3.II.20D

Quantum Mechanics | Part IB, 2004

A one-dimensional system has the potential

V(x)={0x<02U2m0<x<L0x>LV(x)= \begin{cases}0 & x<0 \\ \frac{\hbar^{2} U}{2 m} & 0<x<L \\ 0 & x>L\end{cases}

For energy E=2ϵ/(2m),ϵ<UE=\hbar^{2} \epsilon /(2 m), \epsilon<U, the wave function has the form

ψ(x)={aeikx+ceikxx<0ecoshKx+fsinhKx0<x<Ldeik(xL)+beik(xL)x>L\psi(x)= \begin{cases}a e^{i k x}+c e^{-i k x} & x<0 \\ e \cosh K x+f \sinh K x & 0<x<L \\ d e^{i k(x-L)}+b e^{-i k(x-L)} & x>L\end{cases}

By considering the relation between incoming and outgoing waves explain why we should expect

c2+d2=a2+b2|c|^{2}+|d|^{2}=|a|^{2}+|b|^{2}

Find four linear relations between a,b,c,d,e,fa, b, c, d, e, f. Eliminate d,e,fd, e, f and show that

c=1D[b+12(λ1λ)sinhKLa]c=\frac{1}{D}\left[b+\frac{1}{2}\left(\lambda-\frac{1}{\lambda}\right) \sinh K L a\right]

where D=coshKL12(λ+1λ)sinhKLD=\cosh K L-\frac{1}{2}\left(\lambda+\frac{1}{\lambda}\right) \sinh K L and λ=K/(ik)\lambda=K /(i k). By using the result for cc, or otherwise, explain why the solution for dd is

d=1D[a+12(λ1λ)sinhKLb]d=\frac{1}{D}\left[a+\frac{1}{2}\left(\lambda-\frac{1}{\lambda}\right) \sinh K L b\right]

For b=0b=0 define the transmission coefficient TT and show that, for large LL,

T16ϵ(Uϵ)U2e2UϵLT \approx 16 \frac{\epsilon(U-\epsilon)}{U^{2}} e^{-2 \sqrt{U-\epsilon} L}

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