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# IPython Notebook

IPython Notebook

## We now define the time

We now define the time evolution operator1j in Python.U ~ kin(t)in Fourier space. Note that the square root of minus one isIn [ ]:def UkinTilde(t):return exp(-1j * ...)and we defineψ(t)using Fourier transforms.In [ ]:from scipy import fft, ifftdef psi(t):return ifft(...*fft(psi0))In [ ]:P = ...plot(x, psi(P/4).real, x, psi(P/4).imag)figure()plot(x, ..., x, ...)figure()plot(x, ..., x, ...)We can understand this spread as a consequence of the uncertainty principle. We can calculate thewavepacket width ⟨‾‾‾‾‾‾ x 2 (t)⟩‾by the discrete approximation to the integral x 2 x 2 | 2√ ⟨ (t)⟩ ∼ ∑ |ψ(x) dxIn [ ]:def width(t):return sqrt(sum(... * abs(...)**2 * ...))Our initial wavepacket has a RMS widthΔx = a0In [ ]:width(0), a0We know that a minimum uncertainty wavepacket like ours has, so we expect the packet widthto grow like with given by the momentum uncertainty. We plot the comparison...vtvΔxΔp = ħ/2In [ ]:deltaP = ...;v = ...;times = linspace(0,2*P,100);widths = [width(t) for t in times];plot(times, widths, times, ...)

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