35D (Applications of Quantum Mechanics)

For incidence from the left, normalize the asymptotic wavefunction by

ψ (x) \sim e^{ik x} + r e^{- ik x} (x \to - \infty), ψ (x) \sim t e^{ik x} (x \to + \infty) .

For incidence from the right, define

r^{'}, t^{'}

analogously with the directions reversed. Conservation of the probability current gives

∣ r ∣^{2} + ∣ t ∣^{2} = 1, ∣ r^{'} ∣^{2} + ∣ t^{'} ∣^{2} = 1.

Applying the same conserved sesquilinear current to the two scattering solutions gives

t^{' *} r + r^{' *} t = 0.

These are exactly the column orthonormality relations

S^{†} S = I

S

is unitary.

b

ψ (- x) = + ψ (x)

for even parity and

ψ (- x) = - ψ (x)

for odd parity. If

V (- x) = V (x)

Put

κ = \frac{2 m ( V _{0} - E )}{ℏ} .

An even solution is proportional to

cosh (κ x)

inside the potential barrier. Normalize the right exterior region as

ψ (x) = e^{- ik x} + (r + t) e^{ik x} (x > a /2) .

Continuity of

ψ

and

ψ^{'}

x = a /2

gives, with

γ = κ tanh (κa /2)

\frac{ik ( ( r + t ) e ^{ika /2} - e ^{- ika /2} )}{( r + t ) e ^{ika /2} + e ^{- ika /2}} = γ .

Solving,

r + t = e^{- ika} \frac{ik + γ}{ik - γ}, γ = κ tanh (κa /2) .

Its modulus is one, as expected for the even-parity eigenvalue of the unitary S-matrix.

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