Quantum Electron Model Application of Schrodingers wave equation

Quantum Electron Model

Application of Schrodinger’s wave equation Particle confined in one dimensional box or a particle in the infinite square well: Walls are like an infinitely steep hill- no way the particle can escape Inside is just like the free particle, no potential m A particle in this potential is completely free, except at the two ends (x = 0 and x = L), where an infinite force prevents it from escaping.

Outside the well, if (x) = 0 (the probability of finding the particle there is zero). Inside the well, where U(x) = 0, the time-independent or where Equation 1 is the (classical) simple harmonic oscillator equation; the general solution is

where A and B are arbitrary constants. Typically, these constants are fixed by the boundary conditions of the problem. What are the appropriate boundary conditions for (x)? Ordinarily, both (x) and d /dx are continuous, but where the potential goes to infinity only the first of these applies. Continuity of (x) requires that Which is known as boundary conditions. First take: Put in eqn (2), we get so

Now apply: So now either A=0 or Sin L=0 because then wave function will be zero everywhere n = 1, 2, 3, ……. or

The eqn (4) gives the wave function of a particle confined in a one dimensional box of length L. Now the condition for the normalized wave function is

Since Sin 2 nπ=0 Putting this in eqn (4) we get the normalized wave function This is the normalized wave function.

So, energy of the particle: Put value of , Or Where n = 1, 2, 3, 4……. Eqn (5) gives the energy of the particle confined in a one dimensional box of length L.

Wave function Ψn(x) at different energy Levels |Ψ|2 n=4 Finding prob. |Ψ|2 at different energy Level n=3 n=2 n=1

Wave function Ψn(x) at different energy Levels

The electrons inside a three-dimensional box