(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the 1D potential V(x) such that V(x)=0 for x<0 and V(x) = V for x>0 and assume that a wave packet with energy E_{0}=p^{2}_{0}/2m<V is incident on the barrier from the left. Calculate in terms of E0 and V the difference in time between the arrival of the incident packet at the step and the departure of the reflected packet from the step. Relate this time delay to a “distance of travel” within the step.

2. Relevant equations

time independent Schrödinger equation

3. The attempt at a solution

I've calculated the time independent wavefunction to be

[tex]

\Psi(x) = \begin{cases}

A_Ie^{ik_0x} + B_Ie^{-ik_0x} & x<0\\

A_{II}e^{kx} & x>0 \end{cases}

\\

k = \sqrt{\frac{2mE_0}{\hbar^2}}

\\

k = \sqrt{\frac{2m(V-E_0)}{\hbar^2}}

[/tex]

with the relations

[tex]

\frac{B_I}{A_I} = \frac{ik_0+k}{ik0-k}

\\

\frac{A_{II}}{A_I} = \frac{2ik_0}{ik_0-k}.

[/tex]

From that, I was thinking about writing the wavefunction as

[tex]

\Psi(x) = \begin{cases}

e^{ik_0x} + e^{-i(k_0x-\phi)} & x<0\\

ce^{kx+i\theta} & x>0 \end{cases}

[/tex]

where I've removed the normalization constant and use the two phases [itex]\phi,\theta[/itex] given by

[tex]

\tan(\phi) = \frac{2kk_0}{k^2-k_0^2}

\\

c = \frac{2k_0}{k'}

\\

\tan(\theta) = \frac{k}{k_0}

\\

k'^2=k_0^2+k^2=\frac{2mV}{\hbar^2}

[/tex]

I'm stuck with what the question means by calculate the time difference.

One of my classmates said to apply the time evolution operator [itex]e^{-i\hat{H}t/\hbar}[/itex], which because we're acting on one eigenfunction, becomes [itex]e^{-iE_0t/\hbar}[/itex] and the wavefunction is

[tex]

\Psi(x,t) = \begin{cases}

e^{i(k_0x-E_0t/\hbar)} + e^{-i(k_0x-\phi+E_0t/\hbar)} & x<0\\

ce^{kx+i(\theta-E_0t/\hbar)} & x>0 \end{cases}

[/tex]

Where would I go from here?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Grad 1D scattering from step function

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**