Question

By using Schrodinger equation derive an expression for Energy eigenvalue of a particle inside a box of size L along x axis. Ans also the wave function of matter ware.​

Answer 1

Given:

A particle is inside a box of size L along x axis.

To Find:

An expression for Energy eigenvalue of the particle.

Also the wave function of matter.

Solution:

Let the particle be confined to a region 0 < x < a , along the x axis.

• V(x ) = 0 , 0 < x < a
•            ∞ , elsewhere.

Now, the time independent Schrödinger equation or the energy eigenvalue equation is ,

• HUj = EjUj ,
• H is the Hamiltonian inside the box.
• Value of H can be derived as,
• H = -h² d² / 2mdx²

Now

• Uj = 0 , for outside the box and ,
• Uj = e^ikx . k is any number.

Now we need to choose Uj(x=0) = 0 and Uj(x=a) = 0.

Therefore convenient value of Uj = Csinkx

Let k = nπ/a

• Un = Csin(nπx/a)

Putting this in the schrodinger equation,

• -h²/2m   x  ( -n²π²/a²) x C sin kx = EC sin kx

The only solution is

• En = n²π²h²/2ma²

Now lets nomalize ,

• < Un|Un>  = |C|²∫sin²(nπx/a) dx = |C|²a/2

Therefore , Energy eigenvalue of the particle Un = √(2/a) x  sin(nπx/a)