[PYTHON] Read "Quantum computer made in 14 days". the 2nd day

Introduction

This time, we will actually calculate the electron motion using the Schrodinger equation.

2 Electronic wave function

2.1 Representation of electronic wave function

Here, the wave function of the electron when there is no energy due to the potential is obtained. The time-independent Schrodinger equation obtained last time

-\frac{\hbar^2}{2m}\frac{d^2 \phi(x)}{dx^2}=E\phi(x)

Is used. Here, put the constants together

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

With this

\frac{d^2 \phi(x)}{dx^2}=-k^2\phi(x)

By solving the second-order differential equation, the wave function

\psi(x,t)=Ae^{ikx-i\omega t}+Be^{-ikx-i\omega t}

Here about ω

\omega=\frac{E}{\hbar}Than\\
\omega=\frac{\hbar k^2}{2m}

By substituting this into the wave function formula

\psi(x,t)=Ae^{ik[x-\frac{\hbar k}{2m}t]} + Be^{-ik[x+\frac{\hbar k}{2m}t]}

If x is x (t) so that the shoulder of e becomes 0 when A = 1 and B = 0

x(t)=\frac{\hbar k}{2m}t

This represents the velocity of a plane wave. The moving speed of the peak position of a plane wave is called the phase velocity.

v=\frac{\hbar k}{2m}

Also, regarding the wave function when A = 1 and B = 0, the real part and the imaginary part are

Re[\psi(x,t)]=cos(k\left[ x-\frac{\hbar k}{2m}t \right])\\
Im[\psi(x,t)]=sin(k\left[ x-\frac{\hbar k}{2m}t \right])

Animation of electronic wavefunction

When the time of E = 1.00, 0.25 [eV] was shown by animation, the following waveform was obtained.

reference

EMAN Physics

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