Question

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 between x = 0.03a and x = 1.00a ?

Answer #1

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 free particle has the initial wave function Ψ(x, 0) = Ae−ax2
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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
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A particular positron is restricted to one dimension and has a
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Ax between x = 0 and
x = 1.00 nm, and ψ(x)
= 0 elsewhere. Assume the normalization constant A is a
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(a) What is the value of A (in nm−3/2)?
nm−3/2
(b) What is the probability that the particle will be found
between x = 0.290 nm and x = 0.415 nm?
P =
(c) What is the expectation value...

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ψn(x)=2L−−√sinnπxL
For which values of x, if any, in the range from 0 to L is the
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For which v alues of x is the probability highest?Express your...

The wave function of a particle in a one-dimensional box of
length L is ψ(x) = A cos (πx/L).
Find the probability function for ψ.
Find P(0.1L < x < 0.3L)
Suppose the length of the box was 0.6 nm and the particle was an
electron. Find the uncertainty in the speed of the particle.

Normalize the following wave function (3)
Ψ(x, t) = (
Ce−γx+iδt, x ≥ 0
0, x < 0
where γ and δ are some real constants and γ > 0.

Consider the wave function at t = 0, ψ(x, 0) = C sin(3πx/2)
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(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...

Consider the time-dependent ground state wave function
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(b) Show that the decompositions in part (a) are consistent with
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a. Suppose that at time ta the state function of a one particle
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The wave function for a particle confined to a one-dimensional
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