Question

The Potential Step (E > U0):

(a) Express the reflection and transmission probabilities in terms of E and U0. What is the probability

that the incident particle will be reflected for the case U0 = 0.7 E?

(b) Obtain the probability density and the real and imaginary parts of the wavefunction for t = π/2 ω .

Also, plot them.

Answer 1

Part a

The physics of a potential step for the case $E>U_0$

$U(x)=[_{+U_0}^0 \space _{x>0}^{x<0}$

$u(x)=e^{ikx}+Re^{-ikx}$

$k=\sqrt{\frac{2mE}{\hbar^2}}$

$u'(x)=Te^{-ik'x}$

$k'=\sqrt{\frac{2mE-U_0}{\hbar^2}}$

Continuity wave function $=x=0$ implies

$1+R=T$

$[ike^{ikx}-ikRe^{-ikx}]_{x=0}=[ik'Te^{ik'x}]_{x=0}$

$k(1-R)=k'(1+R)$

$(k+k')R=(k-k')$

$R= \frac{k-k'}{k+k'}$

$T=1-R= \frac{2k}{k+k'}$

$u(x)=[^{e^{ikx}} {\frac{k-k'}{k+k'}}e^{-ikx}$ $x<0$

$\frac{2k}{k+k'}e^{ik'x}$ $x>0$

$P_{reflection}=|R|^2=(\frac{k-k'}{k+k'})^2$

$P_{transmission}=1-P_{reflection}=\frac{4kk'}{(k+k')^2}$

The transmission probability goes to 1 $k=k'$ (since there is no step)

The transmission probability goes for 0 $k'=0$ (since the kinetic energy is zero)

Part b

$P(x,x+Δx)≈|Ψ(x,t)|^2δx.$

$P(x,x+Δx)=∫^{x+Δx}_x|Ψ(x,t)|^2dx$

$P(−∞,+∞)=∫^∞_{−∞}|Ψ(x,t)|^2dx=1.$

$P(∞,+∞)=∫^∞_{−∞}|C|2dx=1.$

$P(x=0,L)=∫^L_0|C|^2dx=1.$

$C=\sqrt\frac{1}{L}$ where $L= \frac{\pi}{2}$

$P(x=0,L/2)=∫^{L/2}_0 |\sqrt{\frac{1}{\frac{\pi}{2}}}|^2$

$=(\frac{1}{0.5 \pi}) \frac{0.5 \pi}{2} = 0.25$