Use the matrix provided and show a) the transmission through the barries is give

Question

Use the matrix provided and show

a) the transmission through the barries is give as T = -4iky/2(y^2-k^2)sinh(yd)-4ikycosh(yd) with transmission probability

t*T = 4/ 4cosh^2(yd)+(y/k - k/y)sinh^2(yd)

b) Sketch the transmission probability versus yd as a function of k/y and discuss the results.

4. Use the matrix of problem 3 and show (a) the transmission through the barrier is given as with transmission probability (b) sketch the transmission probability versus yd as a function of k and discuss the results.

Explanation / Answer

from the given matrix we can get expressions to determine the transmission

the matrix provides four expressions:

-R +A +B =1

ikR - A + B =ik

-e^-dA -e^dB +T =0

e^-dA -e^dB +ikT =0

solving the four expressions we get T = 4ik/[2(^2- k^2) {(e^d -e^-d)/2} - 4ik{(e^d +e^-d)/2} ]

now from the given values we know that d and k/ values mean probability is non-zero

again using hyperbolic functions (e^d -e^-d)/2 = 1-e^-2d/2e^-d = Sinhd

and (e^d +e^-d)/2 =e^2d +1/2e^-d = Coshd

we get T = 4ik/[2(^2- k^2) {Sinh(d)} - 4ik{Cosh(d) } ]

if we get non-zero probability that means energyE is less than barrier height V0

the d value proves the same

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