A particle of mass m is in the one-dimensional potential well V(x) = 0 for -a <

Question

A particle of mass m is in the one-dimensional potential well V(x) = 0 for -a < x < a and V(x) = otherwise.

(a) Write down the Schrodinger equation for the region -a < x < a.

(b) Obtain the solutions to this equation for bound states (energies E > 0). This will involve two arbitrary constants.

(c) Apply the boundary conditions at x = ±a. (What is the wave function when the potential is infinite?)

(d) Hence show that the bound state energies are En = (n^2 h^2)/8m(2a)^2 and find the corresponding eigenfunctions.

(e) Repeat (c) and (d) above, but use the property of parity first.

Explanation / Answer

From the given question, the problem has three regions.
Region I : -infinity to – a where v = infinity
Region II : -a < x < a    v = 0
Region III : + a to + infinity v = infinity

Schrödinger equation to the region II is of the form
Hop (x) = E (x) ……………… (1)
Substituting for the H operator (Hop) we have

-^2/2m (d^2 (x))/(dx^2 )+ V(x) (x)= E (x) …….(2)

But in the region II, v = 0. Hence the above equation becomes
               -^2/2m (d^2 (x))/(dx^2 )=E (x)
               (d^2 (x))/(dx^2 )=2m/ ^2 E (x)

                   (d^2 (x))/(dx^2 )=k^2 (x) where k = (2mE)^(1/2) / ……(3)
(a) The equation (3) gives the Schrödinger equation for the region –a < x < a
(b) The solution for the above equation is
(x)=A coskx+ B sin kx
(c) In regions I and III the wave function is identically zero since the potential is infinite.

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