quantam mechanics

A particle is confined to a one-dimensional infinite square well potential. Potential well extends from x = -L/2 to x = +L/2. Find the wave function and the energy levels of the particle.

A particle is confined to a one-dimensional infinite square well potential. Potential well extends from x = -L/2 to x = +L/2. Find the wave function and the energy levels of the particle.

Out side the well, wave function is zero.

Lets solve the schrodinger equation inside the well.

Inside the well V = 0.

Let..... 

 General solutions to this differential equation are of the form of A sin kx + B cos kx

Boundary conditions are:

ψ = 0 at x = + L/2 and –L/2

Substituting in the wave function we get,

Adding these two equations gives,

Subtracting these two equations gives,

We have no k value for which both the above equations are satisfied. Hence we choose,

A = 0 and cos kL/2 = 0 or B = 0 and sin kL/2 = 0

First one gives, kL/2 = n π/2 where n = 1, 3, 5, ….

Second one gives, kL/2 = n π/2 where n = 2, 4. 6,….

In general, and k = n π/L with n = 1, 3, 5,….or

and k = n π/L with n = 2, 4, 6,…..

 (insert missing solution formula's below in area provided)

We defined

(insert missing solution formula's below in area provided)

n = 1, 2, 3, 4,……

These are the quantized energy levels of a particle in an infinite square well.

The green highlight is pretty.

The green highlight is pretty.

sftac

a "particle confined" to a potential "one-dimensional infinite square"

make it two posts instead;score another point

a square might as well be a triangle when its infinite. not like it really matters. considering that its going to have two dimensions either way. besides,a particle confined to a single coordinates will have no measurable wave function.

I did make it two posts : )