Question

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 answer in terms of the variable L. Enter your answers in ascending order separated by commas.

In parts A and B are your answers consistent with Figure 40.12 in the textbook?

Answer #1

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 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 normalized wave functions for the particle is in a 1D box of
length L., with limits on x = 0 and x = L. V (x) = 0 for 0 <= x
<= L and V (x) = Infinity elsewhere. The probability of a
particle being between x = 0 and x = L / 8 in the ground quantum
state (n = 1) should be calculated.

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...

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)...

For the infinite square-well potential, find the probability
that a particle in its third excited state is in each third of the
one-dimensional box:
(0 ≤ x ≤ L/3)
(L/3 ≤ x ≤ 2L/3)
(2L/3 ≤ x ≤ L)

For the infinite square-well potential, find the probability
that a particle in its fourth excited state is in each third of the
one-dimensional box:
a) (0 ≤ x ≤ L/3)
b) (L/3 ≤ x ≤ 2L/3)
c) (2L/3 ≤ x ≤ L)

A particular positron is restricted to one dimension and has a
wave function given by ψ(x)=
Ax between x = 0 and
x = 1.00 nm, and ψ(x)
= 0 elsewhere. Assume the normalization constant A is a
positive, real constant.
(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...

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

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