Find the wave function and its energy by solving the Schrodinger equation below for the three-dimensional box
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Find the wave function and its energy by solving the Schrodinger equation below for the three-dimensional box.
Step by step
Solved in 3 steps
- You are told that for the particle in 1 D box, the wave function is given by φ (x) ∝ x(L − x). Find(a) Expectation value of position, and uncertainty in position.(b) Expectation value of energyFor a "particle in a box" of length, L, draw the first three wave functions and write down the wavelength of each. Confirm that the wavelengths for the nth level is given by 2LUse the variational principle to obtain an upper limit to ground state energy of a particle in one dimensional box.
- Verity by insertion to the radial part of Schrödinger equation that R_{2,1} is a solution and provide the right energy. Provide all the steps!1) a) A particle is in an infinite square well, with ground state energy E1. The wavefunction is 3 *y. Find in terms of E1. (There is an easy way to do this; no actual integrals 4 + 5 required.) b) A particle is in an infinite square well, with ground state energy Ej. Find a normalized wavefunction that has a total energy expectation value equal to 3E1. (It will be a superposition.) Keep all your coefficients real and positive. c) Now time-evolve your answer from part b, to show how the wavefunction varies with time.For a particle with mass m in a 1-D box of length a, show that the first excited statewavefunction is orthogonal to the wavefunction is orthogonal to the second excited statewavefunction.
- Notice for the finite square-well potential that the wave function Ψ is not zero outside the well despite the fact that E < V0. Is it possible classically for a particle to be in a region where E < V0? Explain this resultConsider an electron in a one-dimensional, infinitely-deep, square potential well of width d. The electron is in the ground state. (a) Sketch the wavefunction for the electron. Clearly indicate the position of the walls of the potential well on your sketch. (b) Briefly explain how the probability distribution for detecting the electron at a given position differs from the wavefunction.Find the lowest energy of an electron confined to move in a three dimensional potential box of length 0.48 Å.
- Calculate the average or expectation value of the position of a particle in a one-dimensional box for n=2.the ground state wavefunction of a quantum mechanical simple harmonic oscillator of mass m and frequency, which is given by: Question mw where a = the potential is V(x) = mw²x² and N is given by: N =) 9 ax² ¡Ent Yo (x, t) = Ne ze By substituting into the time-dependent Schrödinger equation, prove that the ground state energy, Eo, is given by: Eo ħw 2. Find the expectation value to find the particle inside the box at n=1 ? M=1