Question

4 Schr¨odinger Equation and Classical Wave Equation

Show that the wave function Ψ(x, t) = Ae^(i(kx−ωt)) satisfies both the time-dependent Schr¨odinger equation and the classical wave equation. One of these cases corresponds to massive particles, such as an electron, and one corresponds to massless particles, such as a photon. Which is which? How do you know?

Answer #1

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.

Problem 7-3. Show that the wave function ψ
given by Eq. (7-6) satisfies the boundary
conditions given by Eq. (7-5) if the values of
kx, ky, and kz are restricted to
those of Eq. (7-7).
Eq. ( 7- 5 ) : ψ( x + L , y, z,t) = ψ(x, y, z,
t),
ψ(x, y + L ,s,t) = ψ(x, y, z, t),
ψ(x, y, z + L , t) = ψ(x, y, z, t).
Eq. ( 7-6 ) :...

Show that x(t) = A sin(wt) sin(kx) satisfies the wave equation,
where w and k are some constants. Find the relation between w, k,
and v so that the wave equation is satisfied.

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)...

A common solution to the wave equation is E(x,t) = A ei(kx+wt).
On paper take the needed derivatives and show that it actually is a
solution. Note that i is the square-root of -1. Upload a photograph
of your work.

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...

The current through a coil as a function of time is represented
by the equation
I(t) = Ae−bt
sin(ωt), where A = 5.25 A, b =
1.75 ✕ 10−2 s−1, and ω = 375 rad/s.
At t = 0.880 s, this changing current induces an emf in a
second coil that is close by. If the mutual inductance between the
two coils is 4.35 mH, determine the induced emf. (Assume we are
using a consistent sign convention for both coils....

Show that the function f(x, t) = x 2 + 4axt−4a 2 t 2 satisfies
the wave equation if one assumes a certain relationship between the
constant a and the wave speed u. What is this relationship?

1) Describe an example of each of the following that may be
found of your kitchen: Explain how your choice falls into this
category, and if there is a chemical name or symbol for it, provide
that as well. Provide a photo of your example with your ID card in
it. a) a compound b) a heterogeneous mixture c) an element (symbol)
Moving to the Caves… Lechuguilla Caves specifically. Check out this
picture of crystals of gypsum left behind in...

Please summarize the below article in approximately 100
words:
Monumental function in British Neolithic burial practices
Ian Kinnes
The high-risk rate of survival for the non-megalithic series of
Neolithic funerary monuments, recently re-emphasized by Piggott
(1973: 34), introduces a further variable into the deductive study
of burial practices. In Britain and Europe the overall distribution
of monumental forms present both lacunae and a marked preponderance
of cairns over earthen mounds which are in ill accord with the
known or predicted...

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