Question

Consider a free, unbound particle with V(x) = 0 and the initial
state wave function: phi(x,0) = Ae^(-a|x|)

1. Construct phi(x,t)

2. Discuss the limiting cases, i.e. what happens to position
and momentum when a is bery small and when a is very large?

Answer #1

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 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.

A particle is described by the wave function ψ(x) = b(a2 - x2)
for -a ≤ x ≤ a and ψ(x)=0 for x ≤ -a and x ≥ a , where a and b are
positive real constants.
(a) Using the normalization condition, find b in terms of a.
(b) What is the probability to find the particle at x = 0.33a in
a small interval of width 0.01a?
(c) What is the probability for the particle to be found...

A particle is described by the wave function ψ(x) = b(a2 - x2)
for -a ≤ x ≤ a and ψ(x)=0 for x ≤ -a and x ≥ a , where a and b are
positive real constants.
(a) Using the normalization condition, find b in terms of a.
(b) What is the probability to find the particle at x = 0.33a in
a small interval of width 0.01a?
(c) What is the probability for the particle to be found...

Recall that |ψ|2dx is the probability of finding the particle
that has normalized wave function ψ(x) in the interval x to x+dx.
Consider a particle in a box with rigid walls at x=0 and x=L. Let
the particle be in the first excited level and use
ψn(x)=2L−−√sinnπxL
For which values of x, if any, in the range from 0 to L is the
probability of finding the particle zero?
For which v alues of x is the probability highest?Express your...

The wave function for a particle confined to a one-dimensional
box located between x = 0 and x = L is given by Psi(x) = A sin
(n(pi)x/L) + B cos (n(pi)x/L) . The constants A and B are
determined to be

It is possible to construct oscillatory wave packets without
using trigonometric functions.
Consider the function y(x) = (64x^6 - 240x^4 + 180x^2 -
15)*e^(-x^2). Wave packets using polynomials occur in quantum
mechanics
as solutions to the simple harmonic oscillator and the hydrogen
atom, as we discuss later in this test.
(a) Sketch this function in the region where it has reasonably
large amplitude.
(b) What is the width of this wave packet? Make a rough estimate
from your sketch.
(c)...

Consider a particle of mass m and energy E approaching the step
potential?
V (x)= 0, x<0
V(x)=V0, x>0
from negative values of x. Consider the case E > V0. a)
Classically, what is the probability of re?ection? b) Quantum
mechanically, what is the probability of re?ection? Express your
result in terms of the ratio V0/E. What is the probability of
re?ection if E = 2V0?

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. Suppose that at time ta the state function of a one particle
system is Ψ = (2/πc2)3/4 e(exp [– (x2 + y2 + z2)/c2)] where c = 2
nm. Find the probability that a measurement of the particle’s
position at ta will find the particle in the tiny cubic region with
its center at x = 1.2 nm, y = -1.0 nm, z = 0 and with edges each of
length 0.004 nm. Note that 1 nm = 10-9...

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