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Wavepackets
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Wavefunctions of the form ψ(x) = Aexp(ipx/h) are referred to as FREE PARTICLE
wavefunctions and are constant throughout space. As such they are somewhat
artificial (but nonetheless very useful) and do not represent a localized
particle. To represent a localized particle it is necessary to construct a
WAVEPACKET by superpositioning basis vectors (plane waves).
Consider the Gaussian distribution:
P(x) = (1/√2πσ_{x}^{2})exp(-[x - x_{o}]^{2}/2σ_{x}^{2})
Construct a wavepacket of the form:
ψ(x) = (1/2πα)^{1/4}exp(-x^{2}/4α)exp(ik_{o}x)
where (1/2πα)^{1/4} is the normalization factor and 4α = 2σ_{x}^{2}.
This looks like:
|ψ(x)> = Σ_{k}A_{k}|exp(ik_{o}x)>
Where,
A(k) = (1/√2π)∫ψ(x)exp(-ikx)dx
= (1/√2π)∫(1/2πα)^{1/4}exp(-x^{2}/4α)exp(ik_{o}x)exp(-ikx)dx
= (2α/π)^{1/4}exp(-α[k - k_{o}]^{2})
So we can write out original wavefunction in an equivalent form as:
ψ(x) = Σ_{k}(2α/π)^{1/4}exp(-α[k - k_{o}]^{2})exp(ikx)
For a free electron E = h^{2}k^{2}/2m = hω
∴ ω = hk^{2}/2m
Phase velocity:
v_{p} = fλ = ω/k since k = 2π/λ
v_{p} = ω/k = hk/2m
Group velocity:
v_{g} = ∂ω/∂k = hk/m