Problem 4 For the wave function φ(x,0) = Ax(a-x) find the expectation value of Hat time...
Problem 1. Wave function An electron is described by a wave function: for x < 0 *(z) = { ce Ce-s/1(1 – e-3/4) for x > 0 : where I is a constant length, and C is the normalization constant. 1. Find C. 2. Where an electron is most likely to be found; that is, for what value of x is the prob: bility for finding electron largest? 3. What is the average coordinate 7 of the electron? 4. What...
Problem Three (1) Write the expression that defines the expectation value of the operator <x> for any function Y. (II) Find the expectation value of <x> for the particle in a box defined by the wave function: Y=N(L x - X°) within limits ( < x < L. which you previously normalized in Problem Two on the previous page.
6. a) Calculate the expectation value of x as a function of time for an electron in a state that is a (normalized) equal mixture of the ground state and 1st excited state of a 1D HO b) Graph x vs time for the case k = 1 eV/nm2. What is its value at t=0? What is the period of the oscillation in femtoseconds? For the one-dimensional (1D) harmonic oscillator (HO) the potential energy function has the form V(a) k2/2,...
3. At time t-0 a particle is represented by the wave function A-if 0 < x<a ψ(x,0) = 0 otherwise where A, a, and b are constants. a) Normalize ψ(x,0). b) Draw (x,0). c) Where is the particle most likely to be found at t-0? d) What is the probability of finding the particle to the left of a? e) What is the expectation value of x?
With what amplitude and frequency does the expectation value of the momentum of a proton (m=1.67x10-27 kg) in a ID box of length 0.02 nm oscillate with time? The particle has the initial wave function Voya- Vioys.
3. If (:) is a wave function in which the expectation value of the momentum is P, then find the expectation value of the momentum i.e. <p> in the state el (po)/ \(r) (5 points).
Problem 5. (20 points) a) Given a periodic wave function of S(x) = ax -1 <x< n that has a period of 27. Determine if f(x) is an even or odd function b) Find Fourier Sine Transform of f(x)=e
A. Momentum space We define the momentum space wave function φ(p) as where Ψ(x)is a solution of the Schrödinger equation in configuration (position) space a) Show that the expectation values of and p can be written in terms of Ф(p) as <p(p)p(p)dp b) Demonstrate that φ(p) is normalized, ie if Ψ(x) is normalized. J ΙΨ(2)12dr-1 c) Show that Ф(p) 2dp can be interpreted as the probability to find a particle with momen tum between p and p+ dp
2. Find the expectation value for <p2 > for the ground-state wave function of the infinite 1-d square well. Here p = -i(hbar) d/dx is the (linear) momentum operator
(10 points) Normalize the wave function: Find the expectation values of (x), (r aj. Ģ) and (p2).