Question

**The state of a particle is completely described by its
wave function Ψ(?,?) One-dimensional Schrodinger Equation-- answer
the following questions:**

**2) Show that when U(x) = 0, and , is a solution to the
one-?=2??/ℏΨ=?sin??dimensional Schrodinger equation.**

**3) Show that when U(x) = 0, and , is a solution to the
one-?=2??/ℏΨ=?cos??dimensional Schrodinger equation.**

**4) Show that where A and B are constants is a solution
to the Ψ=??+?Schrodinger equation when U(x) = 0, and when E =
0.**

Answer #1

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.

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 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

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...

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.

Consider a one-dimensional real-space wave-function ψ(x) and let
Pˆ denote the parity operator such that P ψˆ (x) = ψ(−x).
a)Starting from the Rodrigues formula for Hermitian polynomials,
Hn(y) = (−1)^n*e^y^2*(d^n/dy^n)e^-y^2 with n ∈ N, show that the
eigenfunctions ψn(x) of the one-dimensional harmonic oscillator,
with mass m and frequency ω, are also eigenfunctions of the parity
operator. What are the eigenvalues?
b)Define the operator Π = exp [ iπ (( 1 /2α) *pˆ 2 +
α xˆ 2/ (h/2π)^2-1/2)] ,...

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.

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