1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can'...
Particle in a box Figure 1 is an illustration of the concept of a particle in a box. V=00 V=00 V=0 Figure 1. A representation of a particle in a box, where the potential energy, V, is zero between x = 0 and x = L and rises abruptly to infinity at the walls. The Schrödinger equation for a particle in a box reads t² d²u Y +V(x)y = Ey 2m dx2 + (1) where ħ=h/21 , y represents the...
The wave function of a restricted particle on the x-axis is between x = 0 and x = 1 ψ= ax ^2 and everywhere else ψ = 0. a) Find the value of constant a b) Find the probability that the particle is between x = 0.1 and x = 0.2 c) Find the wait value for the position of the particle
help on all a), b), and c) please!! 1. A particle in an infinite square well has an initial wave function Alsin sin 4 0 < x < L otherwise s(x, t = 0) 0 (a) Find A so that the wavefunction is normalized. (b) Find '(z,t). (c) Find the expectation value(E) of the energy of ψ(x,t = 0). You may use the result mx n 2 0 1. A particle in an infinite square well has an initial wave...
Extra Credit (3 points to Mideterm-2) Q1. A particle is described by the wave function (x) b(a2-x2) for -a sx s a and (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 a. (b) What is the probability to find the particle at x = +a/2 in a small interval ofwidth 0.01 a ? (c) What is the probability for the particle to be...
A particle in an infinite square well has the initial wave function: (x,0)- A sin(x/a) (0 S a (a) (b) Determine A Find$(z,t) (Hint: You will need to break up this wavefunction into a superposition of pure states. Use orthogonality to find the coefficients.) (c) Calculate (x). Is it a function of time? (d) Calculate (H).
A particle is completely confined to one-dimensional region along the x-axis between the points x = ± L The wave function that describes its state is: SP 10 elsewhere where a and b are (as yet) unknown constants that can be expressed in terms of L Use the fact that the wave function must be continuous everywhere to solve for the constant b. The square of the wave function is a probability density, which means that the area under that...
A fellow student proposes that a possible wave function for a free particle with mass \(m\) (one for which the potential-energy function \(U(x)\) is zero ) is$$ \psi(x)=\left\{\begin{array}{ll} e^{-k x}, & x \geq 0 \\ e^{+\kappa x}, & x<0 \end{array}\right. $$where \(\kappa\) is a positive constant. (a) Graph this proposed wave function.(b) Determine the energy of the particle if the proposed wave function satisfies the Schrödinger equation for \(x<\)0.(c) Show that the proposed wave function also satisfies the Schrödinger equation...
Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f be its summation function n sin(nx) b) Show that f E C(R) and that 1 cos(nx) f'(x)= 2-1 c) Show that 「 f#072821) f(x)dx = k=0 Consider the series following series of functions ' sin(nx) 3 n-1 a) Show that the series is absolutely and uniformly convergent on the real axis. Let f...
1l] A particle with mass m and energy E is inside a square tube with infinite potential barriers at x-o, x-a, y 0, y a. The tube is infinitely long in the +z-direction. (a) Solve the Schroedinger equation to derive the allowed wave functions for this particle. Do not try to normalize the wave functions, but make sure they correspond to motion in +2-direction. (b) Determine the allowed energies for such a particle. (c) If we were to probe the...
(C) An electron is described by the wavefunction (x) = 4 cos(2x/L) for the range = 5234 and is zero otherwise. (In other words, v(x) = 0 for 3 and 43 .) A useful integral is S cos? (ax)dx = 1 + sin (2017) (1) What is the probability of finding the electron between x = 0 and x = ? (ii) What is the probability of finding the electron at = 4? (iii) Where is the maximum probability for...