Show that the wavelength predicted for a particle in a one-dimensional box oflength L from the de Broglie relationship matches the wavelength predicted fromsolutions of the corresponding Schrödinger equation where the potential is defined as:6.EV(x)=0(barrier)V(x)=0(well)V(x)=∞(barrier)(Hint: You can do this for any value of n you want although we recommend the n=2 case).

For a box of length L and V=0 in this length, we can write Schrodinger equation -h2mdψ2dx2=Eψ…

Answered: Show that the wavelength predicted for…

Modern Physics

3rd Edition

ISBN:9781111794378

Author:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Publisher:Raymond A. Serway, Clement J. Moses, Curt A. Moyer

Chapter8: Quantum Mechanics In Three Dimensions

Section8.1: Particle In A Three-dimensional Box

Problem 2E

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Similar questions

A particle of mass m is confined to a box of width L. If the particle is in the first excited state, what are the probabilities of finding the particle in a region of width0.020 L around the given point x: (a) x=0.25L; (b) x=040L; (c) 0.75L and (d) x=0.90L.
in solving the schrodinger equation for the particle in a box system, satisfying the boundary condition imposes:
The wavefunction for a quantum particle tunnelling through a potential barrier of thickness L has the form ψ(x) = Ae−Cx in the classically forbidden region where A is a constant and C is given by C^2 = 2m(U − E) /h_bar^2 . (a) Show that this wavefunction is a solution to Schrodinger’s Equation. (b) Why is the probability of tunneling through the barrier proportional to e ^−2CL?
Solve the Schrodinger equation for a quantum particle of massm trapped in a one-dimensional infinite potential well (box) oflength L and obtain the expressions for wave-functions of theparticle.
An electron is bound in a square well of width 1.05 nm and depth U0=6E∞, where E∞ is the ground-state energy for an infinitely deep potential well.If the electron is initially in the ground level, E1=0.625E∞ , and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the well?
An electron having total energy E = 4.50 eV approaches a rectangular energy barrier with U = 5.00 eV and L = 950 pm as shown. Classically, the electron cannot pass through the barrier because E < U. Quantum- mechanically, however, the probability of tunneling is not zero. (a) Calculate this probability, which is the transmission coefficient. (b) To what value would the width L of the potential barrier have to be increased for the chance of an incident 4.50-eV electron tunneling through the barrier to be one in one million?
Using the wave function and energy E, apply the Schrodinger equation for the particle within the box.
Prove that assuming n = 0 for a quantum particle in an infinitely deep potential well leads to a violation of the uncertainty principle Δpx Δx ≥ h/2.
The ground-state energy of an electron trapped in a onedimensional infinite potential well is 2.6 eV.What will this quantity be if the width of the potential well is doubled?
Can we measure the energy of a free localized particle with complete precision?
A 12.0-eV electron encounters a barrier of height 15.0 eV. If the probability of the electron tunneling through the barrier is 2.5 %, find its width.
Can a quantum particle 'escape' from an infinite potential well like that in a box? Why? Why not?

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