Schodinger's equation for one-dimensional motion of a particle whose potential energy is zero is(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{d^2}{dx^2}\psi +(2mE/h^2)^\frac{1}{2}\psi = 0 [/tex]

where [tex] \psi [/tex] is the wave function, m the mass of the particle, E its kinetic energy and h is Planck's constant. Show that

[tex]\psi = Asin(kx) + Bcos(kx)[/tex] ( where A and B are constants) and [tex] k =(2mE/h^2)^\frac{1}{2}[/tex] is a solution of the equation.

Using the boundary conditions [tex]\psi=0[/tex] when x=0 and when x=a, show that

(i) the kinetic energy [tex]E=h^2n^2/8ma^2[/tex]

(ii) the wave function [tex]\psi = A sin(n\pi\times x/a)[/tex] where n is any integer. (Note if [tex] sin(\theta) = 0 then \theta=n\pi[/tex])

My attempt:

A*sin(0) + B*cos(0) = 0, => 0 +B =0 => B = 0.

Therefore

A*sin(k*a)=0, Therefore [tex] (2mE/h^2)^\frac{1}{2}a = n\pi => E=n^2\pi^2h^2/2ma^2[/tex]

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Boundary value problem

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