A stream of quantum particles of energy E is moving freely until it encounters a potential...

A stream of quantum particles of energy E is moving freely until it encounters a potential barrier of infinite height. Show that 100% of the particles are reflected.

Solutions

Expert Solution

Let the height be finite, V. So that the potential is , where u(x) is the unit step function.

Schrodinger eq. on the left , x<0

For x>0

By continuity and differentiability at x=0,

and

Thus,

which simplifies to

Similarly,

The exact value of B is not important (but can be found out by normalisation).

We want to know the fraction of the wavefunction that penetrates region 2 from region 1, it is the following, using the probalistic interpretation of wavefunction

at x=0.

It gives 0. Hence the probability of finding the particle is 0 in region 2. This could also be seen directly from the expression of wavefunction in region 2. Because k' goes to infinity as V goes to infinity, hence

genius_generous answered 1 year ago

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