonometric identities sin (α+β) =sinαcosβ+ cosαsinβand cos (α+β)
= cosαcosβ−sinαsinβto rewriteEqs. (2) and (3) in the form of Eq.
(1). To get full marks, you need to show the connection between the
three sets of parameters: (c1,c2), (A,φ), and (B,ψ).)
From Quantum chemistry By McQuarrie

Consider the time-dependent ground state wave function
Ψ(x,t ) for a quantum particle confined to an
impenetrable box.
(a) Show that the real and imaginary parts of Ψ(x,t) ,
separately, can be written as the sum of two travelling waves.
(b) Show that the decompositions in part (a) are consistent with
your understanding of the classical behavior of a particle in an
impenetrable box.

The purpose of this problem is to compare the time dependencies
for systems in a superposition of two energy eigenstates in an
infinite square well to those in a simple harmonic
oscillator.
Consider two systems (an infinite square well and a simple harmonic
oscillator) that have the same value for their ground state energy
Eground.
1) What is E3, the energy of the 2nd excited
state (the third lowest energy) of the infinite square well system
in terms of Eground?...

The purpose of this problem is to compare the time dependencies
for systems in a superposition of two energy eigenstates in an
infinite square well to those in a simple harmonic
oscillator.
Consider two systems (an infinite square well and a simple harmonic
oscillator) that have the same value for their ground state energy
Eground.
1) What is E3, the energy of the 2nd excited
state (the third lowest energy) of the infinite square well system
in terms of Eground?...

The purpose of this problem is to compare the time dependencies
for systems in a superposition of two energy eigenstates in an
infinite square well to those in a simple harmonic
oscillator.
Consider two systems (an infinite square well and a simple harmonic
oscillator) that have the same value for their ground state energy
Eground.
1) What is E3, the energy of the 2nd excited
state (the third lowest energy) of the infinite square well system
in terms of Eground?...

Consider an infinite-horizon groundwater management problem
where xt is the stock of groundwater at time t, yt is the quantity
of groundwater extracted at time t, g(yt) gives the amount of total
recharge to the aquifer, B(yt) is the benefit of extracting yt, and
C(xt, yt)
2
is the cost of extracting yt from a stock of size xt. Suppose
that there are property rights to groundwater, but a large number
of users. Groundwater used in agriculture is unregulated. Suppose...

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...

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