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I want to solve a barrier potential problem for a Gaussian Wavepacket moving at a velocity v= $\frac{p_0}{m}$.

Normally we consider plain wave functions of the following type $$\Psi (x) = Ae^{ikx} + Be^{-ikx}$$ and so on...
My question is, given a Gaussian wave function $$\psi (x) = \frac{1}{(\pi \sigma^4)^{1/4}}e^{\frac{-(x-x_0)^2}{2 \sigma^2}}e^{\frac {i p_0(x-x_0)}{\hbar}}$$ how can I do the same things i.e. finding the tunneling probability, etc. for this Gaussian wave packet. Here $x_0<0$ and we assume that we have a potential barrier at $x=0$ with width $L$.

I am not even able to find a good reference for these types of problems.

Any help would be appreciated.

Deepanshu Chaudhary
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1 Answers1

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The reason why we use plane waves so often in quantum mechanics is that they are often the eigenfunctions of the Hamiltonian. In general, many prorblems are easier solved in terms of the eigenfunctions of the Hamiltonian (or its principal part).

Gaussian wave packets do not have this convenience (except for some special cases, such as a Harmonic oscillator), so you will likely need to expand the ave packet in terms of plane waves, solve for the plane waves, and then reassemble the solution - it requires taking a few Gaussian integrals.

See also this question: Why do physicists use plane waves so much?

Roger V.
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