Question

3. Suppose a beam of particles of mass m and kinetic energy E is incident from the left on a potential well given by:

U(x) = ?U₀ (for 0 < x < L where U₀ > 0)

U(x) = 0 ( otherwise )

(a) What is the Schrodinger Wave Equation (S.W.E.) for the region x < 0 ? (Hint: include both incident and reflected waves)

(b) What is the S.W.E. for the region x > L ? (Hint: this will only be a transmitted wave)

(c) What is the S.W.E. for the region 0 < x < L ? (Hint: in general there are two terms)

(d) What are the four boundary value equations for this case?

(e) Starting from the boundary value equations, derive the Transmission Probability (T) for this case.

Answer 1

The Schrodinger's equation is

where

a) The solution to the Schrodinger equation in the region x < 0 is

where

b) The solution to the Schrodinger equation for x > Lis

c) The solution in the region 0 < x < L is

where

The suffixes C, L and R stands for center region(0 < x < L), Right region(x > L) and Left region(x < 0)

d) The boundary conditions are that the wave functions and the derivatives must ve continuous across the two boundaries ( at x = 0 and at x = L)

These equations give

e) To calculate the transmission particle, assume

(amplitude is 1 for the incoming particle)

(which gives the reflection coefficient)

(which gives the amplitude transmission coefficient)

For the case

The amplitude transmittance is can then be solved to find:

The transmission probability is just the modulus squared of the transmittance

After plugging in the values of and one finds the transmission probability: