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Time independent Schrodinger equation.

In region I, particles can not exist. Wave function is zero.

In region II, Potential ...

Solution Summary

Solution is in a 5-page word document. I have shown each and every step in obtaining an expression for the quantized energy states and reflection coefficient of a semi infinite potential well for the cases of E < Vo and E > Vo respectively. I have used the equation editor to type mathematical expressions. Please download this solution to learn the techniques of solving similar problems in your quantum mechanics course. I have provided very detailed solution. Thank you.

Consider the wave equation for a semi-infinite string(in the domain x>or =0)
with wave speed c=1, for initial conditions u(x,0)=0 and
(u) subscript (t)(x,0)= (4x)/(1+x^2), x>or =0
Using d'alembert's solution show that the solution of the wave equation for t>or=0
is u(x,t)=In((1+(x+t)^2)/(1+(x-t)^2))
I have to consider t

Consider the semi-infinte square wellpotential shown in diagram of attachment.
a. Given that the energy eigenvalue for the first excited state E2=0 (i.ie u2 is at the edge of being bound) find the width a of the well in terms of m, h and Vo. (Hint consider the continuity of u2 at x=a).
b. Calculate the energy eigenvalue E1 fo

Find the solution to:
PDE: u*xx-c(to the power of negative k)u*tt=0 , 0
ICs: u(x,0)=f(x) and u*t(x,0)=g(x), x>=0
BC: u*x(0,t)=0 , t>=0
This BC corresponds to a string with its end point free to move in a vertical direction.
(Please remember to include to boundary condition in your solution. Thanks very much!)

Solve the eigenvalue problem
as follows:
Let U = ... be a two-component vector whose first component is a twice differentiable function u(x), and whose second component is a real number u1 Consider the corresponding vector space H with inner product
Let S C H be the subspace
....
and let
....
The above eigenvalue

Let F be an extension field of K. If u is an element of F is transcendental over K, then show that every element of K(u) that is not in K in also transcendental over K.
Hint for proof: Suppose y is an element of K(u). Then for some g(x), h(x) elements of K[x], we have y = g(u)/h(u). Assume that y is algebraic over K and think

What is the probability that an electron in the infinite well in the state Un(x) =[(2/L)^.5]*sin(Pi*n*x/L) is found in the region between x = 0 and x = L/2 , where the Un are the eigenfunctions of the infinite-wellpotential?

Solve the given problem by finding the appropriate derivative.
Find the equation of the line normal to the curve of y= 2 cos (1/2)x, where x=pi.
Write the expression using x as a variable. Write the exact answer in terms of pi.

Please specify your notation(if necessary) and explain clearly each step of your solution.
Thank you very much.
7. Find the image of the semi-infinite strip x ≥ 0, 0 ≤ y ≤ π under the transformation _ = ez , and label corresponding portions of the boundaries.

Consider the semi-infinite square well given by V(x)=-Vo<0 for 0<=x<=a and V(x)=0 for x>a. There is an infinite barrier at x=0. A particle with mass m is in a bound state in this potential energy E<=0.
a) solve the Schroedinger Eq to derive phi(x) for x=>0. Use the appropriate boundary conditions and normalized the wave funct