Question

Consider a wave packet of a particle described by the wavefunction ψ(x,0) = Axe^−(x^2/L^2), -∞ ≤ x ≤ ∞.

a) Draw this wavefunction, labeling the axes in terms of A and
L.

b) Find the relationship between A and L that satisfies the
normalization condition.

c) Calculate the approximate probability of finding the particle
between positions x = −L and x = L.

d) What are 〈x〉, 〈x^2〉, and σ_x ? (Hint: use shortcuts where
possible).

e) Find the minimum uncertainty in the momentum of the particle
based on the results of part d and the Heisenberg uncertainty
principle.

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 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...

The wave function of a particle is ψ (x) = Ne (-∣x∣ /
a) e (iP₀x / ℏ). Where a and P0 are
constant; (e≃2,71 will be taken).
a) Find the normalization constant N?
b) Calculate the probability that the particle is between [-a / 2,
a / 2]?
c) Find the mean momentum and the mean kinetic energy of the
particle in the x direction.

7. A particle of mass m is described by the wave function ψ ( x)
= 2a^(3/2)*xe^(−ax) when x ≥ 0
0 when x < 0
(a) (2 pts) Verify that the normalization constant is correct.
(b) (3 pts) Sketch the wavefunction. Is it smooth at x = 0? (c) (2
pts) Find the particle’s most probable position. (d) (3 pts) What
is the probability that the particle would be found in the region
(0, 1/a)? 8. Refer to the...

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.

Consider a wave-packet of the form ψ(x) = e −x 2/(2σ 2 )
describing the quantum wave function of an electron. The
uncertainty in the position of the electron may be calculated as ∆x
= p hx 2i − (hxi) 2 where for a function f(x) the expectation
values hi are defined as hf(x)i ≡ R ∞ −∞ dx|ψ(x)| 2f(x) R ∞ −∞
dx|ψ(x)| 2 . Calculate ∆x for the wave packet given above. [Hint:
you may look up the...

For the wave function ψ(x) defined by
ψ(x)= A(a-x),x<a
ψ(x)=0, x>=a
(a) Sketch ψ(x) and calculate the normalization constant A.
(
b) Using Zettili’s conventions, find its Fourier transform φ(k)
and sketch it.
(c) Using the ψ(x) and φ(k), make reasonable estimates for ∆x
and ∆k. Using p = ħk, check whether your results are compatible
with the uncertainty principle.

Assume the wavefunction Ψ(x)=Axe^(-bx^2) is a solution to
Schrodinger’s equation for an electron in some potential U(x) over
the range -∞<x< ∞.
A) Write an expression which would enable you to find the value
of the constant A in terms of the constant b.
B) What is (x)_avg, the average value of x?
C) Write an expression which would enable you to find (x^2)_avg,
the average value of x^2 in terms of the constant b.
D) Write an expression which...

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)...

An electron is confined between x = 0 and x =
L. The wave function of the electron is
ψ(x) = A sin(2πx/L).
The wave function is zero for the regions x < 0 and
x > L. (a) Determine the normalization
constant A. (b) What is the probability of finding the
electron in the region 0 ≤ x ≤ L/8? {
(2/L)1/2, 4.54%}

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