Question

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 density |Ψ(x, t)|^2 for this state. Is it periodic in the sense that after some time T it will return to its initial state at t = 0? If so, what is T? Sketch, better yet plot (by using some software), |Ψ(x, t)|^2 for 3 or 4 moments of time t between 0 and T that would nicely display the qualitative features of the changes, if any.

(b) Let us now prepare the system in some arbitrary non-stationary state Ψ(x, 0). Is it true that after some time T, the wave function will always return to its original spatial behavior, that is,

Ψ(x, T) = Ψ(x, 0)

(perhaps with accuracy to an inconsequential overall phase factor)? If so, what is this quantum revival time T? Compare to (a). And why do you think it was possible to have it in this system for an arbitrary state?

(c) Think now about the revival time for a classical particle of energy E bouncing between the walls. Assuming the positive answer to (b), if we were to compare the classical revival behavior to the quantum revival behavior, when these times would be equal?

Answer #1

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/√...

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?

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

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 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...

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)...

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...

1, Show that the real and imaginary parts
(separately) of the time-dependent ground state
wave function psi(x,t) for a particle confined to an impenetrable
box can be written as a linear combination of two traveling
waves.
2, Show the decomposition in (1) is consistent with the
classical behavior of a particle in an impenetrable box.

We observe 3 particles and 3 different one-particle states
ψa(x), ψb(x) i ψc(x). How many
different states that describe a system of three particles can we
make:
(a) if the particles can be distinguished;
(b) if the particles are identical bosons;
(c) if the particles are identical fermions?
Note that, unlike the exercise task, there is a possibility that
the particles may be in the same quantum state.
For example, ψ(x1, x2, x3) =
ψa(x1)ψa(x2)ψc(x3)

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?...

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