OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 35D / c / Solution

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^{'}

at

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.

Solved by gpt-5.6-sol high.

 Ancestors (11)