Question

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/√ 2 [ψ1(x) + ψ2(x)].

1) Find the real positive coefficients C1 and C2 to normalize the wavefunctions ψ1(x) and ψ2(x). Show ψ1(x) and ψ2(x) are orthonormal.

2) What is the time-dependent wavefunction Ψ(x, t) of the particle? Draw the wavefunction at t = π/ω and t = 2π/ω .

3) Calculate the probability distribution of the particle as a function of time. Express it as a real function, and comment on its time dependence.

4) Calculate (x)(t).

5) If a measurement is made at t = 2π/13ω to probe the energy of the particle, what values might you get, and what is the probability of getting each of them?

Answer #1

A particle in a simple harmonic oscillator potential V (x) = 1
/2mω^2x^2 has an initial wave function
Ψ(x,0) = (1/ √10)(3ψ1(x) + ψ2(x)) ,
where ψ1 and ψ2 are the stationary state solutions of the ﬁrst and
second energy level. Using raising and lowering operators (no
explicit integrals except for orthonomality integrals!) ﬁnd
<x> and <p> at t = 0

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...

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?

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...

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...

A free particle has the initial wave function Ψ(x, 0) = Ae−ax2
where A and a are real and positive constants. (a) Normalize it.
(b) Find Ψ(x, t). (c) Find |Ψ(x, t)| 2 . Express your result in
terms of the quantity w ≡ p a/ [1 + (2~at/m) 2 ]. At t = 0 plot |Ψ|
2 . Now plot |Ψ| 2 for some very large t. Qualitatively, what
happens to |Ψ| 2 , as time goes on? (d)...

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...

The infinite potential well has zero potential energy between 0
and a, and is infinite elsewhere.
a) What are the energy eigenstates of this quantum system, and
what are their energies? In the case of a discrete spectrum,
explain where the quantization comes from.
b) Suppose we take the wavefunction at a given time to be an
arbitrary function of x that is symmetric around the center of the
well (at x = a/2). Is this a stationary state in...

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...

An electron (a spin-1/2 particle) sits in a uniform magnetic
field pointed in the x-direction:
B = B0xˆ.
a) What is the quantum Hamiltonian for this electron? Express your
answer in terms of B0,
other constants, and the spin operators Sx, Sy and Sz, and then
also write it as a matrix (in z basis).
b) What are the energy eigenvalues, and what are the associated
normalized eigenvectors
(in terms of our usual basis)? You may express the eigenvectors
either...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 11 minutes ago

asked 48 minutes ago

asked 51 minutes 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